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: N
if s40; if s ¼ 0; if s ¼ 1:
ð1:2Þ
In this paper we consider a related question: What is the number IðF ; f ; eÞ of F -isomorphism classes of elements in EðF ; f ; eÞ? Unfortunately, the formulas for IðF ; f ; eÞ seem to be much more complicated than those for NðF ; f ; eÞ: When p2 [e; we are able to determine IðF ; f ; eÞ completely; when p2 jj e; we are able to determine IðF ; f ; eÞ with some additional assumptions on f and e: It is wellknown and elementary that IðF ; f ; eÞ can be computed as a weighted sum over the elements of EðF ; f ; eÞ; IðF ; f ; eÞ ¼
1 fe
X
jAutðK=F Þj:
ð1:3Þ
KAEðF ;e;f Þ
Our method is to use class field theory to determine the groups AutðK=F Þ explicitly. Besides Krasner’s formulas, another motivation for our work is the connections between p-adic fields and finite commutative chain rings. A chain ring is a ring whose ideals form a chain under inclusion. Finite commutative chain rings have applications in finite geometry [10,20] and combinatorics [7,8,16,17]. Since finite commutative chain rings are precisely the nontrivial quotients of rings of integers of p-adic fields, classifying isomorphism classes of finite extensions of Qp is essentially equivalent to classifying isomorphism classes of finite commutative chain rings. In particular, in Section 2 we will show that IðQp ; f ; eÞ is equal to the number of isomorphism classes of finite commutative chain rings with residue field Fp f ; ramification index e; and length s; for all sufficiently large s: The paper is organized as follows. Section 2 is a summary of the connections between p-adic fields and finite commutative chain rings. Section 3 contains some preparatory results about p-adic fields. In particular, we determine the Fp ½/cSmodule structure of K =ðK Þp ; where K is a finite extension of Qp and c is a Qp -automorphism of K: In Section 4 we consider the problem of computing IðF ; f ; eÞ when p[e: Besides calculating IðF ; f ; eÞ; we also collect some facts about tamely ramified extensions of F which will be used later in the paper. In Section 5 we determine IðF ; f ; eÞ in the case p jj e: Sections 6–10 are devoted to calculating IðF ; f ; eÞ in the case p2 jj e; with some additional restrictions on f and e: In Section 6 we outline the computational plan and determine the structures of
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certain Galois groups. The key ingredients in the formula for IðF ; f ; eÞ are computed in Sections 7–9, and the final formula is assembled in Section 10. For KCO a finite extension of Qp ; we let nK ¼ ½K : Qp be the degree of K=Qp : We denote the ring of integers of K by OK ; the maximal ideal of OK by MK ; and the residue field of K by K% ¼ OK =MK : Any generator pK for MK is called a uniformizer for K: We let nK denote the valuation on K normalized so that nK ðpK Þ ¼ 1 for any uniformizer pK : Then nK extends uniquely to a valuation on O which takes values in Q; and is also denoted nK : In particular, we let np ¼ nQp denote the valuation on O which satisfies np ðpÞ ¼ 1: Let L be a finite extension of K: Then the residue degree % of L=K is denoted f ðL=KÞ; and the ramification index nL ðpK Þ of L=K is ½L% : K denoted eðL=KÞ: Finally, let fza : aX1g be a compatible system of primitive roots of unity in O; with za a primitive ath root of unity and zbab ¼ za for every a; bX1:
2. p-adic fields and finite commutative chain rings In addition to the description in terms of p-adic fields given in Section 1, there is another more explicit construction of finite commutative chain rings based on Galois rings; we refer the reader to [18] for more details. Choose a prime p; positive integers n; f ; and a monic polynomial FAðZ=pn ZÞ½X of degree f whose image in ðZ=pZÞ½X is irreducible. The ring GRðpn ; f Þ ¼ ðZ=pn ZÞ½X =ðFÞ is called the Galois ring of characteristic pn and rank f ; it is determined up to isomorphism by p; n; and f : Every finite commutative chain ring is isomorphic to a ring of the form R½X =ðC; pn1 X t Þ; where R ¼ GRðpn ; f Þ is a Galois ring, CAR½X is an Eisenstein polynomial of degree e; and t¼e 1ptpe
if n ¼ 1; if nX2:
ð2:1Þ
The integers p; n; f ; e; t are called the invariants of the finite commutative chain ring [1]. The following proposition summarizes the connections between finite commutative chain rings and p-adic fields. Proposition 2.1. Let K=Qp be a finite extension, with residue degree f and ramification index e; and let k=Qp be the maximal unramified subextension of K=Qp : Let s; t; n be * denote the image of positive integers such that s ¼ ðn 1Þe þ t; with 1ptpe; and let C CAOk ½X in ðOk =pn Ok Þ½X : Then we have the following. % and let (i) Let aAOk be such that k% ¼ ðZ=pZÞ½a; % where a% is the image of a in k; * FAZp ½X be the minimal polynomial of a over Qp : Then the image F of F in * in ðZ=pZÞ½X is irreducible. % of F ðZ=pn ZÞ½X is monic of degree f and the image F Therefore n * Ok =pn Ok DðZ=pn ZÞ½X =ðFÞDGRðp ; f Þ:
ð2:2Þ
ARTICLE IN PRESS X.-D. Hou, K. Keating / Journal of Number Theory 104 (2004) 14–61
17
(ii) The minimal polynomial of pK over k is an Eisenstein polynomial CAOk ½X of degree e such that * pn1 X t ÞDGRðpn ; f Þ½X =ðC; * pn1 X t Þ; OK =psK OK DðOk =pn Ok Þ½X =ðC;
ð2:3Þ
n n n * where CAðO k =p Ok Þ½X DGRðp ; f Þ½X is an Eisenstein polynomial over GRðp ; f Þ: s Thus OK =pK OK is a finite commutative chain ring with invariants
ðp; 1; f ; t; tÞ if n ¼ 1; ðp; n; f ; e; tÞ if n41:
ð2:4Þ
Moreover, every finite commutative chain ring is isomorphic to OK =psK OK for some finite extension K=Qp and some sX1: p þ np ðeÞ e and let L=Qp be another finite extension. Then (iii) Assume s4 p1 OK =psK OK DOL =psL OL if and only if KDL: Proof. Statements (i) and (ii) are well-known.
p þ np ðeÞ e then (iii) We want to prove that if OK =psK OK DOL =psL OL with s4 p1 p KDL: Note that s4 p1 þ np ðeÞ e implies n41: Thus the residue degree and
ramification index of L=Qp are determined by OL =psL OL DOK =psK OK ; and so L=Qp also has residue degree f and ramification index e: We may assume that K and L are both contained in the algebraic closure O of Qp : Then K=Qp and L=Qp have the same maximal unramified subextension k=Qp ; and K=k and L=k are both totally ramified extensions of degree e: We may assume that e41: Let CAOk ½X be the minimal polynomial of pK over k: The assumption OK =psK OK DOL =psL OL implies that * pn1 X t1 ÞDOL =ps OL : ðOk =pn Ok Þ½X =ðC; L
ð2:5Þ
* has a root By Lemma XVII.8 in [18] there exists rAAutðOk =pn Ok Þ such that rC n * =p O Þ½X is an Eisenstein bAOL =psL OL : Let bAOL be a lifting of b: Since rCAðO k k polynomial of degree eos; it follows that nL ðbÞ ¼ 1: Since k=Qp is unramified, the natural homomorphism Galðk=Qp Þ-AutðOk =pn Ok Þ is an isomorphism. Let R be g so the * ¼ RC the element of Galðk=Qp Þ whose image in AutðOk =pn Ok Þ is r: Then rC RC; s g * image of ðRCÞðbÞ in OL =pL OL is ðRC RCÞðbÞ ¼ ðrCÞðbÞ ¼ 0: Therefore nL ððRCÞðbÞÞXs: Q Let r1 ; r2 ; y; re AO be the roots of RC; then ðRCÞðX Þ ¼ ei¼1 ðX ri Þ: We may order the ri so that m ¼ np ðb r1 Þ is as large as possible. Then for 2pipe we have np ðb ri ÞXminfm; np ðr1 ri Þg: If m4np ðr1 ri Þ then np ðb ri Þ ¼ np ðr1 ri Þ; while if mpnp ðr1 ri Þ then by the maximality of m we get np ðb ri Þpnp ðr1 ri Þ: It follows that for 2pipe we have np ðb ri Þpnp ðr1 ri Þ: Since ðRCÞðbÞ ¼ ðb r1 Þ
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ðb r2 Þyðb re Þ; this implies e X s pnp ððRCÞðbÞÞpm þ np ðr1 ri Þ ¼ m þ np ðdkðr1 Þ=k Þ; e i¼2
ð2:6Þ
where dkðr1 Þ=k ¼ ðRCÞ0 ðr1 Þ is the different of the extension kðr1 Þ=k: By Remark 1 of [19, p. 58] we have np ðdK=k Þp1 e1 þ np ðeÞ: Since we are assuming p 1 1 s4 p1 þ np ðeÞ e; this implies m4p1 þ e1 : Using Lemma 2.2 below we get p1 þ e1 Xnp ðr1 ri Þ; and hence m4np ðr1 ri Þ for all 2pipe: It follows by Krasner’s lemma (see [12, p. 224]) that kðbÞ*kðr1 Þ: Since ½kðbÞ : k ¼ ½kðr1 Þ : k ¼ e; we get L ¼ kðbÞ ¼ kðr1 ÞDK: & Lemma 2.2. Let k be a finite extension of Qp ; let CAk½X be an Eisenstein polynomial ofdegree e; and let r1 ; r2 ; y; re be the roots of C: Then for every 2pipn we have r1 ri 1 : np r1 pp1 Proof. Let E ¼ kðr1 Þ: The lemma may be rephrased as a statement about the higher ramification theory of the extension E=k; which need not be Galois; for the ramification theory of non-Galois extensions, see for instance [3, III, Section 3], or the appendix to [2]. In fact, the integers nE
r1 ri r1
are the lower ramification breaks
for the extension E=k: The lemma is equivalent to the statement that these breaks are 1 nE ðpÞ: Our method is to reduce to the case of a Galois bounded above by p1 extension, where the lemma is well-known (see for instance [19, Exercise 3(c), p. 72]). Let F CO be the splitting field of C; and set G ¼ GalðF =kÞ; H ¼ GalðF =EÞ: Let D ¼ F G1 be the fixed field of the wild ramification subgroup of G; let e1 be the ramification index of D=k; and let e2 be the ramification index of ED=E: Then p[e1 ; and hence p[e2 : It follows that the Hasse-Herbrand functions for the extensions D=k and ED=E are given by fD=k ðxÞ ¼ x=e1 and fED=E ðxÞ ¼ x=e2 for xX0: Using the composition rule for towers of extensions we get fE=k ðxÞ ¼ e11 fED=D ðe2 xÞ: Since the largest lower ramification break of E=k is inffx : f0E=k ðxÞ ¼ 1=eg; it suffices to prove the lemma for the extension ED=D: Since G1 ¼ GalðF =DÞ is a p-group, there is a ð0Þ ð1Þ ð2Þ ðn1Þ ðnÞ refinement G1 ¼ G1 XG1 XG1 X?XG1 XG1 ¼ f1g of the ramification ðiÞ
ðiÞ
ðiþ1Þ
filtration of G1 such that G1 JG1 and jG1 =G1 be the largest integer such that ð jÞ G1 ðH-G1 Þ:
ð jÞ G1
j ¼ p for all 0pipn 1: Let j
is not contained in H; and let D0 be the subfield
of F fixed by Then the largest lower ramification break of ED=D is the same as the largest lower ramification break of ED=D0 : Since ð jÞ ð jÞ jG1 ðH-G1 Þ=ðH-G1 Þj ¼ p; we have H-G1 IG1 ðH-G1 Þ: Therefore the extension ED=D0 is Galois. It follows that the lemma holds for ED=D0 ; and hence also for E=k: &
ARTICLE IN PRESS X.-D. Hou, K. Keating / Journal of Number Theory 104 (2004) 14–61
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Let Cðp; n; f ; e; tÞ be the number of isomorphism classes of finite commutative chain rings with invariants ðp; n; f ; e; tÞ: Then Proposition 2.1(iii) implies that
p þ np ðeÞ e: Cðp; n; f ; e; tÞ ¼ IðQp ; f ; eÞ when ðn 1Þe þ t4 ð2:7Þ p1 When p[e; the number Cðp; n; f ; e; tÞ was first determined by Clark and Liang [1]. A different formula for this quantity was given in [9]. Theorem 2.3 (Clark and Liang [1] and Hou [9]). Let p be a prime and let n; f ; e; t be positive integers such that nX2; 1ptpe; and p[e: Then Cðp; n; f ; e; tÞ ¼
X cjðe;p f 1Þ
f 1 fðcÞ 1 X ði; f Þ ¼ ðp 1; eÞ; tðcÞ f i¼0
ð2:8Þ
where f is the Euler function, ða; bÞ is the greatest common divisor of a and b; and tðcÞ is the smallest positive integer m such that pm 1 ðmod cÞ: From Proposition 2.1(iii) and Theorem 2.3 it follows that when p[e; IðQp ; f ; eÞ ¼
X cjðe;p f 1Þ
f 1 fðcÞ 1 X ði; f Þ ¼ ðp 1; eÞ: tðcÞ f i¼0
ð2:9Þ
In Section 4, we derive a third formula for IðQp ; f ; eÞ in the case p[e: In the other direction, our formulas for IðF ; f ; eÞ allow us to compute Cðp; n; f ; e; tÞ in the following two cases (cf. (2.7), Theorem 5.6, and Theorem 10.1): 1 (i) p jj e and n43 þ p1 et ; 1 (ii) p42; p2 jj e; n44 þ p1 et ; and ðp f 1; eÞ ¼ 1:
3. Preparatory results about p-adic fields Proposition 3.1. Let F CK be finite extensions of Qp such that K=F is totally ramified of degree pi s; with p[s: Then for each positive integer d j s; there is a unique field Kd such that F CKd CK and ½Kd : F ¼ d: Furthermore, for d1 j s and d2 j s; we have Kd1 CKd2 if and only if d1 j d2 : Proof. Let L=F be the Galois closure of K=F ; and set G ¼ GalðL=F Þ and H ¼ GalðL=KÞ: Let G0 be the inertia subgroup and G1 the wild inertia subgroup of G (so G1 is the unique Sylow p-subgroup of G0 ). Then G0 =G1 is a cyclic group whose order is prime to p and divisible by s: Since jG=Hj ¼ pi s factors as the product of jG1 H=Hj ¼ jG1 =ðG1 -HÞj; which is a power of p; and jG=G1 Hj ¼ jG0 =ðG0 -G1 HÞj;
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which is prime to p; we have jG0 =ðG0 -G1 HÞj ¼ s: Let N ¼ G0 -G1 H: Then N is the unique subgroup of G0 of index s which contains G1 ; so NIG: Since K=F is a totally ramified extension we have G0 H ¼ G: Therefore G1 H=N maps isomorphically onto G=G0 ; and hence G=N is a semidirect product of G1 H=NDG=G0 acting on G0 =N: This implies that for each d j s there is a unique subgroup Sd pG of index d such that Sd XG1 H: The fixed field of Sd acting on L is Kd : & Proposition 3.2. Let F be a finite extension of Qp and set f ðF =Qp Þ ¼ f0 : Let e ¼ pi s with p[s; and let KAEðF ; f ; eÞ: ðiÞ There is a unique field LK such that F CLK CK and K=LK is totally ramified of degree pi : ðiiÞ If ðp f0 f 1; sÞ ¼ 1; there is a unique field EK such that F CEK CK and EK =F is totally ramified of degree s: Moreover, we have EK CLK and AutðK=F Þ ¼ AutðK=EK Þ: Proof. (i) This is a special case of Proposition 3.1. (ii) Let k=F be the maximal unramified subextension of K=F : Since p[s; there are uniformizers pLK for LK and pF for F such that psLK =pF AO
k (see [3, II, Proposition 3.5]). Since ððp f0 f 1Þ p; sÞ ¼ 1; we have psLK =pF ¼ bs for some bAO
k: s Then ðpLK =bÞ ¼ pF ; and hence EK ¼ F ðpLK =bÞ is a totally ramified extension of F of degree s which is contained in LK : To prove the uniqueness of EK ; assume that we have F CECK with E=F totally ramified of degree s: Then there is a uniformizer pE s s
s for E such that psE =pF AO
k : As above we get dAOk with d ¼ pE =pF and ðpE =dÞ ¼ s s f0 f pF ¼ ðpLK =bÞ : Thus pE =d ¼ zpLK =b for some zALK with z ¼ 1: Since ððp 1Þ p; sÞ ¼ 1 we must have z ¼ 1; and hence E ¼ kðpE =dÞ ¼ kðp=bÞ ¼ EK : To prove the last statement, we note that for any rAAutðK=F Þ we have rðEK Þ ¼ EK by the uniqueness of EK : Thus rjEK AAutðEK =F Þ: We have already seen that EK ¼ F ðpEK Þ for some pEK AEK such that psEK is a uniformizer for F : Since F does not contain any nontrivial sth root of unity, it follows that AutðEK =F Þ ¼ fidg: Thus rAAutðK=EK Þ: & Let K=Qp be a finite extension with f ðK=Qp Þ ¼ f ; eðK=Qp Þ ¼ e; and let cAAutðK=Qp Þ: Let K c denote the subfield of K fixed by /cS and put eðcÞ ¼ eðK=K c Þ and f ðcÞ ¼ f ðK=K c Þ: The elementary abelian p-group VK ¼ K =ðK Þp can be viewed as a module over the group ring Fp ½/cS; where Fp ¼ Z=pZ: Our approach in Section 5 depends on knowing the Fp ½/cS-module structure of VK in the case p[eðcÞ: The rest of this section is devoted to the determination of this Fp ½/cSmodule structure. The first step is to factor K as K D/pK S /zp f 1 S U1 ;
ð3:1Þ
where U1 ¼ 1 þ pK OK : We have then VK Dð/pK S=/ppK SÞ ðU1 =U1p Þ:
ð3:2Þ
ARTICLE IN PRESS X.-D. Hou, K. Keating / Journal of Number Theory 104 (2004) 14–61
21
Let F =K c be the maximal unramified subextension of K=K c ; then F is the fixed field of /c f ðcÞ S; and K=F is a totally ramified cyclic extension of degree eðcÞ: Since p[eðcÞ; the field F contains an eðcÞth root of unity, and there is a uniformizer pK for eðcÞ eðcÞ c K such that pK AF : In fact we have pK ¼ Zp for some ZAO
F and pAK : Since
eðcÞ O
is generated by roots of unity, we may assume that Z is a root of unity, of F =ðOF Þ eðcÞ
t
eðcÞ
order prime to p: It follows that cðpK Þ ¼ Zq 1 pK ; where tX0 and q is the cardinality of the residue field of K c : Since Zq1 AðK Þp ; this implies that c acts eðcÞ trivially on the image of pK in VK ; and hence also on the image of pK in VK : Since U1 is clearly stabilized by c; this implies that the factors in (3.2) are Fp ½/cSsubmodules of VK : Thus it remains only to determine the Fp ½/cS-module structure of U1 =U1p : Let oðcÞ denote the order of c: Proposition 3.3. (i) If zp eK; there is an isomorphism of Zp ½/cS-modules U1 DZp ½/cSnK =oðcÞ :
ð3:3Þ
Hence there is an isomorphism of Fp ½/cS-modules U1 =U1p DFp ½/cSnK =oðcÞ :
ð3:4Þ
(ii) If zp AK and p[eðcÞ; there is an isomorphism of Fp ½/cS-modules U1 =U1p D/zp S Fp ½/cSnK =oðcÞ :
ð3:5Þ
Proof. (i) We use Theorem 4(b) in [21]. This theorem implies that as long as U1 contains no roots of unity, the Zp ½/cS-isomorphism class of U1 is determined by its Zp -rank. Since /cS is cyclic, this means that we can replace the extension K=K c with an extension K 0 =k0 such that K 0 =k0 and k0 =Qp are unramified, nk0 ¼ nK c ; and GalðK 0 =k0 ÞDGalðK=K c Þ ¼ /cS: The logarithm gives an isomorphism between the Zp ½GalðK 0 =k0 Þ-modules U10 ¼ 1 þ pOK 0 and pOK 0 : Since K 0 =k0 is unramified, pOK 0 is free over Zp ½GalðK 0 =k0 Þ of rank nK 0 =oðcÞ: It follows that U10 is free over Zp ½GalðK 0 =k0 Þ of rank nK 0 =oðcÞ; and hence that U1 is free over Zp ½/cS of rank nK =oðcÞ: & The Zp ½/cS-module structure of U1 cannot be described as simply when U1 contains roots of unity. In fact, when p[eðcÞ it follows from another theorem of Gruenberg and Weiss (Theorem 6.1(a) of [4]) that U1 is cohomologically trivial as a Zp ½/cS-module. Since the Zp -torsion subgroup /zps S of U1 is not in general cohomologically trivial, /zps S need not be a direct Zp ½/cS-summand of U1 : Therefore to prove Proposition 3.3(ii), we work directly with U1 =U1p : (ii) The group U1 has a filtration U1 *U2 *?; where Ui ¼ 1 þ piK OK : This induces a filtration on U1 =U1p whose ith filtrant is U% i ¼ Ui U1p =U1p : Put r ¼ pe=ðp 1Þ: Since we are assuming zp AK; we have ðp 1Þ j e and hence rAZ:
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Define I ¼ fiAZ : 1pipr and p[ig:
ð3:6Þ
Our strategy is to first determine the Fp ½/cS-module structure of the quotients U% i =U% iþ1 for iAI; and then use this information to reconstruct U1 =U1p : There are isomorphisms of Fp ½/cS-modules U% i =U% iþ1 D Ui U1p =Uiþ1 U1p D Ui =ðUi -ðUiþ1 U1p ÞÞ:
ð3:7Þ
For iAI we have Ui -ðUiþ1 U1p Þ ¼ Uiþ1 ; while if ieI and iar we have Uiþ1 U1p *Ui : Therefore if iAI we have U% i =U% iþ1 DUi =Uiþ1 DpiK OK =piþ1 K OK ;
ð3:8Þ
while if ieI and iar we have U% i =U% iþ1 ¼ f1g: Finally, we have jU% r =U% rþ1 j ¼ p since jU1 =U1p j ¼ p feþ1 ; jIj ¼ e; and Urþ1 CU1p : Lemma 3.4. Let iX0 and let c# denote the automorphism of piK OK =piþ1 K OK induced by c: 0 1 f ðcÞ1 % Then c# ; c# ; y; c# are linearly independent over K: Proof. If not then there is a monic polynomial PðX Þ ¼ X m þ am1 X m1 þ ? þ a1 X þ a0
ð3:9Þ
% such that Pð#cÞ ¼ 0: We assume that m is as small as of degree mof ðcÞ in K½X possible; then a0 a0; since c# is invertible. Since 1pmof ðcÞ; there exists aAK% such that cm ðaÞaa: Since Pð#cÞ v ¼ 0 for all vApiK OK =piþ1 cÞ av ¼ 0 for K OK ; we have Pð# all v as well. This implies that QðX Þ ¼ cm ðaÞX m þ am1 cm1 ðaÞX m1 þ ? þ a1 cðaÞX þ a0 a
ð3:10Þ
% is a polynomial of satisfies Qð#cÞ ¼ 0: Therefore RðX Þ ¼ QðX Þ cm ðaÞPðX ÞAK½X degree om with nonzero constant term such that Rð#cÞ ¼ 0: This violates the minimality of m; and therefore proves the lemma. & It follows from Lemma 3.4 that c# 0 ; c# 1 ; y; c# f ðcÞ1 are linearly independent over K% c ; and hence that the degree of the minimal polynomial of c# over K% c is Xf ðcÞ: Since iþ1 i %c piK OK =piþ1 K OK has dimension f ðcÞ over K ; this implies that pK OK =pK OK is a cyclic c K% ½/cS-module generated by some vi ; i.e., %c piK OK =piþ1 K OK ¼ K ½/cS vi :
ð3:11Þ
ARTICLE IN PRESS X.-D. Hou, K. Keating / Journal of Number Theory 104 (2004) 14–61
23
Since c f ðcÞ generates the inertia group of the tamely ramified extension K=K c ; the image x of c f ðcÞ ðpK Þ=pK in K% is a primitive eðcÞth root of unity. On the other hand, class field theory gives an onto homomorphism r : ðK g Þ -/gS such that the inertia subgroup /g f ðgÞ S of /gS is the image of the unit group of K g : Hence ðK g Þ contains an element of order eðgÞ: Therefore K% c contains a primitive eðcÞth root of unity, so we have xAK% c : In particular, K% c vi is a K% c ½/c f ðcÞ S-submodule of piK OK =piþ1 K OK : i i f ðcÞ i # p Let p% iK denote the image of piK in piK OK =piþ1 O : Then c ð p Þ ¼ x : % % K K K Since K f ðcÞ c c f ðcÞ % % /c S acts trivially on K ; it follows that there is a K ½/c S-module isomorphism K% c ½/c f ðcÞ SD
eðcÞ1 M
K% c vi :
ð3:12Þ
i¼0
Therefore K% c vi is a projective K% c ½/c f ðcÞ S-module. Using (3.11) we get %c %c piK OK =piþ1 K OK DK ½/cS#K% c ½/c f ðcÞ S K vi :
ð3:13Þ
%c It follows that piK OK =piþ1 K OK is projective over K ½/cS; and hence also over Fp ½/cS: Using (3.8) we get an Fp ½/cS-module isomorphism ! U1 =U1p D
M
% piK O=piþ1 K OK "Ur :
ð3:14Þ
iAI
Write e ¼ dpt with p[d; then eðcÞ j d and ðp 1Þ j d: Partition I into I-½0; I-½1; y; I-½d 1; where ½i is the congruence class of i modulo d: Then each subset I-½i contains either de ; de 1; or de þ 1 elements. However, using the fact that ðp 1Þ j d; one can show that jI-½ij ¼ de 71 if and only if jI-½pij ¼ de 81: If i j ðmod dÞ then there is an Fp ½/c f ðcÞ S-module isomorphism K% c vi DK% c vj : On the other hand, if vpi is chosen suitably, x/xp gives an Fp ½/c f ðcÞ S-module isomorphism between K% c vi and K% c vpi : Therefore we have an Fp ½/c f ðcÞ S-module isomorphism M
K% vi D c
iAI
d1 M
K% vi
!e=d
c
:
ð3:15Þ
i¼0
Using (3.12) we get M iAI
K% vi D c
eðcÞ1 M
K% vi
!e=eðcÞ
c
i¼0
D K% c ½/c f ðcÞ Se=eðcÞ D Fp ½/c f ðcÞ SnK =oðcÞ :
ð3:16Þ
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Therefore by (3.13) and (3.16) there are Fp ½/cS-module isomorphisms M i
M
iAI
iAI
pK OK =piþ1 K OK D D
ðK% c ½/cS#K% c ½/c f ðcÞ S K% c vi Þ
M
ðFp ½/cS#Fp ½/c f ðcÞ S K% c vi Þ
iAI
D Fp ½/cS#Fp ½/c f ðcÞ S
M
K% c vi
!
iAI
D Fp ½/cS#Fp ½/c f ðcÞ S Fp ½/c f ðcÞ SnK =oðcÞ D Fp ½/cSnK =oðcÞ :
ð3:17Þ
To complete the proof of Proposition 3.3(ii), we need only determine the Fp ½/cSmodule structure of U% r : Lemma 3.5. There is an Fp ½/cS-module isomorphism U% r D/zp S: Proof. Define a group homomorphism c : Ur -Fp by setting cð1 þ pðzp 1ÞxÞ ¼ TrK=F % % p ðxÞ;
ð3:18Þ
% We claim that cðUr -ðUrþ1 U1p ÞÞ ¼ 0: If where xAOK and x% is the image of x in K: p yAUr -ðUrþ1 U1 Þ; then y ð1 þ ðzp 1ÞzÞp ðmod prþ1 K Þ for some zAOK : We have ! ðzp 1Þp1 p p ð1 þ ðzp 1ÞzÞ 1 þ pðzp 1Þ z þ z ðmod prþ1 ð3:19Þ K Þ: p Since ðzp 1Þp1 p ðmod ppK Þ; we have TrK=F % p
ðzp 1Þp1 p z z% þ p
! p ¼ TrK=F % p ðz% z% Þ ¼ 0;
ð3:20Þ
so cðyÞ ¼ 0: Since c is nontrivial, it follows that c induces a group isomorphism between U% r DUr =ðUr -ðUrþ1 U1p ÞÞ and Fp : Suppose that cðzp Þ ¼ zm p : Then cð1 þ pðzp 1ÞzÞ ¼ 1 þ pðzm p 1ÞcðzÞ ÞcðzÞ ¼ 1 þ pðzp 1Þð1 þ ? þ zm1 p 1 þ pðzp 1ÞmcðzÞ ðmod prþ1 K Þ:
ð3:21Þ
% r DFp by raising to the Since TrK=F % p ðmcðzÞÞ ¼ mTrK=F % p ðz%Þ; we see that c acts on U % power m: Therefore the Fp ½/cS-modules Ur and /zp S are isomorphic. &
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25
Using (3.14), (3.17), and Lemma 3.5, we get an Fp ½/cS-module isomorphism U1 =U1p D/zp S Fp ½/cSnK =oðcÞ ; and the proof of Proposition 3.3(ii) is complete.
ð3:22Þ
&
Corollary 3.6. (i) If zp eK; there is an isomorphism of Fp ½/cS-modules K =ðK Þp DFp Fp ½/cSnK =oðcÞ :
ð3:23Þ
(ii) If zp AK and p[eðcÞ; there is an isomorphism of Fp ½/cS-modules K =ðK Þp DFp /zp S Fp ½/cSnK =oðcÞ :
ð3:24Þ
4. The case p[e In this section we determine IðF ; f ; eÞ in the cases where p[e: Thus we are restricting our attention to tamely ramified extensions of the p-adic field F ; which are in general well-understood. Therefore we only give outlines for the arguments in this section, most of which are not new. Besides calculating IðF ; f ; eÞ; we also collect some facts about tamely ramified extensions of F which will be useful in the next section. We start by listing the elements of EðF ; f ; eÞ: Let q denote the cardinality of the residue field of F and set g ¼ ðq f 1Þe: Let pF be a uniformizer of F ; let pE AO be an eth root of pF ; and define E ¼ F ðzg ; pE Þ: The extension E=F is Galois, with GalðE=F Þ ¼ /r; s : re ¼ sdf ¼ 1; srs1 ¼ rq S;
ð4:1Þ
where df ¼ ½F ðzg Þ : F : The actions of r and s on E are given by rðpE Þ ¼ ze pE ;
rðzg Þ ¼ zg ;
sðpE Þ ¼ pE ;
sðzg Þ ¼ zqg :
ð4:2Þ
For 0phoe; we define Kh ¼ F ðzq f 1 ; ph Þ; where ph ¼ zhg pE ; equivalently, Kh is the subfield of E fixed by /rh s f S: Then we have EðF ; f ; eÞ ¼ fKh : 0phoeg:
ð4:3Þ
We now describe the F -automorphisms of Kh : The inertia subgroup AutðKh =F ðzq f 1 ÞÞ of AutðKh =F Þ is a cyclic group of order b ¼ ðe; q f 1Þ generated
ARTICLE IN PRESS X.-D. Hou, K. Keating / Journal of Number Theory 104 (2004) 14–61
26
by an element l such that lðzq f 1 Þ ¼ zq f 1 and lðph Þ ¼ zb ph : We need to determine which elements of GalðF ðzq f 1 Þ=F Þ can be extended to automorphisms of Kh : Let q be the Frobenius automorphism of F ðzq f 1 Þ=F : For cX0 we attempt to extend qc to f
an element mc of AutðKh =F Þ: Since pgh ¼ pgE ¼ pqF 1 AF ; we have mc ðph Þ ¼ eph for some eAKh such that eg ¼ 1: But since p[g; we must have e ¼ zxq f 1 for some xAZ: Therefore c
hq e e x h e zexþh q f 1 pF ¼ ðzq f 1 ph Þ ¼ m c ðph Þ ¼ m c ðph Þ ¼ m c ðzq f 1 pF Þ ¼ zq f 1 pF ;
ð4:4Þ
which implies that ex ðqc 1Þh ðmod q f 1Þ:
ð4:5Þ
Conversely, if x satisfies (4.5), then qc can be extended to m c AAutðKh =F Þ such that mc ðph Þ ¼ zxq f 1 ph : Congruence (4.5) can be solved for x if and only if b ¼ ðe; q f 1Þ divides c ðq 1Þh: Let ch be the smallest positive integer c satisfying this condition. Then ch is
b the order of q in Z=ðb;hÞ Z : Let uAZ satisfy eu b ðmod q f 1Þ and set mh ¼ 1 ch bðq
1Þhu: Then x ¼ mh is a solution to (4.5) with c ¼ ch : The elements in GalðF ðzq f 1 Þ=F Þ which can be extended to automorphisms of Kh =F are ðqch Þi for 0piof =ch : Let mAAutðKh =F Þ be the extension of qch defined by h p : Then m generates the group mðph Þ ¼ zm q f 1 h AutðKh =F Þ=GalðF ðzq f 1 Þ=F Þ ¼ AutðKh =F Þ=/lS
ð4:6Þ
and satisfies m f =ch A/lS: The actions of l and m on Kh ¼ F ðzq f 1 ; ph Þ are given by lðzq f 1 Þ ¼ zq f 1 ;
lðph Þ ¼ zb ph ;
ch
mðzq f 1 Þ ¼ zqq f 1 ;
h mðph Þ ¼ zm p: q f 1 h
ð4:7Þ
c
Using (4.7) we find that m f =ch ¼ luh and mlm 1 ¼ lq h : Therefore c
AutðKh =F Þ ¼ /l; m : lb ¼ 1; m f =ch ¼ luh ; mlm 1 ¼ lq h S:
ð4:8Þ
In particular, jAutðKh =F Þj ¼ bf =ch : Combining this fact with (1.3), we get the following proposition.
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27
Proposition 4.1. Assume that p[e: Then IðF ; f ; eÞ ¼
e1 bX 1 ; e h¼0 ch
ð4:9Þ
where b ¼ ðe; q f 1Þ; q is the cardinality of the residue field of F ; and ch is the smallest positive integer such that b divides ðqch 1Þh: Remark 4.2. The method of Corollary 4.3 in [9] gives the alternative formula IðF ; f ; eÞ ¼
f 1 1 X ði; f Þ ðq 1; eÞ: f i¼0
ð4:10Þ
On the other hand, Proposition 4.1 with F ¼ Qp gives a third formula (in addition to those in Theorem 2.3) for Cðp; n; f ; e; tÞ when p[e and nX2; Cðp; n; f ; e; tÞ ¼
e1 bX 1 : e h¼0 ch
ð4:11Þ
Remark 4.3. For future use we note that l is the restriction to Kh of re=b and m is the restriction to Kh of rvh sch ; where vh ¼
emh þ hð1 qch Þ : qf 1
ð4:12Þ
For each cAAutðKh =F Þ; let Khc denote the subfield of Kh fixed by /cS: By (4.8), we can write c uniquely in the form c ¼ li m j with 0piob and 0pjof =ch : The smallest power of c which lies in the inertia subgroup /lS of AutðKh =F Þ is f
f
cð f ;ch jÞ ¼ ðli m j Þð f ;ch jÞ ¼ ltðcÞ ;
ð4:13Þ
where tðcÞ is computed using (4.8) to be tðcÞ ¼
qlcmð f ;ch jÞ 1 uhch j iþ : qc h j 1 ð f ; ch jÞ
ð4:14Þ
This implies that the extension Kh =Khc has residue degree f ðcÞ ¼ f =ð f ; ch jÞ and ramification index eðcÞ ¼ b=ðb; tðcÞÞ: The order oðcÞ ¼ ½Kh : Khc of c is given by oðcÞ ¼ eðcÞ f ðcÞ ¼
b f : ðb; tðcÞÞ ð f ; ch jÞ
ð4:15Þ
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5. The case p jj e In this section we assume e ¼ pe0 with p[e0 : We use the notation of Section 4 with e0 in place of e: In particular, 8 q ¼ the cardinality of the residue class field of F ; > > > > > g ¼ ðq f 1Þe0 ; > > > > > b ¼ ðe0 ; q f 1Þ; > > > > > < ch ¼ the smallest positive integer such that b j ðqch 1Þh; uAZ satisfies e0 u b ðmod q f 1Þ; > > > > ðqch 1Þh > > > u; mh ¼ > > b > > > > e0 mh þ hð1 qch Þ > > : : vh ¼ qf 1
ð5:1Þ
Let LAEðF ; f ; eÞ: Then by Proposition 3.1, there is a unique KAEðF ; f ; e0 Þ which is contained in L: It follows from (1.3) and (4.3) that IðF ; f ; eÞ ¼
¼
1 fe
X
X
jAutðL=F Þj
KAEðF ;f ;e0 Þ LAEðK;1;pÞ
0 1 1 eX fe h¼0
X
jAutðL=F Þj:
ð5:2Þ
LAEðKh ;1;pÞ
For the time being, we fix K ¼ Kh and concentrate on evaluating the inner sum of (5.2). Let LAEðK; 1; pÞ: Then since KAEðF ; f ; e0 Þ is uniquely determined by L; restriction induces a homomorphism AutðL=F Þ-AutðK=F Þ: Let HL CAutðK=F Þ be the image of this homomorphism. Then AutðL=F Þ is an extension of HL by AutðL=KÞ: Thus if L=K is not Galois then AutðK=F Þ ¼ HL ; while if L=K is Galois then AutðK=F Þ is an extension of HL by a cyclic group of order p: For each cAAutðK=F Þ; let S1 ðcÞ ¼ fLAEðK; 1; pÞ : cAHL and L=K is Galoisg;
ð5:3Þ
S2 ðcÞ ¼ fLAEðK; 1; pÞ : cAHL and L=K is not Galoisg;
ð5:4Þ
and define mi ðcÞ ¼ jSi ðcÞj: Then by counting the elements in the set fðL; dÞ : LAEðK; 1; pÞ; dAAutðL=F Þg
ð5:5Þ
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29
in two different ways we find that X
jAutðL=F Þj ¼
X
ðp m1 ðcÞ þ m2 ðcÞÞ:
ð5:6Þ
cAAutðK=F Þ
LAEðK;1;pÞ
Lemma 5.1. Let cAAutðK=F Þ; let K c be the subfield of K=F fixed by /cS; and let dðcÞ ¼ ðoðcÞ; p 1Þ: Then 8 nF e0 f n F e0 f > oðcÞ 1 oðcÞ þ1 1 > p p > > > þ 1 if zp eK; ðdðcÞ 1Þ > > p1 p1 > > > > nF e0 f n F e0 f < oðcÞ 1 oðcÞ þ1 1 p p m1 ðcÞ ¼ ðdðcÞ 2Þ þ2 1 if zp AK\K c ; > > p1 p1 > > > > nF e0 f n F e0 f > þ2 > > > p oðcÞ 1 p oðcÞ 1 > : ðdðcÞ 1Þ þ 1 if zp AK c : p1 p1
ð5:7Þ
Proof. By class field theory, cyclic extensions L=K of degree p such that cAHL correspond to /cS-invariant subgroups of VK ¼ K =ðK Þp of index p: The unramified degree-p extension of K is excluded from S1 ðcÞ; and so m1 ðcÞ is one less than the number of /cS-invariant subgroups of VK of index p: For each of the dðcÞ homomorphisms c : /cS-F
p ; the largest quotient on which /cS acts through c is VK ðcÞ ¼ VK =ðc cðcÞÞVK : On the other hand, if H is a /cSinvariant subgroup of VK of index p then /cS acts on VK =H through some homomorphism c : /cS-F
p : Thus HXðc cðcÞÞVK and H=ðc cðcÞÞVK is a subgroup of VK ðcÞ of index p: In fact, H2H=ðc cðcÞÞVK gives a one-to-one correspondence between the set of /cS-invariant subgroups of VK of index p and the set [
fUCVK ðcÞ : ½VK ðcÞ : U ¼ pg:
ð5:8Þ
c : /cS-F
p
If zp eK; it follows from Corollary 3.6(i) that there is an isomorphism of Fp ½/cSmodules VK DFp Fp ½/cS
n F e0 f oðcÞ :
1 Thus dimFp VK ðcÞ ¼ oðcÞ nF e0 f for ca1; and
1 dimFp VK ð1Þ ¼ oðcÞ nF e0 f þ 1: The formula for m1 ðcÞ in the case zp eK follows from this. nF e0 f
Now assume zp AK: By Corollary 3.6(ii) we have VK DFp /zp S Fp ½/cS oðcÞ : If zp eK c then cðzp Þ ¼ zm p for some mc1 ðmod pÞ: Therefore dimFp VK ðcÞ ¼ 1 oðcÞ nF e0
1 f when cðcÞef1; mg; and dimFp VK ðcÞ ¼ oðcÞ nF e0 f þ 1 when cðcÞAf1; mg:
1 In the case where zp AK c we have dimFp VK ðcÞ ¼ oðcÞ nF e0 f when cðcÞa1 and
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1 dimFp VK ðcÞ ¼ oðcÞ nF e0 f þ 2 when cðcÞ ¼ 1: The remaining formulas for m1 ðcÞ follow from these observations. &
Lemma 5.2. We have 8 n F e0 f > n F e0 f oðcÞ 1 > p > þ2 > > p2 þ p pdðcÞ < p oðcÞ p1 m2 ðcÞ ¼ nF e0 f > > n F e0 f > oðcÞ 1 > p > : ðp2 pÞp oðcÞ p2 þ p pdðcÞ p1
if zp eK; ð5:9Þ if zp AK:
Proof. Suppose LAS2 ðcÞ: Since AutðL=F ÞDHL ; there is a unique c* AAutðL=F Þ which extends c: Let Lc* be the subfield of L fixed by /*cS: Then Lc* =K c is a ramified extension of degree p such that Lc* K ¼ L: Conversely, let M=K c be a ramified extension of degree p such that the compositum MK is not Galois over K: Then M and K are linearly disjoint over K c and so cAGalðK=K c Þ can be uniquely extended to an element c* AAutðMK=MÞ: Therefore MKAS2 ðcÞ: Thus M2MK gives a bijection between the set fMAEðK c ; 1; pÞ : MK=K not Galoisg and S2 ðcÞ: Let YðcÞ ¼ fMAEðK c ; 1; pÞ : M=K c is Galoisg ZðcÞ ¼ fMAEðK c ; 1; pÞ : M=K c is not Galois but MK=K is Galoisg:
ð5:10Þ ð5:11Þ
Then we have m2 ðcÞ ¼ jEðK c ; 1; pÞj jYðcÞj jZðcÞj:
ð5:12Þ
By Krasner’s formula (1.1) we have jEðK c ; 1; pÞj ¼ p
nF e0 f oðcÞ þ2
p2 þ p;
and by class field theory we have 8 nF e0 f þ1 > > p oðcÞ 1 > > > 1 < p1 jYðcÞj ¼ nF e0 f > > > oðcÞ þ2 1 > p > : 1 p1
ð5:13Þ
if zp eK c ; ð5:14Þ if zp AK c :
It remains to determine jZðcÞj: Let MAZðcÞ: Then cAGalðK=K c Þ lifts to c* AGalðMK=MÞ; so MK=K c is Galois and GalðMK=K c Þ is the semidirect product of GalðMK=MÞ ¼ /*cS acting on GalðMK=KÞDZ=pZ: This action is nontrivial since M=K c is not Galois. On the other hand, suppose that L=K is a cyclic extension of degree p such that L=K c is Galois and GalðL=K c Þ is nonabelian. Then c can be lifted
ARTICLE IN PRESS X.-D. Hou, K. Keating / Journal of Number Theory 104 (2004) 14–61
31
to an automorphism c* of L; which must satisfy oð*cÞ ¼ oðcÞ; since otherwise GalðL=K c Þ ¼ /*cS is abelian. Therefore GalðL=K c Þ is a semidirect product of /*cS acting nontrivially on GalðL=KÞ: For such an L; the group GalðL=K c Þ contains p different subgroups which map isomorphically onto GalðK=K c Þ; so there are p elements MAZðcÞ such that MK ¼ L: Therefore we have jZðcÞj ¼ pjWðcÞj; where WðcÞ ¼ fL : KCLCO; ½L : K ¼ p; L=K c is Galois and nonabeliang:
ð5:15Þ
By class field theory, elements in WðcÞ correspond to subgroups HpVK of index p such that H is invariant under the action of /cS and such that /cS acts nontrivially on VK =H: Using Corollary 3.6, we find that 8 n F e0 f > > oðcÞ 1 p > > ðdðcÞ 1Þ > if zp eK or zp AK c ; < p1 ð5:16Þ jWðcÞj ¼ n F e0 f n F e0 f > þ1 > > oðcÞ oðcÞ > p 1 p 1 > : ðdðcÞ 2Þ þ if zp AK but zp eK c : p1 p1 Eq. (5.9) now follows from (5.12)–(5.16). & It follows from Lemmas 5.1 and 5.2 that 8 nF e0 f > < ðp2 þ pÞp oðcÞ p2 if z eK c ; p p m1 ðcÞ þ m2 ðcÞ ¼ ð5:17Þ nF e0 f > þ2 : 2 c p if zp AK : 2p oðcÞ P In order to write down explicit formulas for LAEðK;1;pÞ jAutðL=F Þj; we change notation slightly: We restore the subscript h to K; and instead of oðcÞ; tðcÞ; K c ; we write oðh; i; jÞ; tðh; i; jÞ; Khij ; where c ¼ li m j : If zp eKh ; then by combining (5.6) with (5.17) we get the following result. Proposition 5.3. If zp eKh then f
X
jAutðL=F Þj ¼
i¼0
LAEðKh ;1;pÞ
1
b1 cX h X
n F e0 f
ððp2 þ pÞpoðh;i;jÞ p2 Þ
j¼0 f
1
b1 cX nF e0 f h X p2 bf ¼ þ ðp2 þ pÞ poðh;i;jÞ ; ch i¼0 j¼0
ð5:18Þ
where oðh; i; jÞ ¼
f b ; ð f ; ch jÞ ðb; tðh; i; jÞÞ
ð5:19Þ
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tðh; i; jÞ ¼
qlcmð f ;ch jÞ 1 uhch j iþ : qc h j 1 ð f ; ch jÞ
ð5:20Þ
In order to evaluate (5.2), we need to be able to tell when zp AKh ; and when zp AKhij : Let fp be the residue degree and ep the ramification index of the extension F ðzp Þ=F : A necessary condition for Kh to contain zp is that fp j f and ep j e0 : Therefore, in what follows, we will assume fp j f and ep j e0 : Then E ¼ F ðzg ; pE Þ contains all the fields in EðF ; fp ; ep Þ; including F ðzp Þ: Therefore F ðzp Þ is the fixed field of a normal subgroup H of GalðE=F Þ: Since the residue degree of E=F ðzp Þ is f =fp ; and the ramification index is e=ep ; we easily see that H ¼ /rep ; rl s fp S for some l: The following lemma shows that we can assume l ¼ 1: Lemma 5.4. There is an automorphism C of GalðE=F Þ ¼ /r; sS such that (i) C maps the inertia group /rS onto itself. (ii) C acts trivially on the quotient GalðE=F Þ=/rS: (iii) CðHÞ ¼ /rep ; rs fp S: Proof. Since GalðE=F Þ=HDGalðF ðzp Þ=F Þ is cyclic, it is generated by ra s for some aAZ: Define an automorphism C1 of GalðE=F Þ by setting C1 ðrÞ ¼ r and C1 ðsÞ ¼ ra s: Then C1 ðHÞ ¼ /rep ; rd s fp S; where d¼l
q fp 1 a: q1
ð5:21Þ
Furthermore, s ¼ C1 ðra sÞ generates the quotient GalðE=F Þ=C1 ðHÞ: This implies that ðep ; dÞ ¼ 1; so there is kAZ such that dk 1 ðmod ep Þ: In addition, since the homomorphism ðZ=e0 ZÞ -ðZ=ep ZÞ is onto, we may choose k so that ðe0 ; kÞ ¼ 1: Define an automorphism C2 of GalðE=F Þ by setting C2 ðrÞ ¼ rk and C2 ðsÞ ¼ s: Then C ¼ C2 3C1 satisfies the given conditions. & By Lemma 5.4 we have H ¼ /ðrs Þep ; rs ðrt sÞ fp S for some s; tAZ such that ðs; e0 Þ ¼ 1: Let r* ¼ rs ; s* ¼ rt s; p* E ¼ zt * F ¼ zt * F is a uniformizer of e0 ðq1Þ pE ; and p q1 pF : Then p e0 F and p* E ¼ p* F : Furthermore, rð * p* E Þ ¼ zse p* E ;
rðz * g Þ ¼ zg ;
s* ðp* E Þ ¼ p* E ;
s* ðzg Þ ¼ zqg :
ð5:22Þ
It follows that by replacing pF ; pE ; r; s with p* F ; p* E ; r; * s* we may assume that H ¼ /rep ; rs fp S: Under this assumption, for every x; yAZ the element rx sy fixes zp if and only if rx sy A/rep ; rs fp S: By (4.1) this is equivalent to fp j y and x
qy 1 ðmod ep Þ: q fp 1
ð5:23Þ
ARTICLE IN PRESS X.-D. Hou, K. Keating / Journal of Number Theory 104 (2004) 14–61
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Since Kh is the subfield of E fixed by /rh s f S; we see that zp AKh if and only if h
qf 1 ðmod ep Þ: q fp 1
ð5:24Þ
To determine whether zp is in Khij we recall that by Remark 4.3, l is the restriction of re0 =b to Kh and m is the restriction of rvh sch to Kh : It follows that c ¼ li m j is the restriction to Kh of ðre0 =b Þi ðrvh sch Þ j ¼ rr sch j ;
ð5:25Þ
where r¼
e 0 i qc h j 1 þ c vh : b q h 1
ð5:26Þ
Thus zp AKhij if and only if fp j ch j and e 0 i qc h j 1 qc h j 1 þ c v h fp ð5:27Þ ðmod ep Þ: b q h 1 q 1 P Using (5.6), (5.17), and (5.27) we get a formula for LAEðKh ;1;pÞ jAutðL=F Þj in the case zp AKh : Proposition 5.5. If zp AKh then f
X LAEðKh ;1;pÞ
2
p bf jAutðL=F Þj ¼ þ ðp2 þ pÞ ch X
þ ðp2 pÞ
1
b1 cX h X i¼0
nF e0 f
poðh;i;jÞ
j¼0
X
nF e0 f
poðh;i;jÞ ;
ð5:28Þ
0pjof =ch iARh; j fp jch j
where Rh; j denotes the set of integers 0piob satisfying congruence (5.27). By combining Propositions 5.3 and 5.5 with Eq. (5.2), we get the main result of this section. Theorem 5.6. Let F be a finite extension of Qp ; let f and e be positive integers such that p jj e; and set e0 ¼ e=p: Then the number of F -isomorphism classes of extensions of F with residue class degree f and ramification index e is 0 1 f
IðF ; f ; eÞ ¼
1 fe
eX 0 1B h¼0
2
B p bf þ @ ch
1
b1 cX h X i¼0
j¼0
nF e0 f C ðp2 þ ohij Þpoðh;i;jÞ C A;
ð5:29Þ
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where (i) (ii) (iii) (iv) (v) (vi) (vii)
q ¼ the cardinality of the residue field of F ; b ¼ ðe0 ; q f 1Þ; uAZ satisfies e0 u b ðmod q f 1Þ; ch is the smallest positive integer such that b j ðqch 1Þh; lcmð f ;ch jÞ
1 h j i þ ðuhc f ;ch jÞ; qch j 1 b ; oðh; i; jÞ ¼ ð f ;cf h jÞ ðb;tðh;i;jÞÞ 8 2 < p if eðF ðzp Þ=F Þ j e;
tðh; i; jÞ ¼ q
ohij ¼
:
p
f ðF ðzp Þ=F Þ j f ; f ðF ðzp Þ=F Þ j ch j; and h; i; j satisfy ð5:27Þ; otherwise:
6. The case p2 jj e For the remainder of the paper we consider the case p2 jj e: Let f1 ¼ f ðF =Qp Þ and e1 ¼ eðF =Qp Þ; and write e ¼ p2 e0 with p[e0 : We make the following simplifying assumptions: ðp f1 f 1; e0 Þ ¼ 1;
f ðF ðzp Þ=F Þ[f
or
eðF ðzp Þ=F Þ41:
ð6:1Þ
Some consequences of these assumptions are given in the following proposition. Proposition 6.1. Assume the conditions in (6.1), and let KAEðF ; f ; eÞ: Then: ðiÞ There is a unique field LK such that F CLK CK and K=LK is totally ramified of degree p2 : ðiiÞ There is a unique field EK such that F CEK CK and EK =F is totally ramified of degree e0 : ðiiiÞ EK is a subfield of LK : ðivÞ AutðK=F Þ acts trivially on EK : ðvÞ zp eK: Proof. Conditions (i)–(iv) follow from Proposition 3.2. To prove (v), note that eðF ðzp Þ=F Þ j ðp 1Þ and ðp 1; eÞ ¼ 1: Thus if eðF ðzp Þ=F Þ41 then eðF ðzp Þ=F Þ[e: Since zp AK implies eðF ðzp Þ=F Þ j e and f ðF ðzp Þ=F Þ j f ; we must have zp eK: & The approach we take here is somewhat different from that of Sections 4 and 5. Even with the conditions in (6.1), the computations that we will face are quite lengthy. To control the overall length of the paper and maintain readability, we will
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35
describe the reasoning behind our computations but omit the details. All the computations in these section have been checked using Mathematica [22]. For each positive integer d; let Bd ¼ fKAEðF ; f ; eÞ : d j jAutðK=F Þjg: P Using (1.3) and the fact that djn fðdÞ ¼ n we get IðF ; f ; eÞ ¼
1 X fðdÞjBd j; fe d40
ð6:2Þ
ð6:3Þ
where f is the Euler function. For d40 and i ¼ 0; 1; 2; put Cid ¼ fKABd : (F CNCK such that K=N is Galois; eðK=NÞ ¼ pi ; and d j ½K : Ng:
ð6:4Þ
By Proposition 6.1(iv) we have Bd ¼ C2d ,C1d ,C0d : Therefore jBd j ¼ jC2d j þ jC1d \C2d j þ jC0d \ðC1d ,C2d Þj:
ð6:5Þ
More precisely, C2d consists of the fields KABd such that the ramification index of K over the fixed field of AutðK=F Þ is p2 ; C1d \C2d consists of the fields KABd such that the ramification index of K over the fixed field of AutðK=F Þ is p; and C0d \ðC1d ,C2d Þ consists of the fields KABd such that the ramification index of K over the fixed field of AutðK=F Þ is 1. We will determine jC2d j; jC1d \C2d j; and jC0d \ðC1d ,C2d Þj separately in Sections 7–9. In our computations, we will frequently encounter a tower of finite extensions Qp CTCLCK; where K=T is Galois, L=T is unramified of degree d; K=L is abelian of degree pi ; and zp eL: We need to give an explicit description of GalðK=TÞ in terms of GalðK=LÞ: Let L=T be an unramified extension of degree d and define KðT; L; pi Þ ¼ fK : LCKCO; K=L abelian of degree pi ; K=T Galoisg: For positive integers m and n; put 2 0 6 60 6 60 6 6 CðmÞ ¼ 6 ^ 6 60 6 6 40 1
1
0
y 0
0 0
1 0
y 0 y 0
^
^
^
0 0
0 0
y 1 y 0
0
0
y 0
0
ð6:6Þ
3
7 07 7 07 7 7 ; ^7 7 07 7 7 15 0 m m
ð6:7Þ
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2
CðmÞ
0m m
60 6 m m Dðm; nÞ ¼ 6 4 ^
CðmÞ ^
0m m
0m m
Eðm; nÞ ¼
39 0m m > > > = y 0m m 7 7 7 n; & ^ 5> > > ; y CðmÞ y
1
01 mn
0mn 1
Dðm; nÞ
ð6:8Þ
! :
ð6:9Þ
We have an isomorphism of groups i
L =ðL Þp D
/pL S i /ppL S
1 þ pL O L ð1 þ pL OL Þ
pi
DðZ=pi ZÞ1þnL :
ð6:10Þ
Let r be the Frobenius map of L=T: Then by Proposition 3.3(i), there is a ðZ=pi ZÞi
basis S for L =ðL Þp DðZ=pi ZÞ1þnL such that the matrix of r with respect to S is Eðd; nT Þ: Define Hðm; n; pi Þ to be the set of all Eðm; nÞ-invariant subgroups of ðZ=pi ZÞ1þmn of index pi : By class field theory, there is a bijection between the set of all r-invariant i
subgroups of L =ðL Þp of index pi and KðT; L; pi Þ: This bijection induces a bijection between Hðd; nT ; pi Þ and KðT; L; pi Þ; which is denoted by H/KH : Furthermore, GalðKH =LÞDðZ=pi ZÞ1þnL =H for each HAHðd; nT ; pi Þ: For each uAðZ=pi ZÞ1þnL =H let xðuÞ be the element of GalðKH =LÞ which corresponds to u under this isomorphism. Let M=T be the maximal unramified subextension of KH =T: Since r is the Frobenius of L=T; it can be extended to the Frobenius r0 of M=T: Let h be an arbitrary extension of r0 to an element of GalðKH =TÞ: Then GalðKH =TÞ is generated by GalðKH =LÞDðZ=pi ZÞ1þnL =H and h; where hxðuÞh1 ¼ xður Þ ¼ xðEðd; nT ÞuÞ for all uAðZ=pi ZÞ1þnL =H; and hd ¼ x
$"a#% a
ð6:11Þ i
for some aAZ=pi ZD/pL S=/ppL S and
i
aAðZ=pi ZÞnL Dð1 þ pL OL Þ=ð1 þ pL OL Þp : It follows from the definition of h that the restriction of hd to M=L is the Frobenius. By class field theory this implies a 1 ðmod pi Þ: To summarize, we have the following description of the structure of GalðKH =TÞ: Proposition 6.2. Let Qp CTCL be finite extensions such that L=T is unramified of degree d and zp eL: Let H/KH be the bijection between Hðd; nT ; pi Þ and KðT; L; pi Þ induced by class field theory. Then for each HAHðd; nT ; pi Þ we have an isomorphism x : ðZ=pi ZÞ1þnL =H-GalðKH =LÞ such that GalðKH =TÞ is generated by GalðKH =LÞ
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and an element h satisfying
! 1 ; h ¼x a d
hxðuÞh1 ¼ xðEðd; nT ÞuÞ where
"1# a
for all uAðZ=pi ZÞ1þnL =H;
ð6:12Þ
AðZ=pi ZÞnL þ1 =H depends only on T and L:
Next, we list the elements H of Hðd; nT ; pi Þ for i ¼ 1; 2: For each such H we will give a more explicit description of the structure of GalðKH =TÞ than that given in Proposition 6.2. These explicit descriptions will be essential in our later calculations. Corollary 6.3. Let Qp CTCLCO be as in Proposition 6.2. Then the elements of Hðd; nT ; pÞ are the groups of the form Hðl; aÞ ¼ a> ¼ fxAðZ=pZÞ1þnL : xt a ¼ 0g;
ð6:13Þ
where 0aaAðZ=pZÞ1þnL satisfies Eðd; nT Þt a ¼ la for some lAZ=pZ such that ld ¼ 1: Furthermore, GalðKHðl;aÞ =TÞ is generated by GalðKHðl;aÞ =LÞ ¼ /jSDZ=pZ and an element h such that hd ¼ jcðaÞ and hjh1 ¼ jl ; where cðaÞ ¼ some fixed aAðZ=pZÞnL :
ð6:14Þ "1#t a
aAZ=pZ for
We omit the proof of Corollary 6.3 since it is a simpler version of the proof of the next corollary. Corollary 6.4. Let Qp CTCLCO be as in Proposition 6.2, and let H be a subgroup of ðZ=p2 ZÞ1þnL of index p2 : ðiÞ If ðZ=p2 ZÞ1þnL =HDZ=p2 Z; then HAHðd; nT ; p2 Þ if and only if H is of the form Hðl; aÞ ¼ a> ¼ fxAðZ=p2 ZÞ1þnL : xt a ¼ 0g; where lAZ=p2 Z satisfies ld ¼ 1 and aAðZ=p2 ZÞ1þnL \ðpZ=p2 ZÞ1þnL Eðd; nT Þt a ¼ la: Furthermore, GalðKHðl;aÞ =TÞ is generated by GalðKHðl;aÞ =LÞ ¼ /jSDZ=p2 Z
ð6:15Þ satisfies
ð6:16Þ
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" #t and an element h such that hd ¼ jcðaÞ and hjh1 ¼ jl ; where cðaÞ ¼ 1a aAZ=p2 Z for some fixed aAðZ=p2 ZÞnL : ðiiÞ If ðZ=p2 ZÞ1þnL =HDðZ=pZÞ2 ; then H*ðpZ=p2 ZÞ1þnL : In this case HAH ðd; nT ; p2 Þ if and only if ðH=ðpZ=p2 ZÞ1þnL Þ> is the column space of some ð1 þ nL Þ 2 matrix A over Z=pZ such that rankðAÞ ¼ 2 and Eðd; nT Þt A ¼ AL for some LAGLð2; pÞ satisfying Ld ¼ I2 : Equivalently, HAHðd; nT ; p2 Þ if and only of H is of the form HðL; AÞ ¼ fxAðZ=p2 ZÞ1þnL : xt A 0 ðmod pÞg;
ð6:17Þ
where A and L are as above. Furthermore, GalðKHðL;AÞ =TÞ is generated by GalðKHðL;AÞ =LÞ ¼ /j1 ; j2 SDðZ=pZÞ2
ð6:18Þ
c ðAÞ c ðAÞ
and an element h such that hd ¼ j11 j22 ; hj1 h1 ¼ jl111 jl212 ; and hj2 h1 ¼ " #t jl121 jl222 ; where ðc1 ðAÞ; c2 ðAÞÞ ¼ 1a AAðZ=pZÞ2 for some fixed aAðZ=pZÞnL and ½lij ¼ L:
Proof. In both cases the necessary and sufficient conditions for H to be an element of Hðd; nT ; p2 Þ are straightforward from the definitions. It remains to show that for HAHðd; nT ; p2 Þ; the structure of GalðKH =TÞ is as described. We will only give the argument for case (i), as case (ii) is quite similar. By Proposition 6.2, GalðKH =TÞ is generated by GalðKH =LÞDðZ=p2 ZÞ1þnL =H and an element h satisfying relations (6.12). We have H ¼ Hðl; aÞ ¼ a> for some lAZ=p2 Z and aAðZ=p2 ZÞnL \ðpZ=p2 ZÞnL such that Eðd; nT Þt a ¼ la and ld ¼ 1: There is a canonical isomorphism c : ðZ=p2 ZÞnL þ1 =a>
-
HomZ=p2 Z ð/aS; Z=p2 ZÞ;
x þ a>
/
/; xS;
ð6:19Þ
where /; S is the standard inner product on ðZ=p2 ZÞnL : It follows that the conjugation action of h on GalðKH =LÞ induces an action of h on HomZ=p2 Z ð/aS; Z=p2 ZÞ: Let f be the unique element of Homð/aS; Z=p2 ZÞ such $" #% " #t that fðaÞ ¼ 1: Then c 1a ¼ 1a a f and h f ¼ lf for all uAZ=p2 Z: Therefore, by identifying GalðKH =LÞ with HomZ=p2 Z ð/aS; Z=p2 ZÞ using c and identifying HomZ=p2 Z ð/aS; Z=p2 ZÞ with Z=p2 Z using the basis ffg; we see that GalðKH =TÞ is
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generated by GalðKH =LÞ ¼ /jSDZ=p2 Z and an element h satisfying the relations " #t hd ¼ jcðaÞ and hjh1 ¼ jl with cðaÞ ¼ 1a a; as claimed. & 7. Determination of jC2d j The goal of this section is to determine jC2d j: We retain the notation of Section 6. In addition, we set n ¼ nF ¼ f1 e1 : Observe that jC2d j ¼ 0 if d[p2 f : Also note that C2d ¼ C2d 0 where d 0 ¼ lcmðp2 ; dÞ: Thus we assume that d j p2 f and p2 j d: Let X be the set of all ðT; L; KÞ in the diagram
such that K=T is Galois. Using Proposition 6.1(i)–(iv), we see that ðT; L; KÞ/K gives a bijection between X and C2d : Hence jC2d j ¼ jXj: Meanwhile, jXj can be computed by counting the elements ðT; L; KÞAX in the order T; L; K: 2
Fix TAE F ; pdf ; e0 ; let L=T be unramified of degree d=p2 ; and let M=L be
unramified of degree p: Then we have jfK : ðT; L; KÞAXgj ¼ jKðT; L; p2 Þj jKðT; M; pÞj:
ð7:1Þ
Using Corollary 6.3, we find that &
& & & d p2 e0 fn & ; p && jKðT; M; pÞj ¼ &H ; p d 1
2
1 2 pd p e0 fn 1 ¼ ðp 1; dÞ þ pd p e0 fn : p1
ð7:2Þ
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Lemma 7.1. We have 8 9 1 2 2 > > > d p e0 fn 1Þ > 1 ðp > 2 > > > ðp 1; dÞ > > > 2 > > 2 > ðp 1Þ > > > > > > > = 1 1 2 2 > p p e fn e fn > 0 0 > d d p ðp 1Þ > if p3 [d; > þ ðp 1; dÞ > > > > > p1 > > > > > > > 1 2 > > > 2p e fn 0 > > 1 d 2 > 1 p 1 > 2p e fn > 2 0 > d ; ðp þ p þ 1; dÞ > > 2 > 2 p 1 > < 9 1 2 2 jKðT; L; p2 Þj ¼ > d p e0 fn 1Þ 1 ðp > 2 > ðp 1; dÞ > > > > > 2 > > 2 ðp 1Þ > > > > > > > > 1 1 > > 2 2 > > p p e fn e fn 0 0 > > d d p ðp 1Þ > = > > þ 2ðp 1; dÞ > if p3 j d: > p 1 > > > > > > 1 2 1 2 > > > > > þ pd p e0 fn ð2pd p e0 fn 1Þ > > > > > > > > 1 > 2 > > 2p e fn > 0 > > d 1 p 1 > > 2 > ; : þ ðp 1; dÞ 2 2 p 1
ð7:3Þ
Proof. First, we have &
& & d p2 e0 fn 2 && & jKðT; L; p Þj ¼ &H 2 ; ; p & ¼ jH1 j þ jH2 j; p d 2
ð7:4Þ
where H1 and H2 are the subsets of H pd2 ; d1 p2 e0 fn; p2 corresponding to the two cases of Corollary 6.4. Put E ¼ E pd2 ; d1 p2 e0 fn : Then H1 ¼ fHðl; aÞ : aAðZ=p2 ZÞ1þe0 fn \ðpZ=p2 ZÞ1þe0 fn ; lAZ=p2 Z; 2
E t a ¼ la; ld=p ¼ 1g;
ð7:5Þ
H2 ¼ fHðL; AÞ : AAMð1þe0 fnÞ 2 ðZ=pZÞ; rankðAÞ ¼ 2; LAGLð2; pÞ; 2
E t A ¼ AL; Ld=p ¼ I2 g:
ð7:6Þ
2
Let lAZ=p2 Z satisfy ld=p ¼ 1: Then jfHðl; aÞAH1 gj ¼
jAl j p2 p
ð7:7Þ
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41
where Al ¼ faAðZ=p2 ZÞ1þe0 fn \ðpZ=p2 ZÞ1þe0 fn : E t a ¼ lag:
ð7:8Þ
X 1 jAl j: p2 p lAZ=pZ
ð7:9Þ
Hence jH1 j ¼
X
jfHðl; aÞAH1 gj ¼
lAZ=pZ 2
2
ld=p ¼1
ld=p ¼1
The cardinality of Al can be easily determined: 8 1 2 1 2 > > p2ð1þd p e0 fnÞ p1þd p e0 fn if l ¼ 1; > < 2 1 jAl j ¼ p1þd p2 e0 fn p1þd p2 e0 fn if l 1 ðmod pÞ but la1; > > > 1 2 : 2 p2 e0 fn pd p e0 fn if lc1 ðmod pÞ: pd Combining (7.9) and (7.10), we find that 1 2
1 2 d pd p e0 fn1 ðpd p e0 fn 1Þ jH1 j ¼ ðp 1; dÞ p; 2 p p1
1 2 d 1 p2 e0 fn1 1 p2 e0 fn þ p; 2 pd ðpd 1Þ þ pd 2p e0 fn : p
ð7:10Þ
ð7:11Þ
To compute jH2 j; we first observe that for HðL; AÞAH2 and QAGLð2; pÞ; we have HðL; AÞ ¼ HðQ1 LQ; AQÞ:
ð7:12Þ
2
Also note that for LAGLð2; pÞ with Ld=p ¼ I; we have HðL; A1 Þ ¼ HðL; A2 Þ if and only if A1 ¼ A2 Q for some Q in the centralizer centðLÞ of L in GLð2; pÞ: Thus for each LAGLð2; pÞ; jAL j ; ð7:13Þ jfHðL; AÞAH2 gj ¼ jcentðLÞj where AL ¼ fAAMð1þe0 fnÞ 2 ðZ=pZÞ : rankðAÞ ¼ 2; E t A ¼ ALg:
ð7:14Þ
By (7.12) and (7.13), we have jH2 j ¼
X L
jAL j ; jcentðLÞj
ð7:15Þ 2
where L runs over the set of canonical forms in M2 2 ðZ=pZÞ with Ld=p ¼ I: We can compute jcentðLÞj using the formula in [6], and jAL j using the well-known formula for the dimension of the solution set of E t A ¼ AL [5, Theorem 4.4.14]. We omit the details of these computations and record the result for jH2 j below.
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9 8 1 2 2 p e0 fn > > d 1 ðp 1Þ > > > > > > ðp 1; dÞ2 > > 2 > > 2 > ðp 1Þ > > > > > > > > 1 1 > 2 2 = > 1þd p e0 fn d p e0 fn > þ ðp 1; dÞp ðp 1Þ > > > > 1 2 1 2 > > > > > > ðpd p e0 fn 1Þp1þd p e0 fn > > > > > > > > 1 2 > > e fn 2p 0 > > d > > 1 p 1 > > 2 ; < þ ðp 1; dÞ 2 2 p 1 jH2 j ¼ 9 > 1 2 > > > > 1 ðpd p e0 fn 1Þ2 > 2 > > > > > ðp 1; dÞ > 2 > > 2 > > ðp 1Þ > > > > > > > = 1 1 2 2 > p p e fn e fn 0 0 > d d > p ðp 1Þ > > þ ðp 1; dÞ > > > > p1 > > > > > > > > 1 2 > > 2p e fn > 0 > d > > 1 p 1 > > 2 : þ ðp 1; dÞ ; 2 2 p 1 Finally, (7.3) follows from (7.4), (7.11), and (7.16). & We now state the main result of this section. Proposition 7.2. Let d 0 ¼ lcmðp2 ; dÞ: Then we have 8 0 > > 9 > 2 > > 1 2 > > 2 0 p e0 fn > > d 1 1Þ > > 2 ðp > > 4 > > ðp 1; dÞ e 0 > > 2 > > 2 > > ðp 1Þ > > > > > > > > > > 1 2 > > 2 p e fn > 0 > d0 > > ðp 1Þ > = > þðp 1; dÞ > > p 1 > > > > > > 1 2 1 2 > > > > 0 p e0 fn 0 p e0 fn > d d > þ p ðp 1Þ > > > > # > > > > > > > > > 1 > 2 1 1 > 2p e fn 2 0 > > þ ðp 1; dÞ 2 ðpd 0 1Þ > < > > ; 2 p 1 jC2d j ¼ > 2 > 9 > 1 2 > 2 > > d 0 p e0 fn 1Þ > 1 ðp > 2 > > > > e0 4 ðp 1; dÞ > > 2 > > 2 > > ðp 1Þ > > > > > > > > > > 1 1 > 2 2 > p e fn p e fn > > 0 0 0 0 > d d > ðp 1Þð2p 1Þ > = > þ ðp 1; dÞ > > > p1 > > > > > 1 2 1 2 > > > > 0 p e0 fn 0 p e0 fn > d d > þ 2p ðp 1Þ > > > > > 3 > > > 1 > 2 > 2p e fn > 0 > 0 > d > 1 p 1 > > 2 > 5 > ðp 1; dÞ þ > > > 2 ; : 2 p 1
if p3 [d;
ð7:16Þ
if p3 j d:
if d[p2 f ;
if d j p2 f and p3 [d;
if d j p2 f and p3 j d:
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Proof. By the comments at the beginning of the section it suffices to prove the proposition under the assumptions d j p2 f and p2 j d: Using (7.1) we get X
jC2d j ¼ jXj ¼
$
1 TAE F ;d p2 f ;e0
now follows &The $ proposition %& & E F ; 1 p2 f ; e 0 & ¼ e 0 : & d
%
ðjKðT; L; p2 Þj jKðT; M; pÞjÞ:
from
(7.2),
(7.3),
and
the
ð7:17Þ
fact
that
8. Determination of jC1d \C2d j In this section we compute the cardinality of C1d \C2d : Recall that C1d \C2d consists of the fields KAEðF ; f ; eÞ such that d j jAutðK=F Þj and the ramification index of K over the fixed field NK of AutðK=F Þ is p: Thus jC1d \C2d j ¼ 0 if d[pf : Furthermore, setting d 0 ¼ lcmðp; dÞ; we have C1d \C2d ¼ C1d 0 \C2d 0 : Therefore we may assume that d j pf and p j d: Let X be the set of all ðM; E; KÞ in the diagram
such that K=M is Galois. The cardinality of X can be calculated by counting the elements& ðM; & order of M; E; K; when M and E are fixed, the number E; KÞAX inthe & & d 1 2 of K is &H p; d p e0 fn; p & 1; which is computed in (7.2). It turns out that 2
jXj ¼
1 pe0 ðp1þd pe0 fn
3 1 2 1 2 d p e0 fn 1 p þ pd p e0 fn 15: p þ 1Þ4ðp 1; dÞ p1
ð8:1Þ
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On the other hand, X ¼ X1 ,X ’ 2 ; where X1 ¼ fðM; E; KÞAX : KAC1d \C2d g;
ð8:2Þ
X2 ¼ fðM; E; KÞAX : KAC1d -C2d g:
ð8:3Þ
Lemma 8.1. For each KAC1d \C2d ; there are unique subfields MCECK such that ðM; E; KÞAX:
Proof. The existence of such an ðM; EÞ follows from the definition of C1d : To see the uniqueness of ðM; EÞ; we assume that there are ðM1 ; E1 Þ and ðM2 ; E2 Þ such that ðM1 ; E1 ; KÞAX and ðM2 ; E2 ; KÞAX: Then K=E1 -E2 is totally ramified, and also Galois since both K=E1 and K=E2 are Galois. Therefore by Proposition 6.1, ½K : E1 -E2 j p2 : If E1 aE2 ; we must have ½K : E1 -E2 ¼ p2 : However, this would imply that KAC2d ; contrary to assumption. Thus E1 ¼ E2 : In the diagram
the extension K=M1 -M2 is Galois since both K=M1 and K=M2 are Galois. If eðK=M1 -M2 Þ4p then eðK=M1 -M2 Þ ¼ p2 ; which implies KAC2d ; contrary to assumption. Thus E1 =M1 -M2 is unramified, and hence M1 ¼ M2 : & Lemma 8.1 implies that ðM; E; KÞ/K gives a bijection between X1 and C1d \C2d : Hence jC1d \C2d j ¼ jX1 j ¼ jXj jX2 j:
ð8:4Þ
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In order to calculate jX2 j; we let Y be the set of all ðT; L; M; E; KÞ in the diagram
such that T ¼ L-M and K=T is Galois. Put Z1 ¼ fðT; L; M; E; KÞAY : E=L is unramifiedg;
Z2 ¼ fðT; L; M; E; KÞAY : E=L is ramified but K=E is unramifiedg:
ð8:5Þ
ð8:6Þ
Lemma 8.2. The map Z:
Y\ðZ1 ,Z2 Þ
-
ðT; L; M; E; KÞ /
X2 ðM; E; KÞ
ð8:7Þ
is a bijection. Proof. For each ðT; L; M; E; KÞAY\ðZ1 ,Z2 Þ; the extension K=L is totally ramified of degree p2 : By Proposition 6.1(i), L is uniquely determined by K: Consequently, T ¼ L-M is determined by K and M; so Z is one-to-one.
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On the other hand, for each ðM; E; KÞAX2 ; we have a diagram
with K=M Galois. Let L=F be the unique subextension of K=F such that K=L is totally ramified of degree p2 : Since KAC2d ; the extension K=L is Galois. Clearly, LCE; and in the diagram
the extension K=L-M is Galois. Hence ðL-M; L; M; E; KÞAY\ðZ1 ,Z2 Þ: This proves that Z is onto. & It follows from Lemma 8.2 that jX2 j ¼ jYj jZ1 j jZ2 j:
ð8:8Þ
The cardinality of Z1 can be calculated by counting the elements &ðT;$L; M; E; KÞ %in & the order T; E; K; L; M: Fix T and E: Then the number of K is &H d; d1 pe0 fn; p &; and the number of ðL; MÞ is 0 if ðp; d=pÞa1; and 1 if ðp; d=pÞ ¼ 1: It follows that 3 8 2 1 > 1 d pe0 fn 1 > p < e 4ðp 1; dÞ þ pd pe0 fn 5 if p2 [d; 0 p1 jZ1 j ¼ > > : 0 if p2 j d:
ð8:9Þ
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Lemma 8.3. We have 8 2 9 1 2 > > 1 > d pe0 fn 1Þ pðp > > 2 pe fn 0 > > > e0 4ðp 1; dÞ þ pd > > > 2 > > = ðp 1Þ > > > > 3 if p2 [d; > > 1 1 > > pe pe fn 1þ fn > 0 0 > ðpd 1Þðp d þ 1Þ5 > > > > > > þ ðp 1; dÞ > > ; > p 1 > > > > 2 > 9 > 1 > pe0 fn > > > 1Þ2 > < e 4ðp 1; dÞ2 pðpd > > 0 > 2 > jYj ¼ > ðp 1Þ > > > > > > > > > 1 1 > > > 1þd pe0 fn d pe0 fn > > > p ðp 1Þ > = > þ 2ðp 1; dÞ > > > if p2 j d: p 1 > > > > > > 1 1 > > > > > þ p1þd pe0 fn ðpd pe0 fn 1Þ > > > > > 3 > > > 1 1 > > > pe0 fn pe0 fn > > d d > ðp 1Þðp pÞ > > > 5 > > > > 2 ; : p 1
ð8:10Þ
$ % Proof. First fix TAE F ; d1 pf ; e0 and let L=T be unramified of degree d=p: Put H ¼ H dp; d1 pe0 fn; p2 ; and let H/KH be the bijection between H and KðT; L; p2 Þ induced by class field theory. For each HAH; the number of ðM; EÞ such that ðT; L; M; E; KH ÞAY is equal to jJðHÞj; where JðHÞ ¼ fJpGalðKH =TÞ : jJj ¼ d; GalðKH =TÞ ¼ GalðKH =LÞ Jg:
ð8:11Þ
Alternatively, jJðHÞj is the number of subextensions M=T of KH =T such that ½M : T ¼ p and L-M ¼ T: For any such M; the compositum E ¼ LM is the only field such that ðT; L; M; E; KH ÞAY: Therefore jfðM; E; KÞ : ðT; L; M; E; KÞAYgj ¼
X
jJðHÞj:
ð8:12Þ
HAH
For each HAH; the structure of GalðKH =TÞ and the position of GalðKH =LÞ in GalðKH =TÞ are explicitly described in Corollary 6.4. In fact, in the notation of Corollary 6.4(i), JðHðl; aÞÞ ¼ f/jp ; jc hS : cAZ=p2 Z; ðjc hÞd=p A/jp Sg;
ð8:13Þ
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and /jp ; jc hS ¼ /jp ; jc hS if and only if c0 cApZ=p2 Z: Therefore 1 jJðHðl; aÞÞj ¼ jfcAZ=p2 Z : ðjc hÞd=p A/jp Sgj p &( )& ! & d 1 t 1&& 1 & 2 2 p aApZ=p Z & ¼ & cAZ=p Z : ð1 þ l þ ? þ l Þc þ & p& a 8 p if la1; > > > > > 1 if l ¼ 1; p2 [d; > > > ! < 1 t aApZ=p2 Z; ¼ p if l ¼ 1; p2 j d; and > a > > > ! > > 1 t > 2 > : 0 if l ¼ 1; p j d; and aepZ=p2 Z: a
ð8:14Þ
Suppose instead we are in the situation of Corollary 6.4(ii). Then H ¼ HðL; AÞ for some L and A satisfying the conditions of the corollary. For each JAJðHðL; AÞÞ the group B ¼ J-/j1 ; j2 S is a normal subgroup of J of order p; and jc11 jc22 hAJ for 2 some ðc1 ; c2 ÞAðZ=pZÞ h i: It follows that B is invariant under conjugation by h; so
B ¼ /jb11 jb22 S with
b1 b2
an eigenvector of Lt : Therefore J ¼ /jb11 jb22 ; jc11 jc22 hS; with c0
c0
ðjc11 jc22 hÞd=p AB: Moreover, /jb11 jb22 ; jc11 jc22 hS ¼ /jb11 jb22 ; j11 j22 hS if and only if c0 c1 c02 c2 j2 AB:
j11
Hence
1 jJðHðL; AÞÞj ¼ jfðW ; ðc1 ; c2 ÞÞ : W is a 1-dimensional eigenspace of Lt p and ðjc11 jc22 hÞd=p A/jb11 jb22 Sgj &( 1 && ¼ & ðW ; ðc1 ; c2 ÞÞ : W is a 1-dimensional eigenspace of Lt p& )& ! & d 1 t 1 & p and ðc1 ; c2 ÞðI2 þ L þ ? þ L Þ þ AAW &: & a
ð8:15Þ
Thus jJðHðL; P AÞÞj can be determined from the canonical P form of L: This allows us to compute HAH jJðHÞj; we omit the details. Since HAH jJðHÞj is independent of the choice of ðT; LÞ; by (8.12) we have &
& X & & pf & jYj ¼ &E F ; ; e0 && jJðHÞj; d HAH
and the formula (8.10) for jYj follows.
&
ð8:16Þ
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Lemma 8.4. We have 1
pðpd pe0 fn 1Þ : ð8:17Þ jZ2 j ¼ e0 ðp 1; dÞ p1 $ % Proof. Fix TAE F ; d1 pf ; e0 ; let L=T be unramified of degree d=p; and let L0 =L be $ % unramified of degree p: Let H/KH be the bijection between H ¼ H d; d1 pe0 fn; p and KðT; L0 ; pÞ induced by class field theory. For each HAH; put JðHÞ ¼ fðB; JÞ : BpJpGalðKH =TÞ; jBj ¼ p; jJj ¼ d; and GalðKH =TÞ ¼ GalðKH =L0 Þ Jg:
ð8:18Þ
When KH =L0 is ramified, the number of ðM; EÞ such that ðT; L; M; E; KH ÞAZ2 is equal to jJðHÞj: In the following diagram we have T ¼ L0 -M; and hence GalðKH =TÞ ¼ GalðKH =L0 Þ GalðKH =MÞ:
Therefore jfðM; E; KÞ : ðT; L; M; E; KÞAZ2 gj ¼
X
jJðHÞj:
ð8:19Þ
HAH KH =L0 ramified
Corollary 6.3 allows us to compute jJðHÞj for each HAH; and hence to compute (8.19). Omitting the details, we get X HAH KH =L0 ramified
1
pðpd e0 fn 1Þ : jJðHÞj ¼ ðp 1; dÞ p1
ð8:20Þ
Using (8.19) we &see$ that jZ2%j &is equal to (8.20) multiplied by the number of pairs ðT; LÞ; which is &E F ; d1 pf ; e0 & ¼ e0 : & By (8.4) and (8.8) we have jC1d \C2d j ¼ jXj jYj þ jZ1 j þ jZ2 j;
ð8:21Þ
where jXj; jYj; jZ1 j; and jZ2 j are given in (8.1), (8.10), (8.9), and (8.17). Hence we have a formula for jC1d \C2d j:
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Proposition 8.5. Let d 0 ¼ lcmðp; dÞ: Then we have 8 0 > > " > 9 > > 1 > p > 2 2 > pe fn > 0 > > ðpd 0 1Þ > e0 ðp 1; dÞ > 2 > > > > ðp 1Þ > > > > > > > > 1 2 > > p > > 0 p e0 fn > = d ðp 1 þ ðp 1; dÞ > > > p 1 > > > > 1 1 2 1 > > > > > þ ðpd 0 pe0 fn 1Þðp1þd 0 p e0 fn pd 0 pe0 fn p þ 1ÞÞ > > > > > # > > > > > > 1 1 > 2 > > 1þ pe fn p e fn > 0 0 0 0 > > d d þ pðp p þ 1Þðp 1Þ > ; > > > > > " 9 > > < > > 1 p > 2 2 > jC1d \C2d j ¼ d 0 pe0 fn 1Þ e ðp 1; dÞ ðp > 0 > 2 > > > ðp 1Þ > > > > > > > > > > > 1 1 > p > pe fn pe fn > 0 0 0 0 > > d d ððp þ ðp 1; dÞ 1Þð2p 1Þ > > > > > p 1 > > > > > > > 1 1 2 = > 1þ pe fn p e fn > 0 0 0 0 d d > p þ 1Þðp 1ÞÞ þ ðp > > > > > 1 1 2 > > > > > > þ pðp1þd 0 pe0 fn p þ 1Þðpd 0 p e0 fn 1Þ > > > > > > > 1 1 > > > 1þ pe fn pe fn 0 0 0 0 > > d d ðp 1Þ p > > > > > > 3 > > 1 1 > > > pe fn pe fn > 0 0 0 0 > > d d ðp 1Þðp pÞ > > > > 5 > þ > > > 2 ; : p 1
if d[pf ;
if d j pf ; p2 [d;
if d j pf ; p2 j d:
9. Determination of jC0d \ðC1d ,C2d Þj Recall that C0d \ðC1d ,C2d Þ consists of those fields KAEðF ; f ; eÞ such that d j jAutðK=F Þj and the ramification index of K over the fixed field NK of AutðK=F Þ is 1. Thus jC0d \ðC1d ,C2d Þj ¼ 0 if d[f ; so we will assume that d j f : Let X be the set of all ðM; KÞ in the diagram
ARTICLE IN PRESS X.-D. Hou, K. Keating / Journal of Number Theory 104 (2004) 14–61
Then
51
&
& & & f 2 & jXj ¼ &E F ; ; p e0 && d 1
1
1
¼ e0 p2 ðp2þd ðpþ1Þe0 fn p2þd pe0 fn þ p1þd pe0 fn p þ 1Þ:
ð9:1Þ
On the other hand, X ¼ X1 ,X ’ 2 ,X ’ 3 ; where X1 ¼ fðM; KÞAX : KAC0d \ðC1d ,C2d Þg;
ð9:2Þ
X2 ¼ fðM; KÞAX : KAC0d -ðC1d \C2d Þg;
ð9:3Þ
X3 ¼ fðM; KÞAX : KAC0d -C2d g:
ð9:4Þ
For each KAC0d \ðC1d ,C2d Þ; we claim that there is a unique M such that ðM; KÞAX: The existence of M follows from the definition of Cid : To see the uniqueness of M; assume to the contrary that we have two different subextensions M1 =F and M2 =F of K=F such that K=M1 and K=M2 are both unramified of degree d: Then K=M1 -M2 is Galois and not unramified. By Proposition 6.1(iv), we must have eðK=M1 -M2 Þ ¼ p or p2 : This means that KAC1d ,C2d ; which is a contradiction. It follows that jC0d \ðC1d ,C2d Þj ¼ jX1 j ¼ jXj jX2 j jX3 j:
ð9:5Þ
We now determine jX3 j: Let Y be the set of all ðT; L; M; KÞ in the diagram
such that T ¼ L-M and K=T is Galois. Write Y ¼ Y1 ,Y ’ 2 ; where Y1 ¼ fðT; L; M; KÞAY : eðK=LÞ ¼ p2 g;
ð9:6Þ
Y2 ¼ fðT; L; M; KÞAY : eðK=LÞ ¼ 1 or pg:
ð9:7Þ
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Then ðT; L; M; KÞ/ðM; KÞ is a bijection between Y1 and X3 (cf. Proposition 6.1). Therefore ð9:8Þ jX3 j ¼ jY1 j ¼ jYj jY2 j: Lemma 9.1. We have 9 8 2 1 2 2 d e0 fn > > > > 1 p ðp 1Þ 2 > > 4 ðp 1; dÞ > > e > 0 > 2 > > 2 > > ðp 1Þ > > > > > > > > > > 1 = > e0 fn > > d pðp 1Þ > if p[d; > þ ðp 1; dÞ > > > > > p1 > > > > 3> > > > 1 > > 2 d 2e0 fn > > 1 p ðp 1Þ > > > þ ðp2 1; dÞ > 5 > > þ 1 > > 21 > ; 2 p < 2 jYj ¼ 9 1 > 2 2 d e0 fn > > > > 1 p ðp 1Þ 2 > > > > e0 4 ðp 1; dÞ > > 2 > > 2 > > ðp 1Þ > > > > > > > > > > 1 1 = > > e e 2þ fn fn 0 0 > d d p ðp 1Þ > if p j d: > þ ðp 1; dÞ > > > > > p1 > > > 3 > > > > > 1 > > 2 d 2e0 fn > 1 2 p ðp 1Þ5 > > > > > > > þ ðp 1; dÞ > > : ; 2 p2 1
ð9:9Þ
Proof. The proof is similar to that of Lemma 8.3. Once again,we only describe the method and omit the computational details. Fix TAE F ; df ; e0 and let L=T be the unramified extension of degree d: Let H/KH be the bijection between H ¼ $ % H d; d1 e0 fn; p2 and KðT; L; p2 Þ induced by class field theory. For each HAH we have jfM : ðT; L; M; KH ÞAYgj ¼ jJðHÞj;
ð9:10Þ
where JðHÞ ¼ fJpGalðKH =TÞ : jJj ¼ d; GalðKH =TÞ ¼ GalðKH =LÞ Jg:
ð9:11Þ
Using (9.10), we see that jfðM; KÞ : ðT; L; M; KÞAYgj ¼
X
jJðHÞj:
ð9:12Þ
HAH
P The sum HAH jJðHÞj can be computed using Corollary 6.4, and the result is independent of ðT; LÞ (cf. the proof of Lemma 8.3). Therefore we get & of the & choice & & P f jYj ¼ &E F ; d ; e0 & HAH jJðHÞj: &
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Lemma 9.2. We have 3 8 2 1 e0 fn > d > pðp 1Þ < e 4ðp 1; dÞ þ 15 0 p1 jY2 j ¼ > > : 0
if p[d;
ð9:13Þ
if p j d:
Proof. By (9.7) we have jY2 j ¼ 0 if p j d: Thus we may assume p[d: Fix f TAE F ; d ; e0 ; let L=T be unramified of degree d; and let L0 =L be unramified of $ % degree p: Let H/KH be the bijection between H ¼ H dp; d1 e0 fn; p and KðT; L0 ; pÞ induced by class field theory. Then for each HAH we have jfM : ðT; L; M; KH ÞAY2 gj ¼ jJðHÞj;
ð9:14Þ
JðHÞ ¼ fJpGalðKH =TÞ : jJj ¼ dg:
ð9:15Þ
where
It follows that jfðM; KÞ : ðT; L; M; KÞAY2 gj ¼
X
jJðHÞj;
ð9:16Þ
HAH
which can be computed 6.3. The lemma then follows from (9.16) and & using Corollary & P & & f the formula jY2 j ¼ &E F ; d ; e0 & HAH jJðHÞj: & Since jX3 j is determined by jYj and jY2 j; the only part of (9.5) that remains to be computed is jX2 j: Let Z be the set of all ðT; L; M; KÞ in the diagram
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such that T ¼ L-M and K=T is Galois. In addition, define Z1 ¼ fðT; L; M; KÞAZ : eðK=LÞ ¼ pg;
ð9:17Þ
Z2 ¼ fðT; L; M; KÞAZ1 : K=LK is Galoisg;
ð9:18Þ
where LK is the unique field between F and K such that K=LK is totally ramified of degree p2 : We claim that c:
-
Z1 \Z2
ðT; L; M; KÞ /
X2 ðM; KÞ
ð9:19Þ
is a bijection. By the definitions of X2 ; Z1 ; and Z2 ; it is clear that c is onto. Suppose that c is not one-to-one. Then there are elements ðT1 ; L1 ; M; KÞ and ðT2 ; L2 ; M; KÞ of Z1 \Z2 such that ðT1 ; L1 ÞaðT2 ; L2 Þ: Since T1 ¼ L1 -M and T2 ¼ L2 -M; we must have L1 aL2 : Since K=L1 and K=L2 are Galois and totally ramified of degree p; we see that K=L1 -L2 is Galois with eðK=L1 -L2 Þ4p: Thus by Proposition 6.1(iv) we get eðK=L1 -L2 Þ ¼ p2 : This implies that ðT1 ; L1 ; M; KÞAZ2 ; which is a contradiction. It follows that jX2 j ¼ jZ1 j jZ2 j:
ð9:20Þ
Lemma 9.3. We have 1
1
p2 ðp1þd e0 fn p þ 1Þðpd pe0 fn 1Þ : jZ1 j ¼ e0 ðp 1; dÞ p1
ð9:21Þ
Proof. Observe that jZ1 j ¼ jZj jfðT; L; M; KÞAZ : K=L unramifiedgj;
ð9:22Þ
and that & 8 &&
< &E F ; f ; pe && if p[d; 0 & d jfðT; L; M; KÞAZ : K=L unram:gj ¼ & : 0 if p j d; ( 1 1þd e0 fn p þ 1Þ if p[d; ¼ pe0 ðp
ð9:23Þ if p j d: Meanwhile jZj can be computed as before: Fix TAE F ; df ; pe0 and let L=T be $ % unramified of degree d: Let H/KH be the bijection between H ¼ H d; d1 pe0 fn; p 0
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and KðT; L; pÞ induced by class field theory. Then for each HAH we have jfM : ðT; L; M; KH ÞAZgj ¼ jJðHÞj;
ð9:24Þ
where JðHÞ ¼ fJpGalðKH =TÞ : jJj ¼ d; GalðKH =TÞ ¼ GalðKH =LÞ Jg: Consequently,
X
jfðM; KÞ : ðT; L; M; KÞAZgj ¼
jJðHÞj;
ð9:25Þ
ð9:26Þ
HAH
&
& X & & f & jZj ¼ &E F ; ; pe0 && jJðHÞj: d
ð9:27Þ
HAH
Using (9.26) and Corollary 6.3 we get 2 3 8 1 pe0 fn > 1 d pðp 1Þ > 1þd e0 fn > > þ 15 if p[d; p þ 1Þ4ðp 1; dÞ > < e0 pðp p1 jZj ¼ > 1 1 > > > p2 ðp1þd e0 fn p þ 1Þðpd pe0 fn 1Þ > : e0 ðp 1; dÞ if p j d: p1 Eq. (9.21) now follows from (9.22), (9.23), and (9.28).
ð9:28Þ
&
Lemma 9.4. We have 9 8 2 1 2 2 > d e0 fn 1Þ > > > 1 p ðp þ 1Þðp 2 > > > > e0 4 ðp 1; dÞ > > 2 > > 2 > > ðp 1Þ > > > > > > > > > > 1 1 = > e e 2þ fn fn > 0 0 > d d p ðp 1Þ > if p[d; > ðp 1; dÞ > > > > > p1 > > > > 3 > > > > 1 > > 2 d 2e0 fn > > 1 p ðp 1Þ > > 2 > > 5 > > þ ðp 1; dÞ > > > ; 2 p 1 < 2 jZ2 j ¼ 9 1 > 2 2 > d e0 fn 1Þ > > > 1 p ðp þ 1Þðp 2 > > > > e0 4 ðp 1; dÞ > > 2 > > 2 > > ðp 1Þ > > > > > > > > > > 1 1 = > e e 2þ fn fn > 0 0 > d d p ðp 1Þ > if p j d: > þ ðp 1; dÞ > > > > > p1 > > > > 3 > > > > 1 > > 2 d 2e0 fn > > 1 2 p ðp 1Þ5 > > > > > > þ ðp 1; dÞ > > : ; 2 p1
ð9:29Þ
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Proof. For each ðT; L; M; KÞAZ2 ; we have a diagram
in which K=S is Galois, and S and L0 are determined by ðT; L; M; KÞ: Thus if we let W denote the set of all ðS; L0 ; T; L; M; KÞ in this diagram such that K=S is Galois, we have jZ2 j ¼ jWj: To compute jWj; we fix SAE F ; df ; e0
and let L0 =S be
unramified of degree d and N=L0 unramified of degree p: Write K ¼ KðS; L0 ; p2 Þ and K0 ¼ fKAK : NCKg ¼ KðS; N; pÞ: For each KAK; let GðKÞ ¼ fðB; JÞ : BpGalðK=L0 Þ; jBj ¼ p; JpGalðK=SÞ; jJj ¼ d; GalðK=SÞ ¼ GalðK=L0 Þ Jg:
ð9:30Þ
Note that if ðS; L0 ; T; L; M; KÞAW; then KAK\K0 : Also note that for each KAK\K0 we have jfðT; L; MÞ : ðS; L0 ; T; L; M; KÞAWgj ¼ jGðKÞj:
ð9:31Þ
More precisely, ðT; L; MÞ2ðGalðK=LÞ; GalðK=MÞÞ is a bijection between the two sets in (9.31). Therefore jfðT; L; M; KÞ : ðS; L0 ; T; L; M; KÞAWgj ¼
X KAK
jGðKÞj
X
jGðKÞj:
ð9:32Þ
KAK0
P To compute KAK jGðKÞj; let H/KH be the bijection between H ¼ $ 1 % 2 H d; d e0 fn; p and K induced by class field theory, and let H1 and H2 be the subsets of H corresponding to the two cases of Corollary 6.4. We have GalðKH =L0 ÞDZ=p2 Z if HAH1 and GalðKH =L0 ÞDðZ=pZÞ2 if HAH2 :
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Therefore jGðKH Þj ¼
jJðHÞj ðp þ 1ÞjJðHÞj
if HAH1 ; if HAH2 ;
ð9:33Þ
where JðHÞ ¼ fJpGalðKH =SÞ : jJj ¼ d; GalðKH =SÞ ¼ GalðKH =L0 Þ Jg:
ð9:34Þ
Thus X KAK
jGðKÞj ¼
X HAH
jGðKH Þj ¼
X HAH1
jJðHÞj þ ðp þ 1Þ
X
jJðHÞj
ð9:35Þ
HAH2
can be computed as before. P We now compute KAK0 jGðKÞj: We claim that if p j d; then jGðKÞj ¼ 0 for all KAK0 : Suppose to the contrary that there exists ðB; JÞAGðKÞ for some KAK0 : Let M denote the subfield of K fixed by J: Then we have the diagram
with p j f ðM=SÞ: Thus L0 -MaS; so GalðK=SÞaGalðK=L0 Þ J; contrary to the definition of GðKÞ: Therefore $ we may% assume 0p[d: Let H/KH be the and K induced by class field bijection between H0 ¼ H dp; d1 e0 fn; p 0 theory. By Corollary 6.3, every HAH is of the form Hðl; aÞ ¼ a> for some $ %t eigenvector a of E dp; d1 e0 fn with eigenvalue l: Furthermore, GalðKHðl;aÞ =SÞ is generated by GalðKHðl;aÞ =NÞ ¼ /jSDZ=pZ and an element h such that " #t hdp ¼ jcðaÞ and hjh1 ¼ jl ; where cðaÞ ¼ 1a aAZ=pZ for some fixed aAðZ=pZÞpe0 fn : Moreover, GalðKHðl;aÞ =L0 Þ is generated by j and hd (cf. the proofs of Proposition 6.2 and Corollary 6.4). These explicit descriptions allow P us to compute each jGðKHðl;aÞ Þj; and hence to compute 0 jGðKH Þj ¼ HAH P KAK0 jGðKÞj:
ARTICLE IN PRESS 58
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Now jfðT; L; M; KÞ : ðS; L0 ; T; L;& M; KÞAWgj & can be computed using (9.32). & & f After multiplying the result by &E F ; d ; e0 & ¼ e0 ; we get formula (9.29) for jWj ¼ jZ2 j: & From (9.5), (9.8), and (9.20), we have jC0d \ðC1d ,C2d Þj ¼ jXj jZ1 j þ jZ2 j jYj þ jY2 j;
ð9:36Þ
where jXj; jYj; jY2 j; jZ1 j; and jZ2 j are given in (9.1), (9.9), (9.13), (9.21), and (9.29). Thus we obtain the main result of this section.
Proposition 9.5. We have 8 0 if d[f ; > > 2 > 9 > 1 > e0 fn > > > > 1Þ2 > 2 pðpd > 2 41 > > ðp 1; dÞ p e > 0 > 2 > > > 2 > ðp 1Þ > > > > > > > > > > 1 > > > 2e0 fn > > d > 1 pðp 1Þ > > 2 > > > ðp þ 1; dÞ > > > 2 > > 2 p 1 > > > > > > > = 1 1 > ðp 1; dÞ e e fn fn > 0 0 > d d ðp ðp 1Þ > if d j f ; p[d; > p1 > > > > > > > 1 1 > > > > > þ ðp1þd e0 fn p þ 1Þðpd pe0 fn 1ÞÞ > > > > > > > > > 1 1 > > ðpþ1Þe pe 2þ fn 2þ fn > 0 > > d d 0 > p þ p > > > > # > > > > > > > > 1 > < > 1þd pe0 fn > þp pþ1 0 1 2 > ; jCd \ðCd ,Cd Þj ¼ > > > 2 9 > > 1 > 2 e0 fn > > d > 1 pðp 1Þ > > > > > e0 p2 4 ðp 1; dÞ2 > > > 2 > > 2 > ðp 1Þ > > > > > > > > > > 1 > > > 2e0 fn > > d > 1 pðp 1Þ > > 2 > > > ðp þ 1; dÞ > > 2 > > 2 p 1 > > > = > > 1 1 > pe e fn 1þ fn > 0 0 : > ðpd 1Þðp d p þ 1Þ > if d j f ; p j d: > > > ðp 1; dÞ > > > > > p1 > > > > > > > 1 1 > > > ðpþ1Þe pe 2þ fn 2þ fn 0 0 > > d d > þ p p > > > > # > > > > > > > > > 1 > > 1þd pe0 fn > > > þp pþ1 > > > ; :
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10. Conclusion of the case p2 jj e With jC2d j; jC1d \C2d j; and jC0d \ðC1d ,C2d Þj computed in Sections 7–9, we are ready to state our main result in the case p2 jj e: Theorem 10.1. Let F =Qp be a finite extension of degree n with residue degree f1 and ramification index e1 : Let f ; e; and e0 be positive integers such that e ¼ p2 e0 ; p[e0 ; ðp f1 f 1; e0 Þ ¼ 1; and either f ðF ðzp Þ=F Þ[f or eðF ðzp Þ=F Þ41: Write f ¼ pi t with p[t: Then the number of F -isomorphism classes of finite extensions K=F with residue degree f and ramification index e is given by
"
t
pi e0 n 1 X 1 1Þ2 2 ðpt fðtÞ ðp 1; tÞ IðF ; f ; eÞ ¼ i p t tjt 2 p1 t
t
i
t
i
i
pt p e0 n ðpt p e0 n 1Þ 1 2 ðpt 2p e0 n 1Þ þ ðp 1; tÞ ðp 1; tÞ p1 2 p1 t
t
i
t
i
þ ðp þ 1Þðp1þt ðpþ1Þp e0 n pt p e0 n Þ ðp2 1Þpt p þ
i X
" p
j1
t
ij
ðpt p
e0 n
j¼1 t
þ ðp 1Þðp2þt p t
ijþ1
t
ij
1Þðpt p pþ1
e0 n
þ ðp2 1Þðp1þt ðpþ1Þp
ij
t
e0 n1
ijþ1
þ pt p e0 n
t
e0 n
ij
pt p
e0 n
t
i
þ pt 2p e0 n
1Þ t
ij
þ pt 2p
e0 n
iþ1
e0 n
Þ
Þ
t ij t ij 1 1 þ ðp 1; tÞ2 ðpt p e0 n 1Þ2 þ ðp2 1; tÞðpt 2p e0 n 1Þ 2 2
## :
Proof. By (6.3) and (6.5) we have
IðF ; f ; eÞ ¼
1 X fðdÞðn0 ðdÞ þ n1 ðdÞ þ n2 ðdÞÞ; fe d40
ð10:1Þ
where n0 ðdÞ ¼ jC0d \ðC1d ,C2d Þj; n1 ðdÞ ¼ jC1d \C2d j; and n2 ðdÞ ¼ jC2d j: Since n2 ðdÞ ¼ n2 ðlcmðp2 ; dÞÞ and n2 ðdÞ ¼ 0 when d[p2 f ; it follows that X d40
fðdÞn2 ðdÞ ¼
X djp2þi t
fðdÞn2 ðdÞ
ARTICLE IN PRESS X.-D. Hou, K. Keating / Journal of Number Theory 104 (2004) 14–61
60
¼
X
fðtÞ fð1Þn2 ðtÞ þ fðpÞn2 ðptÞ þ fðp2 Þn2 ðp2 tÞ
tjt 2þi X
þ
! j
j
fðp Þn2 ðp tÞ
j¼3
¼
X
2
2
fðtÞ p n2 ðp tÞ þ ðp 1Þ
!
2þi X
p
j1
j
n2 ðp tÞ :
By the same reasoning we have X
fðdÞn0 ðdÞ ¼
d40
X
X
fðtÞ n0 ðtÞ þ ðp 1Þ
i X
fðdÞn1 ðdÞ ¼
X
! p j1 n0 ðp j tÞ ;
ð10:3Þ
j¼1
tjt
d40
ð10:2Þ
j¼3
tjt
fðtÞ pn1 ðptÞ þ ðp 1Þ
1þi X
! p
j1
j
n1 ðp tÞ :
ð10:4Þ
j¼2
tjt
It follows that IðF ; f ; eÞ ¼
1
X
p2þi te0
tjt
þ ðp 1Þ
fðtÞ n0 ðtÞ þ pn1 ðptÞ þ p2 n2 ðp2 tÞ i X
! p
j1
j
ðn0 ðp tÞ þ pn1 ðp
jþ1
2
tÞ þ p n2 ðp
jþ2
tÞÞ :
ð10:5Þ
j¼1
Using Propositions 7.2, 8.5 and 9.5 to write out n0 ðp j tÞ þ pn1 ðp jþ1 tÞ þ p2 n2 ðp jþ2 tÞ explicitly for 0pjpi; we obtain the final formula for IðF ; f ; eÞ: &
References [1] W.E. Clark, J.J. Liang, Enumeration of finite commutative chain rings, J. Algebra 27 (1973) 445–453. [2] P. Deligne, Les corps locaux de caracte´ristique p; limites de corps locaux de caracte´ristique 0, in: Representations of Reductive Groups Over a Local Field, Hermann, Paris, 1984, pp. 119–157. [3] I.B. Fesenko, S.V. Vostokov, Local Fields and their Extensions. A Constructive Approach, American Mathematical Society, Providence, RI, 1993. [4] K.W. Gruenberg, A. Weiss, Galois invariants for local units, Quart. J. Math. Oxford 47 (1996) 25–39. [5] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1991. [6] X. Hou, GLðm; 2Þ acting on Rðr; mÞ=Rðr 1; mÞ; Discrete Math. 149 (1996) 99–122. [7] X. Hou, q-ary bent functions constructed from chain rings, Finite Fields Appl. 4 (1998) 55–61. [8] X. Hou, Bent functions, partial difference sets and quasi-Frobenius local rings, Des. Codes Cryptogr. 20 (2000) 251–268. [9] X. Hou, Finite commutative chain rings, Finite Fields Appl. 7 (2001) 382–396. [10] W. Klingenberg, Projective und affine Ebene mit Nachbarelementen, Math. Z. 60 (1960) 384–406.
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[11] M. Krasner, Nombre des extensions d’un degre´ donne´ d’un corps p-adique: e´nonce´ des re´sultats et pre´liminaires de la de´monstration (espace des polynomes, transformation T), C. R. Acad. Sci. Paris 254 (1962) 3470–3472. [12] M. Krasner, Nombre des extensions d’un degre´ donne´ d’un corps p-adique: suite de la de´monstration, C. R. Acad. Sci. Paris 255 (1962) 224–226. [13] M. Krasner, Nombre des extensions de degre´ donne´ d’un corps de nombre p-adique: les conditions ðnÞ
ðnÞ
d’Ore et la caracte´risation de Ek;j ; pre´liminaires du calcul de Nk;j;s ; C. R. Acad. Sci. Paris 255 (1962) 1682–1684. ðnÞ
[14] M. Krasner, Nombre des extensions de degre´ donne´ d’un corps p-adique: calcul de Nk;j;s ; de´monstration du the´ore`me 1, C. R. Acad. Sci. Paris 255 (1962) 2342–2344. [15] M. Krasner, Nombre des extensions de degre´ donne´ d’un corps p-adique: comple´ments au the´ore`me 1 dans le cas non p-adique; de´monstration du the´ore`me 2, C. R. Acad. Sci. Paris 255 (1962) 3095–3097. [16] K.H. Leung, S.L. Ma, Constructions of partial difference sets and relative difference sets on p-groups, Bull. London Math. Soc. 22 (1990) 533–539. [17] K.H. Leung, S.L. Ma, Partial difference sets with Paley parameters, Bull. London Math. Soc. 27 (1995) 553–564. [18] B.R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974. [19] J.-P. Serre, Local Fields, Springer, New York, 1979. [20] G. To¨rner, F.D. Veldkamp, Literature on geometry over rings, J. Geom. 42 (1991) 180–200. [21] A. Weiss, Multiplicative Galois Module Structure, American Mathematical Society, Providence, RI, 1996. [22] http://www.wolfram.com/
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 62–74
http://www.elsevier.com/locate/jnt
On the pair correlation of the eigenvalues of the hyperbolic Laplacian for PSLð2; ZÞ\H C.J. Mozzochi Princeton, NJ 08542, USA Received 21 February 2002 Communicated by D. Goss
Abstract Let H be the upper half plane and X ¼ PSLð2; ZÞ\H the corresponding modular surface. The eigenvalues of the hyperbolic Laplacian, D; on X are denoted by lj ¼ 1=4 þ t2j : For a40 and TX2 let Eða; TÞ ¼
X
t2j
eT 2 cosðaTtj Þ:
0otj
In this paper, we evaluate Eða; TÞ by means of the Selberg trace formula. Since the contribution to the Selberg trace formula from the hyperbolic conjugacy classes is not manageable, we have the result that the Selberg trace formula, the seemingly natural tool to use to attack the pair correlation problem, does not yield meaningful results when used with a natural choice of test function. In combining the results in this paper with the results in Part II, substantial evidence is presented to indicate that the Selberg trace formula may not be the natural tool to use to attack the problem, after all. r 2003 Elsevier Inc. All rights reserved.
1. Introduction We consider G ¼ PSLð2; ZÞ: Hence, we have that l0 ¼ 0pl1 pl2 pl3 p? are the eigenvalues E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00150-1
ARTICLE IN PRESS C.J. Mozzochi / Journal of Number Theory 104 (2004) 62–74
63
(i.e. the point spectrum) of the hyperbolic Laplacian associated with G and lj ¼ sj ð1 sj Þ with sj ¼ 12 þ itj ; so lj ¼ 14 þ t2j
j ¼ 1; 2; 3; y:
with tj 40;
The Weyl–Selberg formula states that for G ¼ PSLð2; ZÞ; X 1 MðTÞ ¼ mðtj Þ ¼ 12 T 2 þ c1 T log T þ OðTÞ; 0otj pT
where mðtj Þ denotes the multiplicity of lj ; which is equal to the dimension of the eigenspace of lj (cf. [4, p. 142]). This estimate follows from the Selberg trace formula by an appropriate use of test functions, say hðtÞ; gðxÞ; where (as in every such application of the trace formula) Z 1 N gðxÞ ¼ hðtÞeixt dt; 2p N hðtÞ
is holomorphic in the strip jIm tjp12 þ e;
hðtÞ5ðjtj þ 1Þ2e
in the strip jIm tjp12 þ e;
and hðtÞ is even: Then for the group G the Selberg trace formula states X i mðtj Þhðtj Þ ¼ I þ E þ P þ H; h þ 2 tj where I is the contribution of the identity and E; P and H; respectively, are the contributions of the conjugacy classes of the elliptic, parabolic, and hyperbolic elements of G: Specifically, Z 1 N I¼ t tanhðptÞ hðtÞ dt; 12 N
E¼
X m1 X fRg k¼1
1 2m sin pk m
Z
e2ptk=m hðtÞ dt; 2pt N 1 þ e N
ARTICLE IN PRESS 64
C.J. Mozzochi / Journal of Number Theory 104 (2004) 62–74
where fRg denotes an elliptic class in G; and m ¼ ordfRg ¼ 2; 3; Z 1 N G0 1 G0 þ it þ ð1 þ itÞ hðtÞ dt P ¼ PPSLð2;ZÞ ¼ 2p N G 2 G N X p LðnÞ gð2 log nÞ; þ gð0Þlog þ 2 2 n n¼1
H¼
N X X fPg
k¼1
log NP ðNPÞ
k=2
ðNPÞk=2
gðk log NPÞ;
where fPg denotes a primitive hyperbolic class in G and NP denotes its norm. All series and integrals converge absolutely. Iwaniec gives a nice presentation of the Selberg trace formula in [2]. The result in this paper is the following: Theorem 1.1. For fixed a40 and TX2; let Eða; TÞ ¼
X
t2j
eT 2 cosðaTtj Þ:
0otj
Then Eða; TÞ ¼ e
aT
1 2 þ4T 2
aT 1 þ 2 þ O e 2 4T
as T-N;
where the implied constant depends only on a: To establish Theorem 1.1 we employ the Selberg trace formula with the appropriate test functions. For each a; T let t2
t2
hðtÞ ¼ hðt; a; TÞ ¼ eT 2 eiaTt þ eT 2 eiaTt : Clearly, t2
hðtÞ ¼ hðt; a; TÞ ¼ 2eT 2 cosðaTtÞ;
ð1:1Þ
and the hðt; a; TÞ satisfy the Selberg conditions for the Selberg trace formula. Also, 1 1 aT aT i ð1:2Þ h ¼ e4T 2 e 2 þ e4T 2 e 2 ; 2
ARTICLE IN PRESS C.J. Mozzochi / Journal of Number Theory 104 (2004) 62–74
1 gðx; a; TÞ ¼ 2p
Z
65
N
hðt; a; TÞeixt dt:
ð1:3Þ
N
Clearly, gðx; a; TÞ ¼
1 2p
Z
N N
1 þ 2p Lemma 1.1.
Z
N
t2
eT 2 cosððx aTÞtÞ dt Z
N
t2
eT 2 cosððx þ aTÞtÞ dt:
2 2
eq t cosðptÞ dt ¼
N
Proof. [1, p. 480, 3.896(2)]. Hence,
and
ð1:4Þ
N
pffiffiffi p2 p 4q2 e : q
&
T 2 x2 aT 3 T a2 T 4 T 2 x2 þaT 3 x 4 2 x 4 4 2 e þe gðx; a; TÞ ¼ pffiffiffie 2 p
ð1:5Þ
T 2 2 aT 3 T a2 T 4 gðx; a; TÞppffiffiffie 4 e 4 x þ 2 x ¼ g1 ðx; a; TÞ: p
ð1:6Þ
2. The hyperbolic classes contribution Consider primitive binary quadratic forms, f ðx; yÞ ¼ ax2 þ bxy þ cy2 ;
a; b; cAZ;
ða; b; cÞ ¼ 1: Let d ¼ dðf Þ ¼ b2 4ac40: We will use ja; b; cj to denote such a form. Two forms ja; b; cj and ja0 ; b0 ; c0 j are called equivalent (in the narrow sense) if there is a unimodular transformation g such that
0 a b0 =2 a b=2 t g: ¼g b0 =2 b=2 c c0 Let hðdÞ ¼ the number of equivalence classes.
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Let D ¼ fd40 j d 0; 1ðmod 4Þ; d not a square g: Let ðxd ; yd Þ be the fundamental solution of the Pellian equation x2 dy2 ¼ 4: pffiffiffi Let ed ¼ 12fxd þ d yd g: Let mðnÞ denote the multiplicity of distinct primitive hyperbolic classes ðNPn Þ in G with trace nX3: It is well-known that n þ ðn2 4Þ1=2 2
ðNPn Þ ¼
!2 :
Theorem 2.1 (Sarnak [3]). The norms ðNPn Þ of the conjugacy classes with trace nX3 of primitive hyperbolic transformations of G are e2d where dAD with multiplicity hðdÞ: Theorem 2.2 (Sarnak [3]). X
HðxÞ ¼
hðdÞ ¼ Liðx2 Þ þ Oðx3=2 ðlog xÞ2 Þ
fdAD;ed pxg
as x-N; where
LiðuÞ ¼
Z m 2
1 dt: log t
Lemma 2.1. Z 0
N
h pffiffiffiffii pffiffiffiffiffiffi x2 exp gx dx ¼ pb expðbg2 Þ 1 f g b ; 4b
where Re b40 and 2 fðxÞ ¼ pffiffiffi p
Proof. [1, p. 307, 3.322(2)].
Z
&
0
x
2
et dt
ðthe error functionÞ:
ARTICLE IN PRESS C.J. Mozzochi / Journal of Number Theory 104 (2004) 62–74
67
We now decompose HT ¼
N X
!
logðNPn Þ
gðlogðNPn ÞÞ ðNPn Þ1=2 ðNPn Þ1=2 ! N X logðNPn Þ mðnÞ þ gð2 logðNPn ÞÞ ðNPn Þ ðNPn Þ1 n¼3 mðnÞ
n¼3
^ N X
þ
mðnÞ
!
N X
logðNPn Þ
k¼5
ðNPn Þk=2 ðNPn Þk=2
n¼3
gðk logðNPn ÞÞ
¼ S1 þ S2 þ S3 þ S4 þ S5 : ð1þeÞaT Lemma 2.2. Si ¼ Oð1Þ for i ¼ 2; 3; 4; and below I2 ¼ Oð1Þ and I3 ¼ O e 4 ; where the implied constant depends only on a and e: Proof. By Theorem 2.1, X
S1 ¼ Limit A-N
hðdÞ
2oed oA1=2
2 log ed gð2 log ed Þ; ed e1d
so by Theorem 2.2 Z
S1 ¼
N 2
Since Dx Liðx2 Þ ¼
x log x
S1 ¼
Z 2
þ
2 log x gð2 log xÞdHðxÞ: x x1
; we have N
Z 2
N
Z
N
2gð2 log xÞ dx x2 1 2 2 log x 2 3=2 gð2 log xÞ d O x ðlog xÞ x x1
2gð2 log xÞ dx þ
¼ I1 þ I2 þ I3 :
ð2:1Þ
By Theorem 2.1, S2 ¼ Limit A-N
X 2oed oA1=2
hðdÞ
2 log ed gð4 log ed Þ; e2d e12 d
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so by Theorem 2.2, S2 ¼
Z
N
0
Since Dx Liðx2 Þ ¼
x log x
2 log x gð4 log xÞ dHðxÞ: x2 x12
; we have
Z N 2gð4 log xÞ 2gð4 log xÞ dx þ dx x xðx4 1Þ 2 2 Z N 2 log x þ gð4 log xÞd O x3=2 ðlog xÞ2 1 2 x2 2 x ¼ Ia þ Ib þ Ic :
S2 ¼
Z
N
ð2:2Þ
For each fixed a40 and e40 I2 ¼ Oð1Þ; Ia ¼ Oð1Þ;
ð1þeÞaT I3 ¼ O e 4 ; Ib ¼ Oð1Þ;
Ic ¼ Oð1Þ;
where the implied constant depends only on a and e: This follows from Lemma 2.1 and partial integration. The proof for S3 and S4 is constructed in the same way. & Lemma 2.3. 2 aT 1 1 aT 1 I1 ¼ e4T 2 1 þ f þ e2 2 2T 2 2 aT 1 12 aT 1 4T þ þ e 1f e 2 þ Oð1Þ; 2 2T 2 where f is the error function and the implied constant depends only on a: Proof. This follows from Lemma 2.1 and the change of variable t ¼ log x: & Lemma 2.4. For each fixed a40;
2 aT 1 aT 1 12 aT 1 e4T 1 þ f þ e 2 ¼ O e 2 þ4T 2 : 2 2T 2 Proof. This follows by straightforward application of L’Hoˆpital’s rule.
&
ARTICLE IN PRESS C.J. Mozzochi / Journal of Number Theory 104 (2004) 62–74
Lemma 2.5. For each fixed a40;
2 1 aT aT 1 12 aT 1 e4T 1 f þ Limit e 2 e 2 e4T 2 ¼ 0: T-N 2 2T 2 Proof. Straightforward.
&
Lemma 2.6. S5 ¼ Oð1Þ; where the implied constant depends only on a: Proof. X
S5 ¼
hðdÞ
15aT ed pe 64 dAD
þ
X
N X 2 log ed gð2k log ed Þ ek ek d k¼5 d
hðdÞ
15aT e 64 oed dAD 0 ¼ S5 þ S500 :
N X 2 log ed gð2k log ed Þ ek ek d k¼5 d
We first consider S50 : Let Sd ¼
N X 2 log ed gð2k log ed Þ; ek ek d k¼5 d
Sd 5
N X log ed g1 ð2k log ed Þ: ekd k¼1
Let AðxÞ ¼
x X
1:
n¼1
Then AðxÞ ¼ x þ Oð1Þ ¼ y1 ðxÞ þ y2 ðxÞ: Hence, by (1.6)
Z N a2 T 4 2 T 2 2 3 log ed eT x log ed þðaT 1Þx log ed dy1 ðxÞ Sd 5 pffiffiffi e 4 p 1 Z 2 T a 2 T 4 N 2 2 3 þ pffiffiffi e 4 log ed eT x log ed þðaT 1Þx log ed dy2 ðxÞ p 1 ¼ I1 þ I2 :
69
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70
By Lemma 2.1 with b ¼ 4T1 2 and g ¼ ðaT 3 1Þ and with the change of variable t ¼ x log ed we have aT
I1 5e 2 : Let a2 T 4 2 2 2 3 T Aðx; a; T; ed Þ ¼ pffiffiffi e 4 T x log ed þðaT 1Þx log ed ; p
I2 ¼ 2T 2 log3 ed
Z
N
xy2 ðxÞAðx; a; T; ed Þ dx Z N ðaT 3 1Þlog2 ed y2 ðxÞAðx; a; T; ed Þ dx 1
1
Að1; a; T; ed Þy2 ð1Þlog ed : By Lemma 2.1 and with the change of variable t ¼ x log ed together with the fact that x5eex and ed pe
15aT 64
we have
T a2 T 4 15aT 2 1 2 4 3 jAð1; a; T; ed Þy2 ð1Þlog ed jppffiffiffie 4 log ed eðaT Þlog ed p pffiffiffi e64a T p 64 p and aT
aT
1
2
4
I2 5a2 T 4 e 2 ð1eÞ þ ða2 T 4 aTÞe 2 þ aT 2 e64a T : 2
x But by Theorem 2.2 and the fact that Liðx2 Þ5log x we have
X
15aT
hðdÞ5
15aT ed pe 64
e 32 aT
so that by considering the cases 0oao1 and 1pa it follows that S50 ¼ Oð1Þ: We now consider S500 : S500 5
X 15aT e 64 oed dAD
hðdÞ
log ed Sd ; e5d
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where Sd ¼
N X
g1 ð2k log ed Þ:
k¼5
Since e
15aT 64 oed
and 5pk; elementary manipulations with inequalities yields ! 3 1 aT k log e d T 2 k2 log2 ed 1 T 2 k2 log2 ed 1 ð5Þ 15 2 k2 log2 e T 64 d : pe e
Hence, by 3.321(3) in [1], a2 T 4 T Sd 5pffiffiffi e 4 p
Z
N
11
e75T
dx ¼ O log1 ed :
x log2 ed
2 2
0
Consequently, by Theorem 2.2 we have Z N X 1 1 S500 5 hðdÞ 5 ¼ d HðxÞ 5 x e 2 d ed Z N Z N 1 1 dx þ dðOðx3=2 ðlog xÞ2 ÞÞ; ¼ 4 x log x x5 2 2 and the result follows by straightforward evaluation of these integrals.
&
3. The elliptic classes contribution Lemma 3.1. E ¼ Oð1Þ; where the implied constant depends only on a: Proof. Z
k
e2ptm hðtÞ ¼ 2pt N 1 þ e N
Z
N
k
e2ptm hðtÞdt þ
0
Z
N
e2pt
k 1m
0
The result follows by partial integration.
&
4. The identity class contribution Lemma 4.1. I ¼ Oð1Þ where the implied constant is absolute. Proof. 1 I¼ 6
Z
N
t hðtÞdt þ Oð1Þ:
0
The result follows by partial integration.
&
hðtÞdt þ Oð1Þ:
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5. The parabolic classes contribution Lemma 5.1. P ¼ Oð1Þ where the implied constant depends only on a: Proof. Follows immediately from the following two lemmas and the easily established result, gð0Þlog p2 ¼ Oð1Þ: & R 1 N G0 G0 Lemma 5.2. 2p N G ð1=2 þ itÞ þ G ð1 þ itÞ hðtÞdt ¼ Oð1Þ; where the implied constant depends only on a: Proof. This follows by using the expansion G0 1 1 ðsÞ ¼ log s þ O 2s G jsj2 and partial integration. Lemma 5.3. S ¼ 2 on a:
!
&
PN
LðnÞ n¼1 n gð2 log nÞ
¼ Oð1Þ; where the implied constant depends only
Proof. aT
Sp 2
e4 X
g1 ð2 log nÞ þ 2
n¼1
N X LðnÞ g1 ð2 log nÞ n aT n¼e 4
¼ S1 þ S2 : It is easy to determine that g1 ðx; a; TÞ is concave downward and achieves its maximum at x ¼ aT: Hence, aT aT aT aT S1 p 2e 4 g1 2 log e 4 ¼ 2e 4 g1 2 2 4 a T 2T ¼ pffiffiffie 16 ¼ Oð1Þ: p Let x X LðnÞ 1 ¼ ðlog x þ cÞ þ O AðxÞ ¼ n logDx n¼1
¼ y1 ðxÞ þ y2 ðxÞ:
! for all DX1
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Hence, S2 ¼ 2
Z
N aT e4
g1 ð2 log xÞdy1 ðxÞ þ 2
Z
N
aT e4
g1 ð2 log xÞdy2 ðxÞ
¼ I1 þ I2 :
I1 ¼ 2 Using the fact that
Z
N aT e4
g1 ð2 log xÞ dx5 x
Z
N
aT e4
g1 ð2 log xÞdx:
aT 4 plog x;
choosing D ¼ 4 and noting that 2T 2 log x aT 3 þ g01 ð2 log xÞ ¼ g1 ð2 log xÞ x x
we have by integrating by parts once that
1 1 I2 5 4 2 þ 3 a T a T
Z
N
aT e4
0
1 a2 T 4 16 e 2g1 ð2 log xÞdx þ @ 4 3 A: aT
1 But by Lemma 2.1 with b ¼ 4T and g ¼ aT 2 and the change of variable t ¼ log x we have
I1 51; and using that estimate for the integral in the above inequality we have S2 ¼ Oð1Þ:
&
6. Proof of the theorem Theorem 1.1 follows immediately from (1.2), Lemmas 2.2, 2.4–2.6, 3.1, 4.1 and 5.1.
Acknowledgments I thank Henryk Iwaniec for suggesting that I investigate the pair correlation problem for the modular group by means of the Selberg trace formula and for several helpful conversations. I thank Peter Sarnak for several helpful conversations. I also thank the referee for calling my attention to a subtle error in the initially submitted paper.
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References [1] I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals Series and Products, Academic Press, New York, 1980. [2] H. Iwaniec, Introduction to the Theory of Automorphic Forms, Bibl. Rev. Mat. Iberoamericana, Madrid (1995), p. 167. [3] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982) 229–247. [4] A.B. Venkov, Spectral theory of automorphic functions, the Selberg zeta function, and some problems of analytic number theory and mathematical physics, Russian Math. Surveys 34 (3) (1979) 79–153.
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Journal of Number Theory 104 (2004) 75–99
http://www.elsevier.com/locate/jnt
Elliptic fibers over non-perfect residue fields Michael Szydlo RSA Laboratories, Bedford, MA 01730, USA Received 8 March 2002; revised 23 April 2003 Communicated by D. Goss
Abstract Kodaira and Ne´ron classified and described the geometry of the special fibers of the Ne´ron model of an elliptic curve defined over a discrete valuation ring with a perfect residue field. Tate described an algorithm to determine the special fiber type by manipulating the Weierstrass equation. In the case of non-perfect residue fields, we discover new fiber types which are not on the Kodaira–Ne´ron list. We describe these new types and extend Tate’s algorithm to deal with all discrete valuation rings. Specifically, we show how to translate a Weierstrass equation into a form where the reduction type may be easily determined. Having determined the special fiber type, we construct the regular model of the curve with explicit blow-up calculations. We also provide tables that serve as a simple reference for the algorithm and which succinctly summarize the results. r 2003 Published by Elsevier Inc. Keywords: Tate’s algorithm; Kodaira symbol; Non-perfect residue field; Resolution of singularities; Blow-up; Elliptic scheme; Ne´ron model; Reduction type; Discrete valuation ring; Flat regular model
1. Introduction Useful models of elliptic curves reflect the curve’s arithmetic in the geometry of the special fibers. One such model, defined over a discrete valuation ring (DVR), is the Ne´ron model [18]. This model is a flat regular minimal model of an elliptic curve whose fibers contain some information about the structure of the rational points on the curve. The Ne´ron model is defined in terms of a universal mapping property [6], which makes it automatically a group scheme. Kodaira and Ne´ron classified the possible geometries of the special fibers of these regular schemes. Given a Weierstrass E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Published by Elsevier Inc. doi:10.1016/j.jnt.2003.06.004
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equation defining an elliptic curve over a DVR, Tate presents an algorithm to determine the reduction type of its Ne´ron model. This algorithm may be interpreted as a recipe for the resolution of the singularities on a one-dimensional elliptic scheme. However, both the Kodaira–Ne´ron classification and Tate’s algorithm only apply to discrete valuation rings with perfect residue fields. This paper focuses on the construction of regular elliptic schemes defined over DVRs whose residue field is not perfect. Surprisingly, some of the resulting special fibers are not on the Kodaira–Ne´ron list. We first provide a list of the additional special fiber types, and describe the geometry of each new type. Next, given a Weierstrass equation with coefficients in any DVR, we provide an algorithm to determine which standard or new reduction type it defines, thereby extending Tate’s algorithm to allow non-perfect residue fields. As in the classical case, such an algorithm also determines the sequence of blow-ups required to construct the regular model, and the smooth part of these models is still a Ne´ron model. Elliptic curves are often first considered over number fields, where the residue field associated to each prime ideal is finite and thus perfect. However, it is also natural to consider discrete valuation rings over non-perfect residue fields. For example, to study a one parameter family of elliptic curves defined over Z; one considers an elliptic scheme over the base Z½T: Not all localizations of this ring have a perfect residue field.1 Recently, the new reduction types defined in this paper have been applied to the study of the Grothendieck pairing on the component group of an elliptic curve, in the case of a non-perfect residue field. Bertapelle and Bosch [3], and Lorenzini [16] have shown that this pairing is not always a perfect pairing, as was previously conjectured [8]. With the new reduction types, Lorenzini produces explicit examples on which Grothendieck’s pairing is degenerate. The construction of one-dimensional regular models can also be used as a building block for the construction of flat regular models of elliptic schemes defined over surfaces, or higher-dimensional base schemes. This construction, and the material in this article, was the subject of the author’s Ph.D. thesis [23]. De-singularizing elliptic threefolds in characteristic 0 was studied earlier by Miranda [17]. Organization: The rest of this article is organized as follows. In Section 2, we present background and notation used throughout the paper, and describe our result. In Section 3 we describe the geometry of the new special fiber types. Next, in Section 4, we present a series of conditions on the coefficients of a Weierstrass equation which are sufficient to determine the reduction type. These conditions are succinctly presented in several tables. In Section 5, we prove that a Weierstrass equation can always be translated to a form satisfying such conditions, thus providing an effective algorithm to determine the reduction type. Finally, in Section 6, we discuss how the regular model is constructed using blow-ups, and we carry out the construction for explicit examples which yield new special fiber types. We conclude with suggestions for further related research. 1
Localizing the ring Z½T at the ideal ðTÞ yields a DVR with perfect residue field Q: Localizing at (2), however, yields a DVR with residue field F2 ðTÞ; which is not perfect.
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2. Background and notation Before describing in further detail our results, we present notation and review some background material. Schemes over a DVR: Throughout this paper we let R denote a discrete valuation ring, and let K be its field of fractions. Let m be the unique maximal ideal of R; let k ¼ R=m be the residue field, and let k% be an algebraic closure of k: Let v be the valuation on R; and let p be a uniformizer for R; so that vðpÞ ¼ 1: For every ring we denote its spectrum with SpecðÞ; and throughout this paper we deal with schemes of finite type over the base scheme, S ¼ SpecðRÞ: For each such scheme X =S; we define the generic fiber of X to be the variety X SpecðKÞ; and the special fiber of X to be the variety or scheme X˜ ¼ X SpecðkÞ: For any variety V =K we say that a scheme X =S is a model for V if the generic fiber of X is the variety V : We also call the special fiber X˜ the reduction of V in X : Elliptic schemes: To assist the reader in following our explicit calculations, we recall the standard polynomials associated with elliptic curves. First, a Weierstrass equation is a cubic equation of the form f ¼ y2 þ a1 xy þ a3 y ðx3 þ a2 x2 þ a4 x þ a6 Þ ¼ 0;
ð1Þ
where ai AK: We also define the usual quantities b2 ; b4 ; b6 ; b8 ; c4 ; c6 ; d; j: b2 ¼ a21 þ 4a2 ;
b4 ¼ 2a4 þ a1 a3 ;
b6 ¼ a23 þ 4a6 ;
b8 ¼ a21 a6 þ 4a2 a6 a1 a3 a4 þ a2 a23 a24 ; c4 ¼ b22 þ 24b4 ;
c6 ¼ b32 þ 36b2 b4 216b6 ;
d ¼ b22 b8 8b34 27b26 þ 9b2 b4 b6 ;
j ¼ c34 =d:
ð2Þ ð3Þ ð4Þ ð5Þ
We define an elliptic curve E=K; to be the subvariety of P2 ðKÞ which is cut out by a Weierstrass equation with non-zero discriminant d; in the affine neighborhood SpecðK½x; yÞ: Our main object of interest, however, is two dimensional. Definition 1. An Elliptic Scheme, C=R; is a two-dimensional projective scheme flat over SpecðRÞ whose generic fiber is an elliptic curve E=K: For each Weierstrass equation f ¼ 0 with coefficients in R; ð f AR½x; y; da0Þ; we define the Elliptic Scheme defined by f to be the closure of SpecðR½x; y=ð f ÞÞ in P2 ðSÞ: * and the geometric We denote the special fiber of an elliptic scheme by C˜ ¼ C#R k; special fiber by C% ¼ C#R k: % Regular models: Recall that a local ring of dimension d with maximal ideal mp is a regular local ring if dimðmp =m2p Þ ¼ d: A scheme S is then called regular or
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non-singular if for every point pAS; the local ring Op is regular. If Op is not regular, S is said to have a singularity at p: ˜ (2) the We consider the regularity of several objects: (1) the special fiber C; % geometric special fiber C; and (3) C itself, which is a two-dimensional scheme. The geometric special fiber C% of an elliptic scheme defined by the Weierstrass equation f ¼ 0 is singular if and only if vðdÞ40; in which case the singularity is either a node or a cusp. A more general elliptic scheme C; defined with several coordinate patches, can be more complicated. Considering an affine coordinate patch of the form % are those for which SpecðR½x; y=ðgÞÞ; (where gAR½x; y),2 the singular points pAC; df df % both dx ¼ 0 and dy ¼ 0; ðmod pÞ: Such a singular point in C may, or may not, be ˜ % C˜ can be a regular point of C: k-rational, and if not, its image under the map C˜ Finally, a singular point pA ˜ C; may or may not be a singular point of (the surface) C: This can be tested directly from the definition: it is singular if dimðmp =m2p Þ42: The regular models we construct will also be proper, flat, and minimal. Assuming that k is perfect, the smooth part of a regular, proper, minimal model of an elliptic curve E=K is the Ne´ron model of the curve, and in particular, a group scheme. We refer the reader to [6,9,10,22] for a definition and discussion of these terms. Minimal Weierstrass Equations: An R-translation of a Weierstrass Equation is an R-linear change of variables of the form x/x0 þ a; y/y0 þ bx0 þ g; where a; b; gAR: See [21] for the formulas explaining how the Weierstrass coefficients ai change under an R-translation. Definition 2. A Weierstrass Equation f AR½x; y is Minimal if there is no R-translation for which the translated coefficients a0i satisfy pi ja0i for iAf1; 2; 3; 4; 6g: A minimal Weierstrass equation can be used to construct a minimal model. See [10,22] for a description of the relationship between a minimal Weierstrass equation and a minimal model. Kodaira–Ne´ron special fibers: We now define some terms to describe the geometry of the special fibers, and review Kodaira’s notation for standard reduction types. The special fiber is neither required to be irreducible nor reduced, so we define the components of a curve over k or k% to be the maximal irreducible subschemes. A component is a rational curve if its reduced subscheme, L; is isomorphic to P1 =k: The multiplicity of a rational curve locally defined by a polynomial f is the valuation of f in the local ring OL (which is a DVR). When two distinct components meet at a point p; the intersection defines a zero-dimensional scheme supported at p: The degree of this scheme is called the local intersection multiplicity of the components at p: Two components intersect transversally at a point if the intersection multiplicity is one. Kodaira and Ne´ron classified the special fibers into reduction types according to the geometry (genus and regularity) of the reduced components, the multiplicity of each component, and the local intersection multiplicity between pairs of 2
This is an example. Not every coordinate patch must be in such a form.
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components. The following table presents the Kodaira symbol of each type, and the number of components (when k is algebraically closed). Note that the symbols In ; and In actually denote a family of reduction types, one for each integer nX1: ð6Þ
The familiar types I0 ; I1 ; and II each consist of only a single component, namely an elliptic curve, a nodal cubic, and a cuspidal cubic. Type III has two components meeting tangentially, and each remaining type consists of two or more rational curves which intersect transversally. For further description of these types, including the configuration and multiplicity of the components, we refer the reader to [13,18,22]. Lastly, we remark that when k is not algebraically closed, one or more of the ˜ components in the geometric special fiber C% may be identified in C: Blow-ups: It is conjectured that every scheme is birational to a regular scheme, but this has only been proved for certain classes of schemes [1,11,12], including the arithmetic surfaces C=R considered in this paper. The construction of a regular model of a scheme is called a resolution of singularities. Given that the general resolution of singularities is still open, it is not obvious that general elliptic schemes always have regular models. One technique to create a regular model is to use the blow-up construction. For each subscheme T of S the blow-up of S along T is a scheme S 0 with a birational morphism S 0 -S; which is an isomorphism outside of T: When S ¼ SpecðRÞ is affine, and fgi ARg; cut out T; the fgi g naturally define a rational map S-Pk ðRÞ: In this case we can define the blow-up of S along T to be the closure of the graph of f in S R Pk ðRÞ: In practice, one describes the blow-up scheme in terms of the k þ 1 standard affine coordinate neighborhoods of S R Pk ðRÞ; effectively performing a substitution of variables in each neighborhood. See [7,9,10,22] for several different, but equivalent, complete definitions of a blow-up which apply to more general schemes TCS: By repeating this blow-up construction for wisely chosen subschemes T containing the singular points, one hopes to obtain a regular scheme S 00 and a birational morphism S 00 -S: Tate’s algorithm: It is known that when k is perfect, a flat proper regular minimal model of an elliptic curve always exists. Tate’s algorithm is a procedure used to determine the special fiber type of such a good model, when it is defined by a minimal Weierstrass equation.3 The constructive version of Tate’s algorithm uses the blow-up construction to resolve the singularities on the elliptic scheme SpecR½x; y=ð f Þ; defined by the Weierstrass equation f ¼ 0: 3
When k is perfect, and charðkÞa2; 3; it is well known that the reduction type can be easily determined by examining d; and j:
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At each stage of the algorithm, the scheme is checked for regularity, and if still singular, it is replaced with the blow-up along a certain subscheme T: The resulting scheme, defined locally in multiple coordinate patches, is the regular model sought. Tate conveniently chose to translate the coordinates of the Weierstrass equation in the successive stages of his algorithm, so that the existence and location of the singularities could easily be determined by examining the valuations of the ai ; or bi : The non-constructive version of Tate’s algorithm skips the actual blow-up computations, but only explains how to translate the coordinates of the original Weierstrass equation so that the required sequence of blow-ups, and therefore the reduction type, can be determined directly from the valuations of the ai (or of the quantities b2 ; b4 ; b6 ; b8 ; c4 ; c6 ; and d). Our results: We have extended Tate’s algorithm to the case when k is not perfect. We posed the question: ‘‘Do the blow-ups in Tate’s algorithm still produce a regular model ?’’ We found that they often do, but not always. When they do not, we discovered new special fiber types in the regular minimal model. The first result is that only finitely many new reduction types (and families of reduction types) that arise. In other words, with a few modifications, this sequence of blow-ups still always terminates, and produces a regular scheme. We list and describe these new reduction types. Secondly, we produce a series of tables which help to concisely summarize this extention of Tate’s algorithm. For every reduction type, we record the form of the Weierstrass equation, after it has been translated, thus producing sufficient conditions on the fai g for each type. Next, we present a simple, effective procedure to translate any Weierstrass equation into such a form, thus completing the determination of the special fiber when k is not perfect. Lastly, we return to the construction of the regular model, and compare the required sequence of blow-ups to those specified in Tate’s original algorithm. By verifying the regularity of each kind of resulting scheme, we have a proof that our extention of Tate’s algorithm correctly determines the reduction type.
3. A description of the new reduction types In this section, we list and describe the new special fiber types. To complement the standard Kodaira symbols listed above in (6), we now introduce some new symbols % for the extra reduction types, and record the number of components of each in C: 0 Note that the symbols Kn ; Kn ; and Tn each denote a family of reduction types, indexed by nAZ; nX1: ð7Þ
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Fig. 1. Types X1 ; Y1 ; and Z1 :
˜ We are now going to describe the geometry of these curves C=k: We do this by first % k; describing the geometric special fibers C= % and then describing the behavior of the ˜ % C: singular points or curves under the mapping r : C˜ % If p is a point in C; and p˜ ¼ rðpÞ is its image in C; we say that rðpÞ; ramifies in C% to ˜ has multiplicity k41: We use the same terminology a multiplicity k point if r 1 ðpÞ ˜ Often, singular points in C% become regular points in C: ˜ We also for a curve pA ˜ C: ˜ define the smooth locus of points in C to be the open subscheme containing only points whose inverse image in C% is reduced and non-singular. 3.1. New types with residue characteristic 3 When charðkÞ ¼ 3; the special fiber is either a standard Kodaira type listed above in list (6), or one of the two new types, Z1 ; or Z2 : % which is not Type Z1 : Geometrically, type Z1 is a cubic curve with a cusp pAC; ˜ k-rational. Its image, rðpÞAC; is a regular point which ramifies in a degree three extention of k (Fig. 1). Type Z2 : Geometrically, type Z2 consists of a chain of three rational curves, intersecting transversally, of multiplicity 1, 2, and 3. The first two curves are ˜ is generically regular, and it k-rational, and the last, p; is not. Its image, rðpÞAC; ramifies in a degree three extention of k (Fig. 2).
*
*
3.2. New types with residue characteristic 2 When charðkÞ ¼ 2; the special fiber is either a standard Kodaira type listed above in list (6), or any one of the new types, except Z1 ; or Z2 : % which is not Type X1 : Geometrically, type X1 is a cubic curve with a cusp pAC; ˜ is a regular point which ramifies in an extention of k-rational. Its image, rðpÞAC; k of degree two or four (Fig. 1).4 Type X2 : Geometrically, type X2 consists of four rational curves, intersecting transversally, as follows. A first k-rational curve of multiplicity 2, meets two
*
*
4
The degree depends on the field in which both coordinates of the cusp are rational.
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Fig. 2. Type Z2 :
Fig. 3. Type X2 :
Fig. 4. Types Y2 and Y3 :
*
*
*
*
rational curves of multiplicity 1, and a second curve of multiplicity 2. This last ˜ is generically regular, and ramifies curve, p; is not k-rational. Its image, rðpÞAC; in a degree two extention of k (Fig. 3). % which is not kType Y1 : Type Y1 is geometrically a cubic curve with a cusp pAC; ˜ rational. Its image, rðpÞAC; is a regular point which ramifies in an extention of k of degree two (Fig. 1). Type Y2 : Geometrically, type Y2 consists of two rational curves of multiplicity 1 and 2, intersecting transversally. The second curve, p; is not k-rational. Its ˜ is generically regular, and ramifies in a degree two extention of k image, rðpÞAC; (Fig. 4). Type Y3 : Geometrically, type Y3 consists of a chain of five rational curves of multiplicity 1, 2, 3, 4 and 2, intersecting transversally. The first three curves are krational, but the last two, p and p0 are not. The image of the first, rðpÞAC˜ is of multiplicity 2 over k; and thus everywhere singular. The image of the second, ˜ is generically regular. Both rðpÞ and rðp0 Þ ramify in a degree two rðp0 ÞAC; extention of k (Fig. 4). Types Kn ðn oddÞ: Geometrically, the types Kn for odd n41 consist of a chain of one rational curve, and n 1 2 multiplicity 2 curves, which intersect transversally. Each multiplicity 2 component is not k-rational, but has a regular image in C˜ which ramifies in a degree two extention of k (Fig. 5). Type K1 consists of a single component, a cuspidal cubic with a rational cusp. (It is identical to type II).
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Fig. 5. Types Kn (nAZ; n40), and types Kn0 ; (nA2Z; n40).
Fig. 6. Type Tn (nAZ; n40).
*
*
*
Types Kn ðn evenÞ: Geometrically, the types Kn for even n42 have the same configuration as the types Kn for odd n; except there are n 2 2 multiplicity 2 components and an additional rational curve intersecting the last multiplicity 2 component transversally (Fig. 5). Type K2 consists of two rational curves meeting tangentially at a point. (It is identical to type III). Types Kn0 ðn evenÞ: Geometrically, the types Kn0 for even n have the same configuration as the types Kn for odd n; except there are n2 multiplicity 2 components. The last component contains no k-rational point, except the point at which it intersects the previous component (Fig. 5). Types Tn ðn40Þ: Geometrically, the types Tn consist of two rational curves meeting the first component of a chain of n 1 2 multiplicity 2 curves, all intersecting transversally. All components are k-rational, except the last one, p: Its image, ˜ is generically regular, and ramifies in a degree two extention of k rðpÞAC; (Fig. 6).
Having defined and described these new special fiber types, we now assert that this list is complete. Theorem 3.1 (Reduction types). Let C=R be a flat proper regular minimal model of ˜ is either a standard reduction type in Kodiara’s list (6), E=K: Then the special fiber, C; or one of the new reduction types in list (7), and defined in this section. The proof of Theorem 3.1 will be completed later in Section 6, when we will have constructed the regular model of the elliptic curve defined by any Weierstrass equation.
4. Valuations of Weierstrass coefficients In this section we associate a set of conditions on the Weierstrass coefficients to each reduction type. Although not every Weierstrass equation meets such criteria, at least these conditions are sufficient to determine the reduction type.
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The conditions are succinctly presented in several tables, where each column corresponds to a different reduction type. When k is perfect, such conditions can be expressed completely in terms of the valuations of the Weierstrass coefficients. In general, however, other kinds of conditions will be required, and we list them here. 1. 2. 3. 4. 5. 6.
vðfÞ ¼ a; for fAfa1 ; a2 ; a3 ; a4 ; a6 ; b2 ; b4 ; b6 ; b8 ; c4 ; c6 ; dg: vðfÞXa; for fAfa1 ; a2 ; a3 ; a4 ; a6 ; b2 ; b4 ; b6 ; b8 ; c4 ; c6 ; dg: The reduction ai =pa ðmod pÞ is not a square in k: The reduction ai =pa ðmod pÞ is not a cube in k: A specified auxiliary polynomial has distinct roots; or not: A specified auxiliary curve f 0 ; contains no k-rational points:
Table notations: These six kinds of conditions are described with entries in a table, as follows. Fixing a reduction type, the entry in the row labeled vðfÞ represents one condition, according to the following notation. An integer a means vðfÞ ¼ a: The symbol aþ means vðfÞXa: The symbol ans denotes the condition that f=pa ðmod pÞ is not a square. Similarly, the symbol anq denotes the condition that f=pa ðmod pÞ is not a cube. The symbol ant denotes the condition that the quadratic polynomial Y 2 ¼ a2 X 2 þ pa6a contains no k-rational points. We define the element dF to be a2 a4 a6 ; its valuation determines whether or not the auxiliary polynomial (13) has distinct roots. We present the tables of conditions according the characteristic of k; starting with charðkÞ ¼ 0 or charðkÞX5: 4.1. Chara2; 3; general form Not surprisingly, when charðkÞa2; 3; any set of conditions which suffice to determine the reduction type when k is perfect are also sufficient when k is not perfect. In particular, this implies that when charðkÞa2; 3; none of the new reduction types appear as the special fiber of a regular model, and additionally, the reduction type may be still be determined from vðdÞ; and vð jÞ alone. With this in mind, we still choose to present general conditions on the Weierstrass coefficients without completing the square and assuming that a1 ¼ a3 ¼ 0: This way the data contained in the following tables serves as the base case which will highlight the differences that appear later, when charðkÞ ¼ 2 or 3: These conditions are also a good reference when constructing the models with blow-ups, and are useful when resolving the singularities of elliptic schemes over higher dimensional bases5 [23]. Certain of the conditions in Table 1 have equivalent formulations in terms of the number of roots of certain auxiliary polynomials. For example, the condition vðb6 Þ ¼ 4 which applies to type IV ; is equivalent to specifying that the polynomial X2 þ 5
a3 a6 Xþ 2 p p
ð8Þ
One approach applies Tate’s algorithm to an equation defined over a DVR which may have characteristic zero, yet arises as the localization of a two-dimensional ring in which 2 is not a unit.
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Table 1 Standard Kodaira types for Tate’s algorithm ðcharðkÞa2; 3Þ Type vða1 Þ vða2 Þ vða3 Þ vða4 Þ vða6 Þ vðb2 Þ vðb4 Þ vðb6 Þ vðc4 Þ vðc6 Þ vðdÞ
I0
In þ
0 0þ 1þ 1þ 1þ 0
II þ
1 1þ 1þ 1þ 1
III
IV
þ
þ
1 1þ 1þ 1 2þ
I0
In
þ
1 1þ 1þ 2þ 2þ
þ
1 1þ 2þ 2þ 3þ
1 1 2þ 3þ 4þ 1
IV
III
II
o=w
þ
þ
þ
1þ 2þ 3þ 4þ 6þ
1 2þ 2þ 3þ 4þ
1
0
n
2
4
1 1 2
1 2þ 3þ 4þ 5
3
1 0
1 2þ 3þ 3 5þ
2 3 6
2 4
3
5 3
7þ
4 8
9
5 10
12þ
has distinct roots in k: % The condition vðb6 Þ ¼ 8 which applies to type IV ; is equivalent to specifying that X2 þ
a3 a6 Xþ 4 p2 p
ð9Þ
has distinct roots in k: % Finally, the condition vðc6 Þ ¼ 3 is equivalent to the condition that X3 þ
a2 2 a4 a6 X þ 2X þ 3 p p p
ð10Þ
has three distinct roots in k% (Table 1). 4.2. The families In and In If Table 1 implies that the reduction type is in the family In ; the conditions in Table 2 can be used to determine the exact type. Similarly, if a reduction type is in the family In ; Table 3 can be used to determine the exact type. As before, certain conditions in Table 3 may be formulated in terms of the number of roots of certain auxiliary polynomials. When k is odd, the condition vðb8 Þ ¼ k implies that the polynomial X2 þ
a3 k 1 p 2
Xþ
a6 pk 1
ð11Þ
has distinct roots in k: % If k is even, the polynomial a2 2 a4 a6 X þ k X þ k 1 p p p2
ð12Þ
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Table 2 Reduction types in the family In I1
Type
I2
þ
vða1 Þ vða2 Þ vða3 Þ vða4 Þ vða6 Þ vðb2 Þ vðb8 Þ vðdÞ
I3
þ
0 0þ 1þ 1þ 1 0 1 1
I4
þ
0 0þ 1þ 1þ 2þ 0 2 2
I5
þ
?
þ
0 0þ 2þ 2þ 3 0 3 3
0 0þ 2þ 2þ 4þ 0 4 4
0 0þ 2þ 2þ 5 0 5 5
? ? ? ? ? ? ? ?
Table 3 Reduction types in the family In ; ðcharðkÞa2Þ Type
I1
I2
I3
I4
I5
?
vða1 Þ vða2 Þ vða3 Þ vða4 Þ vða6 Þ vðb8 Þ vðdÞ
1þ 1 2þ 3þ 4þ 5 7
1þ 1 3þ 3þ 5þ 6 8
1þ 1 3þ 4þ 6þ 7 9
1þ 1 4þ 4þ 7þ 8 10
1þ 1 4þ 5þ 8þ 9 11
? ? ? ? ? ? ?
Table 4 Characteristic 3 Reduction types Type vða2 Þ vða4 Þ vða6 Þ vðdÞ
I0
In
Z1
II
III
IV
I0
Z2
In
IV
III
II
o=w
0
0 1þ 1þ n
1þ 1þ 0nq 1þ
1þ 1þ 1 2þ
1þ 1 2þ 3
1þ 2þ 2 4þ
1þ 2þ 3þ 6
2þ 3þ 3nq 7þ
1 3þ 4þ 7þ
2þ 3þ 4 8þ
2þ 3 5þ 9
2þ 4þ 5 10þ
2þ 4þ 6þ 12þ
has distinct roots in k: % These facts will be useful to the reader who wishes to calculate the blow-ups for the types in Table 3. Interestingly, Table 2 turns out to apply in all residue characteristics, and Table 3 still applies when charðkÞ ¼ 3: 4.3. Char 3 residue field Here we treat the case where charðkÞ ¼ 3; and define the conditions on the Weierstrass coefficients in Table 4. Notice the appearance of a new kind of condition which is not needed when charðkÞa2; 3: Namely, one of the conditions for the new
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type Z1 is that a6 ðmod pÞ is not a cube in k: Similarly, one of the conditions for the new type Z2 is that a6 =p3 ðmod pÞ is not a cube in k: Recall that we use the notation 0nq and 3nq ; for these new kinds of conditions in Table 4. There are no new kinds of conditions appearing in the families In and In when charðkÞ ¼ 3: If Table 4 implies that the reduction type is in one of these families, the same tables presented above, Tables 2 and 3, may be used to determine the exact reduction type. 4.4. Char 2 residue field The case where charðkÞ ¼ 2 is the most interesting one. We define the conditions on the Weierstrass equations in Table 5 associated to each reduction type. We see several new kinds of condition in characteristic two. First, for each of the new reduction types X1 ; Y1 ; X2 ; Y2 ; Y3 ; and for each of the new types in the family Kn ; one of the conditions is that a certain element is not a square in k: Recall that in Table 5 the notation ans means that ai =pa ðmod pÞ is not a square in k: A second new kind of condition is present for types I0 : The auxiliary polynomial F ¼ X3 þ
a2 2 a4 a6 X þ 2X þ 3 p p p
ð13Þ
must have distinct roots for type I0 : We define dF ¼ a2 a4 a6 ; the discriminant of this polynomial ðmod pÞ; so that we may succinctly denote this condition by vðdF Þ ¼ 3 in Table 5. Note that the conditions defined in this table imply that for type X2 the polynomial (13) has a double root which is not k-rational. Reading the table: Table 5 contains the most complex set of conditions on the Weierstrass equation, so let us give an example of how to read the table columns. Let f ¼ 0; f AR½x; y be a Weierstrass equation. According to the first column, f is of type I0 if vðdÞ ¼ 0: According to the second column, f belongs to a type in the In family if vða1 Þ ¼ 0; vða2 ÞX0; vða3 ÞX1; vða4 ÞX1; and vða6 ÞX1; and we consult Table 2 to determine exactly which type it is. The third column tells us that f is of type X1 if vða1 ÞX1; vða2 ÞX0; vða3 ÞX1; vða4 Þ ¼ 0; vða6 ÞX0; and a4
Table 5 Characteristic 2 Reduction types Type vða1 Þ vða2 Þ vða3 Þ vða4 Þ vða6 Þ vðdF Þ vðdÞ
I0 þ
0 0þ 0þ 0þ 0þ 0 0
In 0 0þ 1þ 1þ 1þ
X1
Y1
Kn
þ
þ
þ
1 0þ 1þ 0ns 0þ
1 0þ 1þ 1þ 0ns
1 0ns 1þ 1þ 1þ
II þ
1 1þ 1þ 1þ 1
III
IV
Y2
þ
þ
þ
1 1þ 1þ 1 2þ
1 1þ 1 2þ 2þ 4
1 1þ 2þ 2þ 2ns
I0 þ
1 1þ 2þ 2þ 3þ 3
X2 þ
1 2þ 2þ 2ns 4þ
In þ
1 1 2þ 3þ 4þ
IV þ
1 2þ 2 3þ 4þ 8
Y3 þ
1 2þ 3þ 3þ 4ns
III
II
o=w
þ
þ
1þ 2þ 3þ 4þ 6þ
1 2þ 3þ 3 5þ
1 2þ 3þ 4þ 5
12þ
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Table 6 Reduction types in the family Kn Type
K1
K2
K20
K3
K4
K40
K5
K6
K60
vða1 Þ vða2 Þ vða3 Þ vða4 Þ vða6 Þ vðb8 Þ
þ
þ
þ
þ
þ
þ
þ
þ
þ
1 0ns 1þ 1þ 1
1 0ns 1þ 1þ 2þ 2
1 0ns 2þ 2þ 2nt
1 0ns 2þ 2þ 3
1 0ns 2þ 2þ 4þ 4
1 0ns 2þ 2þ 4nt
1 0ns 3þ 3þ 5
1 0ns 3þ 3þ 6þ 6
1 0ns 4þ 4þ 6nt
? ? ? ? ? ? ?
Table 7 Reduction types in the family In Type
I1
T1
I2
T2
I3
T3
I4
T4
?
vða1 Þ vða2 Þ vða3 Þ vða4 Þ vða6 Þ vða6 =a2 Þ vðb8 Þ
1þ 1 2 3þ 4þ
1þ 1 3þ 3þ 4ns
1þ 1 3þ 3 5þ
1þ 1 3 4þ 6þ
1þ 1 4þ 4þ 6
1þ 1 4þ 4 7þ
5
6þ
6
1þ 1 3þ 4þ 5 4ns 7þ
7
8þ
8
1þ 1 4þ 5þ 7 6ns 9þ
? ? ? ? ? ? ?
is not a square in k: Similarly, Table 5 defines conditions for the other reduction types. Consulting subtable. Notice that Table 5 contains conditions for the three infinite families In ; Kn ; and In : If this table implies that the reduction type is in one of these families, a sub-table must be consulted to determine the exact type. For the family In ; the conditions in Table 2 can be used to determine the exact type. For the family Kn ; Table 6, presented below, must be consulted. Finally, for the family In ; the conditions in Table 7, also presented below, can be used to determine the exact type. 4.5. Char 2 Kn detail The family of reduction types Kn is a sequence of new reduction types which appear only when charðkÞ ¼ 2: Each type in this family has the common condition that a2 ðmod pÞ is not a square in k: Table 6 presents the conditions sufficient to determine each reduction type in this family. Note that a new kind of condition, denoted ant ; is used for the types Kn0 ; (n even). Recall that this condition has been defined to mean that the quadratic curve in k½X ; Y Y 2 ¼ a2 X 2 þ
a6 pn
ð14Þ
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contains no k-rational points. The notation ant was chosen because there is no krational point that can be translated. 4.6. Char 2 In detail The subfamily In for charðkÞ ¼ 2; differs from the usual subfamily In when charðkÞa2: Here, the family In also includes the new types Tn ; for n40: In Table 7, we present the conditions on the Weierstrass equations sufficient to determine the type. Notice the conditions for the types Tn : for n odd we 6 require that panþ3 not be a square in k; and for n even we require that a2 pa6nþ2 not be a square in k: 4.7. Translated form definitions Having specified a set of conditions associated to each reduction type, we now make a formal definition. Definition 3. Let R be an arbitrary DVR with residue field k; and let f AR½x; y define a Weierstrass equation. We say that the polynomial f ; and the Weierstrass equation f ¼ 0 is in Translated Form if one of the following holds: * * * * * * *
charðkÞa2; 3; and f meets all conditions for a type in Table 1. for any k; f meets all conditions for a type in Table 2. charðkÞa2; and f meets all conditions for a type in Table 3. charðkÞ ¼ 3; and f meets all conditions for a type in Table 4. charðkÞ ¼ 2; and f meets all conditions for a type in Table 5. charðkÞ ¼ 2; and f meets all conditions for a type in Table 6. charðkÞ ¼ 2; and f meets all conditions for a type in Table 7.
This concludes the presentation of the tables of conditions, and the definition of what it means for a Weierstrass equation to be in translated form. We next discuss what to do with Weierstrass equations which are not in translated form.
5. Translating the Weierstrass equation In this section we prove that every Weierstrass equation may be put into translated form. Theorem 5.1. Let R be an arbitrary DVR with residue field k: Let f be a minimal Weierstrass equation with coefficients in R: There always exist R translations x/x0 þ a and y/y0 þ bx þ g such that the modified Weierstrass equation is in translated form. There is an effective algorithm to compute these R-translations.
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Because we will prove that the conditions presented in Section 4 determine the reduction type, the translations of Theorem 5.1 constitute the extention of Tate’s algorithm to general DVRs. There are several components of the algorithm, each corresponding to a table in Section 4: There is one main procedure for each of the cases: * * *
charðkÞa2; 3: charðkÞ ¼ 3: charðkÞ ¼ 2:
There is also a secondary procedure for each of the four infinite families: * * * *
In : In ðcharðkÞa2Þ: In ðcharðkÞ ¼ 2Þ: Kn :
These procedures all follow a common paradigm, which we briefly discuss. In each procedure, a Weierstrass equation is compared to the conditions described in the columns of the table, until it successfully meets such a set of conditions. These tests proceed in order of the columns (from left to right), and if a test fails, a translation may be made before moving to the next column. So, to fully describe each procedure, we simply need to specify which, if any, translations are to be made when the conditions of a column are not met. If the conditions defined in the rightmost columns of Tables 1, 4, or 5 are met, then we begin anew with a more minimal Weierstrass equation. Once the table of conditions is known, it is not difficult to recover the details of the translations required to complete the definition of each procedure. When charðkÞa2; 3; these translations are identical to those in Tate’s original algorithm. We will provide details for the main procedure when charðkÞ ¼ 2; and also for the secondary procedure for the family Kn : These are the two most interesting procedures, and we leave the rest for the reader to discover, or find in [23]. 5.1. CharðkÞ ¼ 2 main procedure Let f be a Weierstrass equation with coefficients in R: The following 17 step procedure will produce an R translation x/x0 þ a and y/y0 þ bx0 þ g such that the revised coefficients fai g satisfy the conditions of one of the types (or families of types) in Table 5. The reader can most easily verify this proof by following along with Table 5, sequentially examining the conditions and possible translations. 1. First suppose vðdÞ ¼ 0: Then we have type I0 : 2. Suppose instead vðdÞ40 and vða1 Þ ¼ 0: Translate via x ¼ x0 a3 =a1 and compute new fai g: Then vða3 Þ40: The singularity at ðx; yÞ on the special fiber satisfies df dy ¼ 2y0 þ a1 x0 þ a3 ¼ 0 ðmod pÞ: So now x ¼ 0 at the singular point.
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3.
4.
5.
6.
7. 8. 9. 10.
91
Now translate via y ¼ y0 þ a4 =a1 ; and compute new fai g: Now vða4 Þ40: The df ¼ a1 y0 þ a4 ¼ 0 ðmod pÞ: So singularity at ðx0 ; y0 Þ on the special fiber satisfies dx y ¼ 0 at the singular point as well. Next, the fact that f ðx0 ; y0 Þ ¼ 0 ðmod pÞ implies that vða6 Þ40: So, collecting these conditions, vða1 Þ ¼ 0; vða3 Þ40; vða4 Þ40; and vða6 Þ40; we are in one of the cases In : We then consult Table 2 to determine exactly which type it is. Now suppose both vðdÞ40 and vða1 Þ40: Then the condition df dy ¼ 0 ðmod pÞ at the singular point (x0 ; y0 ) implies 2y0 þ a1 x0 þ a3 ¼ a3 ¼ 0 ðmod pÞ: So vða3 Þ40: Assume additionally that a4 is not a square in k: Then we are in the case X1 : If instead a4 is a square in k; translate via x ¼ x0 þ a where a2 ¼ a4 ðmod pÞ and compute new fai g: Now vða4 Þ40: If a6 is not a square in k; we are in the case Y1 : If instead a6 is a square in k; translate via y ¼ y0 þ a where a2 ¼ a6 ðmod pÞ and compute new fai g: Now vða6 Þ40: Assume that a2 is not a square in k: Then we are in one of the cases Kn ; or Kn0 : We then consult Table 6 to determine exactly which type it is. If instead a2 is a square in k; translate via y ¼ y0 þ bx where b2 ¼ a2 ðmod pÞ and compute new fai g: Now vða2 Þ40: These cumulative translations imply vða1 ÞX1; vða2 ÞX1; vða3 ÞX1; vða4 ÞX1; and vða6 ÞX1: Suppose that additionally vða6 Þ ¼ 1: Then we are in the case II : Suppose instead vða6 ÞX2; and additionally vða4 Þ ¼ 1: Then we are in the case III : Suppose instead vða4 ÞX2; and additionally vða3 Þ ¼ 1: Then we are in the case IV : Suppose instead vða3 ÞX2; and additionally pa62 is not a square in k: Then we are in the case Y2 : If instead pa62 is a square in k; translate via y ¼ y0 þ a where a2 ¼ pa62 ðmod pÞ and compute new fai g: Now vða6 ÞX3: These cumulative translations imply vða1 ÞX1; vða2 ÞX1; vða3 ÞX2; vða4 ÞX3; and vða6 ÞX3: So by reducing ap2 ; pa42 ; and pa63 ðmod pÞ; we may form the polynomial in k½X : F ðX Þ ¼ X 3 þ
a2 2 a4 a6 X þ 2 X þ 3: p p p
ð15Þ
If F ðX Þ has distinct roots (or equivalently vða2 a4 þ a6 Þ ¼ 3), then we are in the case I0 : 11. Otherwise F ðX Þ has multiple roots. Suppose that F ðX Þ has a double root and a single root, and that the double root is not rational over k: Then F ðX Þ factors as ðX aÞðX 2 bÞ with b not a square in k: This implies vða4 Þ ¼ 2; and b ¼ pa42 is not a square in k; so we may translate via x/x0 þ ap; so that vða2 ÞX2; and vða6 ÞX4: Then we are in the case X2 :
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12. Suppose instead that F ðX Þ has a double root and a single root, and that the double root is rational over k: Then F ðX Þ factors as ðX aÞðX bÞ2 : Now translate via x ¼ x0 þ bp; and compute new fai g: The new F ðX Þ factors as ðX a0 ÞX 2 ; with a0 a0: Now vða4 ÞX3 and vða6 ÞX4; and vða2 Þ ¼ 1: We are therefore in one of the cases In ; or Tn : We consult Table 7 to determine exactly which type it is. 13. If instead F ðX Þ has a triple root, F ðX Þ factors as ðX aÞ3 with aAk: (A triple root of a cubic in characteristic 2 must always be rational.) Now translate via x ¼ x0 þ ap; and compute new fai g: The new F ðX Þ factors as X 3 ; so vða4 ÞX3; vða6 ÞX4; and vða2 ÞX2: Suppose additionally vða3 Þ ¼ 2: Then we are in the case IV : 14. If instead vða3 ÞX3 and pa64 is not a square in the residue field, then we are in the case Y3 : 15. If pa64 is a square in k; translate via y ¼ y0 þ a where a2 ¼ pa64 ðmod pÞ and compute new fai g: Now vða6 ÞX5: These cumulative translations imply vða1 ÞX1; vða2 ÞX2; vða3 ÞX3; vða4 ÞX3; and vða6 ÞX5: Suppose that additionally vða4 Þ ¼ 3: Then we are in the case III : 16. If instead vða4 ÞX4 and vða6 Þ ¼ 5; then we are in the case II : 17. If vða6 ÞX6; the cumulative conditions are vða1 ÞX1; vða2 ÞX2; vða3 ÞX3; vða4 ÞX4; and vða6 ÞX6: Then the Weierstrass equation is Not Minimal ; so we replace the Weierstrass equation with a more minimal one via the transformation a0i ¼ paii ; and start over at step one. The Weierstrass equation has been transformed into translated form, and the reduction type has been identified, unless the reduction type belongs to one of the families In ; Kn ; Kn0 ; In ; or Tn : In these cases, additional translations may be needed to put f into fully translated form and to determine the exact reduction type. 5.2. Procedure for Kn family Even if an equation meets the conditions of the family Kn according to Table 5, it may still not be in translated form. In this case, the following procedure completes the translations and determines exactly which type it is. Let n ¼ 1: If vða6 Þ ¼ n then we have a type Kn : Let n ¼ n þ 1: If vðb8 Þ ¼ n then we have a type Kn : If Y 2 ¼ a2 X 2 þ a6 =pn ðmod pÞ has no rational points, then we have a type Kn0 : Let X0 and Y0 be two elements in R such that ðX0 ; Y0 Þ reduces to the rational point. Translate via x ¼ x þ X0 pn=2 and y ¼ y þ Y0 pn=2 so that vða6 Þ4n: 7. Let n ¼ n þ 1: 8. Go to step number 2.
1. 2. 3. 4. 5. 6.
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6. Constructing regular models with blow-ups In order to complete the proof of Theorem 3.1 and to prove that the procedures described by Theorem 5.1 always serve as an algorithm to determine the reduction type, we need to verify that the conditions defined in Section 4 are correct. The conditions associated to a particular type are correct if they imply that the sequence of blow-ups produces a special fiber of that type. Thus, explicit calculations of these blow-ups prove the following theorem. (In this section we denote a candidate reduction type with X:) Theorem 6.1. Let R be an arbitrary DVR with residue field k: Let f be a minimal Weierstrass equation with coefficients in R; which is in translated form, meeting the conditions associated to a reduction type X: Let E=K be the elliptic curve defined by f : Then there exists a sequence of blow-ups, (specified in [22,24]), which produce a regular model C; whose generic fiber is E=K; and whose special fiber is of type X: When k is perfect, Theorem 6.1 is a restatement of Tate’s algorithm. One case of this theorem, when k is not perfect, is easy to prove. Namely, if the Weierstrass equation f satisfies the conditions of a standard reduction type, the usual blow-ups of Tate’s algorithm still produce the regular model. By verifying this, one proves the lemma below. This proof amounts to a simple check that, for the standard reduction types, the translations requested by Tate’s algorithm are all k-rational. Lemma 1. Let R be an arbitrary DVR, and let f be a minimal Weierstrass equation with coefficients in R; in translated form, meeting the conditions associated to a standard reduction type X: Then the sequence of blow-ups specified in Tate’s algorithm produces a regular model with the correct reduction type X: The remaining ingredients needed to complete Theorem 6.1 are the case-bycase analyses for each new reduction type. For each of the new reduction types, following Tate’s algorithm directly would entail making a non-k-rational translation. Not being able to complete the sequence of blow-ups might seem to impede the resolution of singularities. However, terminating the sequence of blow-ups at this point already yields a regular scheme. This implies that for each new reduction type, the required blow-ups follow the same sequence as for one of the standard types. The correspondence (16) below gives an example for each new reduction type of a standard type with the same sequence of blow-ups.
ð16Þ
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Having defined the sequence of blow-ups required for each new type, we should now verify that the resulting schemes are all regular. Intuitively, this is true because any would-be singularities are not k-rational. We will now present complete details in two representative cases: type Z1 ; and type Y2 : 6.1. Char 3 Z1 type For our first example, will show that the conditions associated with Type Z1 are correct (i.e. the special fiber is a cuspidal cubic). We begin with the subscheme of R½x; y defined by an equation satisfying the conditions in the column of Table 4 that is labeled Z1 : In other words, the cubic y2 ¼ x3 þ a2 x2 þ a4 x þ a6
ð17Þ
satisfies vða2 Þ40; vða4 Þ40; vða6 Þ ¼ 0; and the requirement that a6 is not a cube in k: We now look for singularities and perform a blow-up if any are found. By setting df dy ¼ 0; we find that any singular point must be contained in the subscheme defined by p ¼ 0; 2y ¼ 0; and x3 þ a6 ¼ 0: Because charðkÞ ¼ 3; we know that x3 þ a6 ¼ ðx þ aÞ3 ðmod pÞ where a3 ¼ a6 : Thus the x-coordinate of the cusp is the cube root of a6 ðmod pÞ; which is not k-rational. Therefore the cusp does not ˜ correspond to a singular point of C: More precisely, the image of the geometric cusp in C˜ is the point defined by the ideal m ¼ ðp; y; x3 þ a6 Þ: This ideal is maximal, and because the polynomial (17) is not zero (mod m2 ), the vector space mm2 is two dimensional, and so this point is a regular point. In fact, because the scheme defined by polynomial (17) was already regular, no blow-ups were required at all. To summarize, the geometric special fiber C% is a cuspidal cubic. However, C˜ is everywhere regular and the cusp is only defined in a degree three extension of k: This phenomenon is only possible because in characteristic three, a cubic polynomial need not be separable. 6.2. Char 2 Y2 type For our second example, we show that the conditions associated with Type Y2 are correct (i.e. its special fiber consists of two components, as in Fig. 4). This time, we begin with a subscheme of R½x; y defined by an equation satisfying the conditions in the column of Table 5 that is labeled Y2 : In other words, the cubic y2 þ a1 xy þ a3 y ¼ x3 þ a2 x2 þ a4 x þ a6
ð18Þ
satisfies vða1 ÞX1; vða2 ÞX1; vða3 ÞX2; vða4 ÞX2; vða6 Þ ¼ 2; and the requirement that pa62 is not a square in k: As usual, we now look for singularities and perform a blow-up if any are found. By inspection, the geometric special fiber C% is a cuspidal cubic curve. However, this
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Table 8 Blowing up a Point on a Surface Co-ordinate patch
Substitution 1
Substitution 2
Patch 1 Patch 2 Patch 3
y ¼ xy1 x ¼ yx1 x ¼ px1
p ¼ xp1 p ¼ yp1 y ¼ py1
time the cusp is k-rational and located at x ¼ 0; y ¼ 0: This point, considered as a point of C=R; is defined by the ideal m ¼ ðx; y; pÞ: By checking that dimðm=m2 Þ ¼ 3; we see that it is a singular point. To resolve this singularity, we blow-up C at the singular point. According to our definition of a blow-up, the revised scheme is the closure of the graph of the morphism to P2 ðRÞ given by the coordinates ðx; y; pÞ: To be explicit, each of the three affine coordinate patches is given by the change of variables in Table 8. We focus on the third coordinate patch, which is the affine subscheme of R½x1 ; y1 defined by y21 þ a1 x1 y1 þ
a3 a4 a6 y1 ¼ x31 p þ a2 x21 þ x1 þ 2 : p p p
ð19Þ
The portion of C% visible in this coordinate patch is the multiplicity 2 curve defined by y21 ¼ pa62 : Of course, this curve is only of multiplicity two because charðkÞ ¼ 2: ˜ however, is generically regular. This can The image of this double line in C; be seen by viewing it as the product of two regular schemes: Specðk½y1 =ðy21 pa62 ÞÞ and Specðk½x1 Þ: We know that the first factor is regular because the polynomial y21 pa62 is irreducible. If one examines the first coordinate patch of the blow-up, the original rational curve can be seen to intersect the double line transversally. To summarize, the geometric special fiber consists of a rational curve intersecting a double line transversally, and the image of this double line in C˜ is regular. Our blow-up has explicitly constructed the reduction type Y2 ; defined in Section 3. The fact that k is not perfect admits the possibility that a quadratic polynomial might not be separable. This example is typical of the new reduction types in characteristic two: a singular point or curve is only defined in a degree two extension of k: For the types requiring more than one blow-up, there may be many coordinate patches to keep track of.
7. Conclusions and further work This concludes our description of the new reduction types, the presentation of the extention to Tate’s algorithm, and the construction of the regular model. We chose
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to center the discussion of the extended Tate’s algorithm around the valuations of the Weierstrass coefficients because this description succinctly describes the geometry and rationality of the special fiber. The tables in this paper describe useful, albeit non-canonical, forms of Weierstrass equations which reveal the special fiber type. The translated form of a Weierstrass equation is also appealing since it automatically suggests the sequence of blow-ups which will produce the regular minimal model. For a particular choice of DVR R; (e.g Z½tð2Þ ), the algorithm may be easily implemented with a computer program, provided one has a procedure to check whether elements of k are squares, or cubes, and whether or not the quadratic curves of form (14) have any rational points. 7.1. Group schemes and Ne´ron models While the focus of this paper is the construction of regular models, it is natural to ask for a description of these models in terms of group schemes and Ne´ron models. When k is perfect, we know that the Ne´ron model of an elliptic curve coincides with the smooth part of a flat proper regular minimal model of the curve. In fact, the following statement does not require the assumption that k is perfect. Proposition 7.1.1. Let R be an arbitrary DVR, let K ¼ fracðRÞ; and let E=K be an elliptic curve. Let C be a flat proper regular minimal model of E=K; and let Csm =R be the smooth part of C=R: Then Csm =R is the Ne´ron model of E=K: Essentially the same techniques used to prove this when k is perfect [2,22], apply more generally. The approach is to first show that the smooth part of the regular model is a group scheme, and then use the assumptions of minimality and smoothness to obtain the Ne´ron mapping property. Because the accounts available in the literature assume that k is perfect, Lorenzini and Liu have explicitly presented a proof in [15]. Proposition 7.1.1 implies that the regular models constructed in this paper yield Ne´ron models once the singular points on the special fiber have been discarded. Focusing on the case when the special fiber is a new reduction type, we see that the special fiber of the Ne´ron model, Csm ; has at most two components. The order of the component group is specified in the following table. ð20Þ While the minimality and smoothness of Csm can be used to show that it is a group scheme, in our case this is also possible with direct calculations. This is not difficult since there are at most two components. Let W denote the smooth part of the elliptic scheme defined by a Weierstrass equation. If the special fiber of the Ne´ron model has only a single component, then W ¼ Csm ; and we check that it is a group scheme by
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examining the elliptic curve’s addition formula. One then checks that the three projective coordinates of the addition map Csm Csm -Csm are never simultaneously zero, and therefore the group law on E=K immediately makes Csm into a group scheme. The special fiber has two components only for reduction types X2 ; K2n ; and Tn : In such a case there exists an R-valued point, P; which reduces to the cusp in W : Translating by P yields an automorphism of Csm of order two. With this translation to identify the generic fibers of two copies of W ; a scheme isomorphic to Csm is produced. By using this simpler description, it is straightforward to extend the group law on E=K to a morphism, proving that Csm is a group scheme. 7.2. Applications and further work Elliptic schemes: The original motivation for this work was the study of elliptic schemes over higher-dimensional base schemes. This work is presented in [23] where schemes of arbitrary dimension are considered, and an algorithm to construct a regular scheme is presented. In this context, most of the DVRs which appear have non-perfect residue fields, and new phenomenon occur due to the fact that the discriminant may not be irreducible in the local rings of the base. We showed that it is possible to always construct flat regular models over high-dimensional bases when char k43: For the other cases it was still true that the algorithms always terminated in our experiments, yet this was not proved for every (regular) base scheme. It would be interesting to explore relationships with the general resolution of singularities [1], and the generalization which allows finite extentions [12]. Extentions and representations: A natural extention of this work is the study of how the reduction types change with finite extentions R0 =R: Contrary to the perfect residue field case, the geometric special fiber of the regular model can change when an unramified extention of k is made. It is also natural to look for an appropriate definition of a conductor in the case when k is not perfect. This would involve finding a good definition for the higher ramification groups needed to define the Swan representation. If a good definition can be made, can we find a relationship between the valuation of the conductor, discriminant, and number of components in the regular model? This would attempt to generalize Ogg’s formula, treated by Ogg [19] when charðkÞa2; and by Saito [20] in general. (Liu specializes this to elliptic curves in [14].) Ogg’s formula is the commonly used tool to compute the exponent in the conductor of an elliptic curve when charðkÞ ¼ 2; or charðkÞ ¼ 3: Component pairings: As described in the introduction, the new reduction types of Section 3 have been applied to the study of the Grothendieck pairing on an elliptic curve. This map pairs a component of the special fiber of the Ne´ron model with a component of the special fiber of the dual Ne´ron model, and produces an element of Q=Z: Bertapelle, Bosch, and Lorenzini [3–5,16]
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compared this pairing with the matrix of intersection multiplicities of the special fiber in a regular proper model of an elliptic curve or Jacobian. With this new tool, they computed the Grothendieck pairing explicitly, discovering cases for which it is not perfect.
Acknowledgments I thank my Ph.D. advisor, Barry Mazur, for guidance on the thesis work, Joe Harris, for teaching me some algebraic geometry, Joe Silverman, for encouraging me to publish this article and for the books that inspired this research, Dino Lorenzini, for his careful reading, Kelly Morgan, for her editorial help, and the anonymous reviewer, for the many detailed comments.
References [1] S.S. Abhyankar, Resolution of Singularities of Arithmetic Surfaces, Harper and Row, New York, 1965, pp. 111–152. [2] M. Artin, Lipman’s Proof of Resolution of Singularities, in: G. Cornell, J. Silverman (Eds.), Arithmetic Geometry, Springer, Berlin, 1986. [3] A. Bertapell, S. Bosch, Weil restriction and Grothendieck’s duality conjecture, J. Algebra Geom. 9 (2000) 155–164. [4] S. Bosch, Q. Liu, Rational points on the group of components of a Ne´ron model, Manus. Math. 98 (1999) 275–293. [5] S. Bosch, D. Lorenzini, Grothendieck’s pairing on component groups of Jacobians, Invent. Math. 148 (2002) 353–396. [6] J. Dieudonne´, A. Grothendieck, Ele´ments de Ge´ome´trie Alge´brique, Publ. Math. IHES 24 (1965) 1–231. [7] D. Eisenbud, Commutative Algebra With a View Toward Algebraic Geometry, in: Graduate Texts in Mathematics, Vol. 150, Springer, Berlin, 1995. [8] A. Grothendieck, Groupes de Monodromie en Ge´ome´trie Alge´brique SGA 7 I, Springer, Berlin, 1972. [9] D. Eisenbud, J. Harris, The Geometry of Schemes, in: Graduate Texts in Mathematics, Vol. 197, Springer, Berlin, 2000. [10] R. Hartshorne, Algebraic Geometry, in: Graduate Texts in Mathematics, Vol. 52, Springer, Berlin, 1977. [11] H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero, Ann. of Math. 79 (1964) 109–326. [12] A.J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. IHES 83 (1996) 51–93. [13] K. Kodaira, On compact analytic surfaces II, Ann. of Math. 77 (1963) 563–626. [14] Q. Liu, Formule d’Ogg d’apre`s Saito, Private Note. [15] Q. Liu, D. Lorenzini, Ne´ron models of elliptic curves, Private Note, 2001. [16] D. Lorenzini, Reduction in the case of imperfect residue fields, Preprint, 2001. [17] R. Miranda, Smooth models for elliptic threefolds, in: R. Friedman, D.R. Morrison (Eds.), The Birational Geometry of Degenerations, Progress in Mathematics, Vol. 29, Birkhauser, Basel, 1983, pp. 85–133. [18] A. Ne´ron, Mode`les Minimaux des Varie´te´s Abe´liennes sur les Corps Locaux et Globaux, Publ. Math. IHES 21 (1964) 361–482. [19] A. Ogg, Elliptic curves and wild ramification, Am. J. of Math. 89 (1967) 1–21.
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[20] T. Saito, Conductor, discriminant, and the Noether formula of arithmetic surfaces, Duke Math. J. 57 (1988) 151–173. [21] J. Silverman, in: The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 106, Springer, Berlin, 1991. [22] J. Silverman, in: Advanced Topics in The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 151, Springer, Berlin, 1996. [23] M. Szydlo, Flat Regular Models of Elliptic Schemes, Ph.D. Thesis, Harvard University, 1999. [24] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in: Modular Functions of One Variable IV, Lecture Notes in Mathematics, Vol. 476, Springer, Berlin, 1975, pp. 33–52.
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 100–117
http://www.elsevier.com/locate/jnt
On some p-adic properties of Siegel–Eisenstein series H. Katsuradaa and S. Nagaokab, a
Faculty of Engineering, Muroran Institute of Technology, Muroran, 050-8585 Japan Department of Mathematics, School of Science and Technology, Kinki University, Higashi-Osaka, 577-8502 Japan
b
Received 13 May 2002; revised 25 November 2002 Communicated by D. Zagier
Abstract We introduce a formula for the p-adic Siegel–Eisenstein series which demonstrates a connection with the genus theta series and the twisted Eisenstein series with level p: We then prove a generalization of Serre’s formula in the elliptic modular case. r 2003 Elsevier Inc. All rights reserved.
1. Introduction We have previously generalized a Serre’s example of p-adic Eisenstein series with p-adic weight ð1; pþ1 2 ÞAZp Z=ðp 1ÞZ to the Siegel modular case [6]. We found that such p-adic Siegel–Eisenstein series becomes a ‘‘true’’ Siegel modular form of weight ðnÞ one for G0 ðpÞ with a character. In fact the formal Fourier expansion coincides with the genus theta series associated with the binary quadratic form of discriminant p: In this note, we study a more general type of p-adic Siegel–Eisenstein series, that is, the p-adic weight is ðk; pþ2k1 Þ: It is easy to see that our p-adic Siegel–Eisenstein series 2 of degree one just coincides with the Eisenstein series of weight k and Nebentypus ð1Þ ðG0 ðpÞ; ðpÞÞ: Therefore, it is natural to ask whether our p-adic Siegel–Eisenstein ðnÞ
series for general k and n becomes a ‘‘true’’ modular form for G0 ðpÞ with a
Corresponding author. E-mail addresses: [email protected] (H. Katsurada), [email protected] (S. Nagaoka). 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00160-4
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character or not (cf. Section 3, ðk; nÞ-question). The main result of this note is to show that our p-adic Siegel–Eisenstein series of degree two can be expressed as a linear combination of two genus theta series and the twisted Eisenstein series (cf. Theorem 4.1). This implies that the series becomes a ‘‘true’’ modular form for ð2Þ G0 ðpÞ with a character in this case.
2. Siegel modular forms and Siegel–Eisenstein series In this section we summarize some of basic facts about Siegel modular forms and the Siegel–Eisenstein series for later use. Concerning the fundamental terminology on Siegel modular forms, the reader should consult the standard reference [1]. Let Hn denote the Siegel upper half space of degree n; and GðnÞ ¼ Spn ðZÞ the Siegel modular group of degree n: Let G be a congruence subgroup of GðnÞ and w is a congruence character (cf. [1, 2.2.2]). We denote by Mk ðG; wÞ the C-vector space of modular forms for G of weight k with character w: If w is trivial, we set Mk ðGÞ ¼ Mk ðG; wÞ: In this note we mainly treat the following two cases: G ¼ GðnÞ ¼ Spn ðZÞ ðthe full modular groupÞ and G¼
ðnÞ G0 ðNÞ
:¼
A C
AG C On D B
ðnÞ
ðmod NÞ :
In both cases, any modular form f ðZÞ has a Fourier expansion of the form X af ðTÞeðtrðTZÞÞ; f ðzÞ ¼ 0pTASymn ðZÞ
where Symn ðZÞ :¼ fT ¼ ðtij ÞASymn ðQÞ j tii AZ; 2tij AZg pffiffiffiffiffiffiffi and eðzÞ :¼ expð2p 1zÞ: We consider a subgroup GðnÞ N
:¼
A C
AG C ¼ On : D B
ðnÞ
The Siegel–Eisenstein series of degree n and weight k is defined as X ðnÞ detðCZ þ DÞk ; ZAHn : Ek ðZÞ :¼ ðnÞ ðnÞ C
D
AGN \G
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If k4n þ 1; this series is absolutely convergent and belongs to Mk ðGðnÞ Þ: Now we consider a more general type of Eisenstein series ðnÞ
Ek ðZ; s; w; NÞ :¼
X
C
D
wðdetðDÞÞ detðCZ þ DÞk jdetðCZ þ DÞjs :
ðnÞ
ðnÞ
AGN \G0 ðNÞ
Here ZAHn ; sAC; kAZ; NAN; and w is a Dirichlet character of modulo N: We naturally assume that wð1Þ ¼ ð1Þk : The series is convergent for ReðsÞ þ k4n þ 1: By a result of Langlands’ theory, it can be continued as a meromorphic function in s to the whole complex plane. Moreover, we define the ‘‘twisted’’ Eisenstein series as ðnÞ
ðnÞ
Fk ðZ; s; w; NÞ :¼ Ek ððNZÞ1 ; s; w; NÞ detðZÞk :
ðnÞ
Proposition 2.1. Suppose kXn þ 1 and wa1: Then both Ek ðZ; s; w; NÞ and ðnÞ
Fk ðZ; s; w; NÞ are holomorphic in s at s ¼ 0: Moreover, we have ðnÞ
ðnÞ
ðnÞ
ðnÞ
ðnÞ
ðnÞ
Ek ðZ; w; NÞ :¼ Ek ðZ; 0; w; NÞAMk ðG0 ðNÞ; w% Þ; Fk ðZ; w; NÞ :¼ Fk ðZ; 0; w; NÞAMk ðG0 ðNÞ; wÞ:
ðnÞ
If k4n þ 1 and w ¼ w0 (the trivial character of conductor 1), then Ek ðZ; w0 ; 1Þ ðnÞ
obviously coincides with Ek ðZÞ: In the following, we derive the Fourier expansion of the above-defined Eisenstein ðnÞ ðnÞ ðnÞ series Ek ðZÞ and Fk ðZ; w; NÞ: First we write the Fourier expansion of Ek ðZÞ as ðnÞ
Ek ðZÞ ¼
X
ðnÞ
ak ðTÞeðtrðTZÞÞ:
0pTASymn ðZÞ ðnÞ
We next derive an explicit expression for the Fourier coefficients ak ðTÞ: We put Symn ðZÞþ :¼ fTASymn ðZÞ j T40g; and for TASymn ðZÞ; let
DðTÞ :¼
8 n < ð1Þ2 detð2TÞ :1 2
n1 ð1Þ 2
detð2TÞ
if n is even; if n is odd:
ð2:1Þ
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When n is even we can write DðTÞ ¼ D0 ðTÞ f ðTÞ2 ;
ð2:2Þ
with D0 ðTÞ the fundamental discriminant and f ðTÞAN: We let D0 ðTÞ wT ðÞ ¼
ð2:3Þ
be the associated primitive quadratic character. Theorem 2.2. For each prime number q and TASymn ðZÞþ ; there exists a computable integral polynomial FqðnÞ ðT; X ÞAZ½X such that
ðnÞ
ak ðTÞ ¼
8 n2 > Bkn;wT Q > 2 n ki > 2 n k > 2 > ð1Þ 2 f ðTÞ2kn1 > > B B B k 2kn i¼1 2k2i > > > Q > ðnÞ > k > > < q : prime Fq ðT; q Þ if n is even; n1 > > > 2 nþ1 nþ1 k Q ki > n > ð1Þ 2 2 jDðTÞjk 2 > > > Bk i¼1 B2k2i > > > Q > ðnÞ > > Fq ðT; qk Þ if n is odd; : q : prime
where Bm ðresp: Bm;w Þ is the mth Bernoulli (resp. generalized Bernoulli) number. ðnÞ
In the above formula, the essential part of the Fourier coefficient ak ðTÞ consists ðnÞ
ðnÞ
of the polynomial factor Fq ðT; qk Þ: The polynomial Fq ðT; X Þ has been studied by several authors, for example, Maass, Kaufhold, and Kitaoka. Recently, the first ðnÞ author derived an explicit formula for Fq ðT; X Þ [4]. We introduce the explicit formula here only for the cases n ¼ 1 and n ¼ 2: The case n ¼ 1: For tAN; put t :¼ ordq ðtÞ: Then we have Fqð1Þ ðt; X Þ ¼
t X
ðqX Þi :
i¼0
The case n ¼ 2: For a non-zero TASym2 ðZÞ; we put eðTÞ :¼ maxfmAN j m1 TASym2 ðZÞg:
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We next define a1 :¼ ordq ðeðTÞÞ;
a :¼ ordq ð f ðTÞÞ;
where f ðTÞ is the integer defined in (2.2). Under this notation, we have Fqð2Þ ðT; X Þ ¼
a1 X
( ðq2 X Þ
j¼0
j
aj X
2 m
ðq3 X Þ wT ðqÞðqX Þ
m¼0
aj1 X
) 2 m
ðq3 X Þ
:
ð2:4Þ
m¼0
Next we consider the Fourier expansion of the twisted Eisenstein series ðnÞ Fk ðZ; w; NÞ: If kXn þ 1 and wa1; we have X
ðnÞ
Fk ðZ; w; NÞ ¼ x1 n;k
ðnÞ
Sk ðw; N; TÞ detðTÞk
nþ1 2 eðtrðTZÞÞ:
ð2:5Þ
TASymn ðZÞþ
Here n1 Y pffiffiffiffiffiffiffi xn;k :¼ ð4pÞnðn1Þ=4 ð2p 1Þnk Gðk ðm=2ÞÞ m¼0
and ðnÞ
Sk ðw; N; TÞ :¼
X
wðnðRÞÞnðRÞk eðtrðRTÞÞ;
RASymn ðQÞ mod 1
where nðRÞ is the common denominator of the elementary divisors of R: Formula (2.5) is already given in [11, p. 338], under the assumption k4n þ 1: However, it also applies for k ¼ n þ 1 (for example, [9, Theorem 7.1], and [2, Proposition 5.2]). Proposition 2.3. Assume that n ¼ 2 and kX3: Then the twisted Eisenstein series ð2Þ Fk ðZ; w; NÞ has a Fourier expansion of the form ð2Þ
Fk ðZ; w; NÞ ¼
X
cðTÞeðtrðTZÞÞ;
TASym2 ðZÞþ 3 1 wT wðpÞp1k Lðk 1; wT wÞ detðTÞk2 cðTÞ ¼ x1 2;k 22k Lðk; wÞzð2k 2Þ 1p Y ð2Þ k
Fq ðT; q wðqÞÞ; q : prime ð2Þ
where Fq ðT; X Þ is the polynomial given in (2.4).
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3. A question on p-adic Siegel–Eisenstein series As stated in the preceding section, any modular form in Mk ðGðnÞ Þ has a Fourier expansion of the form X af ðTÞeðtrðTZÞÞ: f ðZÞ ¼ 0pTASymn ðZÞ
We now introduce the following abbreviation. For T ¼ ðtij ÞASymn ðZÞ; Z ¼ ðzij ÞAHn ; we define qT :¼ eðtrðTZÞÞ ¼
Y 1piojpn
2t
qij ij
n Y
qti i ;
i¼1
where qij :¼ eðzij Þ; qi ¼ qii ; and ti ¼ tii : By the semi-positivity of T; X f ¼ af ðTÞqT AC½q1 ij ; qij ½½q1 ; y; qn ; that is, f AMk ðGðnÞ Þ can be viewed as an element of the formal power series ring ðnÞ C½q1 ij ; qij ½½q1 ; y; qn : In this sense, the Siegel–Eisenstein series Ek is an element of Q½q1 ij ; qij ½½q1 ; y; qn : Following [8], we can similarly define the notion of a p-adic Siegel modular form. A formal power series X f ¼ af ðTÞqT AQp ½q1 ij ; qij ½½q1 ; y; qn 0pTASymn ðZÞ
is called a p-adic Siegel modular form if there exists a sequence of Siegel modular forms ffm g with rational Fourier coefficients satisfying lim fm ¼ f ðp adicallyÞ:
m-N
The precise meaning of the p-adic convergence is as follows. We write the Fourier expansion of fm as X aðmÞ ðTÞqT AQ½q1 fm ¼ ij ; qij ½½q1 ; y; qn : Then, limm-N fm ¼ f means that ordp ð f fm Þ :¼
inf
0pTASymn ðZÞ
ordp ðaf ðTÞ aðmÞ ðTÞÞ- þ Nðm-NÞ:
In this case, we write the weight of fm as km : It follows from the general theory of padic modular forms that the sequence fkm g necessarily has a limit in the space X :¼ lim Z=pm1 ðp 1ÞZ ¼ Zp Z=ðp 1ÞZ: ’
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We now define the notion of the p-adic Siegel–Eisenstein series. We put 1 Bk ðnÞ ðnÞ ðnÞ Gk :¼ zð1 kÞEk ¼ Ek : 2 2k We next take a sequence of positive integers fkm g such that the limit lies in the above space X ; lim km ¼ ðs; tÞAX ¼ Zp Z=ðp 1ÞZ: ðnÞ
In this case, if the corresponding sequence of Eisenstein series fGkm g converges padically, then the limit ðnÞ
ðnÞ
lim Gkm ¼: G˜ ðs;tÞ AQp ½q1 ij ; qij ½½q1 ; y; qn
m-N
is called a p-adic Siegel–Eisenstein series. In the case n ¼ 1; we can define the p-adic Siegel–Eisenstein series for any such fkm g: In [6] we have studied a special class of p-adic Siegel–Eisenstein series, that is, we have calculated the formal Fourier expansion of the p-adic Siegel–Eisenstein series ðnÞ ðnÞ lim G1þpm1 ðp1Þ=2 ¼ G˜ pþ1 :
m-N
ð1; 2 Þ
In order to generalize this result, we fix a pair of integers ðk; pÞ satisfying two conditions: ðiÞ p is a prime number such that p42k; p1 ðmod 2Þ: ðiiÞ k 2
ð3:1Þ
For the pair ðk; pÞ we let km ¼ km ðk; pÞ :¼ k þ
p 1 m1
p : 2
ð3:2Þ
Note that the sequence fkm g has a limit p þ 2k 1 lim km ¼ k; AX ¼ Zp Z=ðp 1ÞZ: 2 Our main object is ðnÞ ðnÞ G˜ pþ2k1 ¼ lim Gkm : ðk;
2
Þ
m-N
From Theorem 2.2, one can get the following result.
ð3:3Þ
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Theorem 3.1. The above limit (3.3) defines a p-adic Siegel–Eisenstein series with rational coefficients: ðnÞ G˜ pþ2k1 AQ½q1 ij ; qij ½½q1 ; y; qn : ðk;
Þ
2
By an argument similar to the computation in [6] (combined with Theorem 2.2), ðnÞ we can calculate the Fourier coefficient of G˜ pþ2k1 : ðk;
2
Þ
In the above, we defined and studied a special class of p-adic Siegel–Eisenstein ðnÞ series G˜ pþ2k1 : A notable fact is that our p-adic Siegel–Eisenstein series sometimes ðk;
2
Þ
becomes a ‘‘true’’ modular form. For example, ð1Þ G˜ pþ2k1 ¼ ðk;
2
Þ
N X Bk;wp X þ wp ðdÞd k1 qt 2k t¼1 0od j t
ð3:4Þ
ð1Þ
is a modular form of weight k for G0 ðpÞ with character wp : 8 p > < if p 1 ðmod 4Þ; wp ðÞ ¼ p > : if p 3 ðmod 4Þ: This shows that ð1Þ ð1Þ G˜ pþ2k1 AMk ðG0 ðpÞ; wp Þ: ðk;
2
Þ
ð3:5Þ
Now we shall pose the following question, called the ðk; nÞ-question, ðk; nÞ-question: Does our p-adic Siegel–Eisenstein series belong to ðnÞ Mk ðG0 ðpÞ; wp Þ? The above example (3.5) indicates that the answer to the ðk; 1Þ-question is affirmative. For QASymm ðZÞþ ; we associate the ordinary theta series by the expression X ðnÞ eðtrðQ½GZÞÞ; ZAHn : yQ ðZÞ :¼ GAMm;n ðZÞ
The genus theta series associated with Q is defined by 0 1, ! h yðnÞ ðZÞ h X X 1 Qj ðnÞ A genus y ðQÞðZÞ :¼ @ ; EðQj Þ EðQj Þ j¼1 j¼1 where fQ1 ; y; Qh g is a complete set of representatives of GLn ðZÞ-equivalence classes in the genus containing Q; and EðQj Þ denotes the order of the unit group of Qj :
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The main result in [6] is as follows: Theorem 3.2. Let p43 be a prime number such that p 3 ðmod 4Þ and SASym2 ðZÞþ a symmetric matrix with detð2SÞ ¼ p: Then, 1 ðnÞ G˜ pþ1 ¼ hðpÞ genus yðnÞ ðSÞ; ð3:6Þ 2 ð1; 2 Þ pffiffiffiffiffiffiffi where hðpÞ is the class number of the field Qð pÞ: In particular ðnÞ ðnÞ G˜ pþ1 AM1 ðG0 ðpÞ; wp Þ:
ð3:7Þ
ð1; 2 Þ
The formula due to Hirzebruch and Zagier [3, Chapter 1, Proposition 3] plays an important role in the proof of (3.6). Obviously this result asserts that the ð1; nÞquestion has an affirmative answer.
4. Main result In this section, we demonstrate that the answer to the ðk; 2Þ-question in Section 3 is affirmative. Throughout this section, we consider the pair of positive integers ðk; pÞ given in (3.1). We take a symmetric matrix F ASym2k ðZÞþ with detð2F Þ ¼ p (the existence of such an F is obvious). We denote by F the adjoint matrix of F (i.e., 2F ¼ pð2F Þ1 ). Of course, both matrices 2F and 2F have the same level p: Theorem 4.1. Using the above notation, if kX2; then we have ð2Þ G˜ pþ2k1 ¼ ðk;
2
Þ
Bk;wp ðgenus yð2Þ ðF Þ pk1 genus yð2Þ ðF ÞÞ 2kðpk1 1Þ þ ð1Þkþ1
ð1 þ p1k Þð1 p2k Þ Bk;wp ð2Þ F ðZ; wp ; pÞ; ð1 p32k Þ pk 2k k
ð2Þ
where Fk ðZ; wp ; pÞ is the twisted Eisenstein series defined in Proposition 2.1. ð2Þ
Remark 1. It should be noted that the Eisenstein series Fk ðZ; wp ; pÞ is defined only for kX3 (see Proposition 2.1). In the case k ¼ 2; the last term of the right-hand side of our main formula vanishes because it involves the factor 1 p2k : So we do not ð2Þ need the weight 2 twisted Eisenstein series F2 ðZ; wp ; pÞ: Our main formula in the case k ¼ 2 can be read as ð2Þ G˜ pþ3 ¼ ð2; 2 Þ
B2;wp ðgenus yð2Þ ðF Þ p genus yð2Þ ðF ÞÞ; 4ðp 1Þ
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and this has been already presented in [7]. The corresponding result in the case k ¼ 1 is covered by the formula (3.6). Remark 2. In [5], Koblitz constructed a p-adic analogue of Cohen’s Eisenstein series of half integral weight. It should be noted that the essential part of the Fourier coefficient of our p-adic Siegel–Eisenstein series can be expressed by his p-adic Cohen–Eisenstein series (e.g. cf. Theorem 4 in [7]). ð2Þ ð2Þ Corollary 4.2. G˜ pþ2k1 belongs to Mk ðG0 ðpÞ; wp Þ: ðk;
2
Þ
This corollary gives an affirmative answer to the ðk; 2Þ-question. The rest of this paper is devoted to proving Theorem 4.1.
5. Local densities In this section, we introduce several results about local densities which are used to calculate the Fourier coefficients of the genus theta series (cf. [10]). Yang’s result [12] plays an important role in our proof. We assume that m and n are positive integers with mXn: Let SASymm ðZÞþ and TASymn ðZÞ: For each prime integer q; let aq ðS; TÞ denote the ordinary local density by aq ðS; TÞ ¼ lim qnðnðnþ1Þ=2mnÞ Aqn ðS; TÞ; n-N
Aqn ðS; TÞ ¼ xfGAMm;n ðZ=qn ZÞ j S½G T mod qn Symn ðZÞg: In the following, we consider the pair of integers ðk; pÞ in (3.1). Let l be an odd integer such that 1plo2k: We put F ð2k;l;uÞ :¼ E2kl1 >u>pEl1 >upAM2k ðZp Þ;
ð5:1Þ
where u is a p-adic unit satisfying 1 if k 0 or 1 ðmod 4Þ; u ¼ p 1 if k 2 or 3 ðmod 4Þ:
Lemma 5.1. Let F a1 ; a2 AZp :
ð2k;l;uÞ
be as above and T ¼
pa a1 0
0 pb a2
ð5:2Þ
with 0papb and
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(1) Assume that a b ðmod 2Þ: Then we have ! a1 u ð1Þðl1Þ=2 ðl1Þ=2 ðaþ1Þð2kÞ ð2k;l;uÞ ap ðF ; TÞ ¼ p p p p
ð1 p22k Þð1 pð32kÞðbaÞ=2 Þ 1 p32k
þ 1þ
22k pl2kþ2 ð1 p1 Þ 22kþlþð32kÞðaþbÞ=2 1 p p : 1 p32k 1 p32k
(2) Assume that acb ðmod 2Þ: Then we have ! ðl1Þ=2 a u ð1Þ 1 ap ðF ð2k;l;uÞ ; TÞ ¼ pðl1Þ=2 pðaþ1Þð2kÞ p p
ð1 p22k Þ þ ð1 p2k ZÞð1 þ p1k ZÞZpð32kÞðba1Þ=2þ1k 1 p32k
þ 1þ
pl2kþ2 ð1 p1 Þ þ p3kþ3þlþð32kÞðaþb1Þ=2 Z 1 p32k
ð1 p2k ZÞð1 þ p1k ZÞ ; 1 p32k
where Z ¼ ða1pa2 Þ:
Proof. See Yang [12, Theorems 3.1 and 7.1].
&
Proposition 5.2. Let F and F be two matrices in Sym2k ðZÞþ given in Section 4 and T a p a1 0 a matrix in Sym2 ðZÞþ which is GL2 ðZp Þ-equivalent to : 0 pb a2 (1) If q is a prime number different from p; then aq ðF ; TÞ ¼ aq ðF ; TÞ ¼ ð1 qk wp ðqÞÞð1 q22k Þð1 q1k wp ðqÞwT ðqÞÞ1 Fqð2Þ ðT; qk wp ðqÞÞ; ð2Þ
where wT is the Dirichlet character in (2.3) and Fq ðT; X Þ is the polynomial in (2.4).
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(2) Assume that q ¼ p: (i) If a b ðmod 2Þ; then ap ðF ; TÞ ¼
a1 u ðaþ1Þð2kÞ ð1 p22k Þð1 pð32kÞðbaÞ=2 Þ p p 1 p32k
þ 1þ
22k p32k ð1 p1 Þ 32kþð32kÞðaþbÞ=2 1 p p ; 1 p32k 1 p32k
ap ðF ; TÞ ¼
a1 u p
þ 1þ
! ð1Þk1 k1 ðaþ1Þð2kÞ ð1 p22k Þð1 pð32kÞðbaÞ=2 Þ p p 1 p32k p
pð1 p1 Þ 1 p22k p1þð32kÞðaþbÞ=2 : 32k 1p 1 p32k
(ii) If acb ðmod 2Þ; then
a1 u ðaþ1Þð2kÞ ap ðF ; TÞ ¼ p p
ð1 p22k Þ þ ð1 p2k ZÞð1 þ p1k ZÞZpð32kÞðba1Þ=2þ1k þ1 1 p32k
þ
2k p32k ð1 p1 Þ ZÞð1 þ p1k ZÞ 43kþð32kÞðaþb1Þ=2 ð1 p þ p Z ; 1 p32k 1 p32k
ap ðF ; TÞ ! a1 u ð1Þk1 k1 ðaþ1Þð2kÞ ¼ p p p p
ð1 p22k Þ þ ð1 p2k ZÞð1 þ p1k ZÞZpð32kÞðba1Þ=2þ1k 1 p32k
þ 1þ
2k pð1 p1 Þ ZÞð1 þ p1k ZÞ 2kþð32kÞðaþb1Þ=2 ð1 p þ p Z ; 1 p32k 1 p32k
where Z ¼ ða1pa2 Þ:
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Proof. (1) The desired formulas are deduced from Katsurada [4]. (2) We recall that the Jordan decomposition of F (resp. F ) at p is just F ð2k;1;uÞ (resp. F ð2k;2k1;uÞ ). So the desired formulas in (i) and (ii) are derived from Lemma 5.1. & From formulas (i) and (ii) in Proposition 5.2, (2), we get the following result, which plays a key role in the proof of the main result (cf. proof of Proposition 6.2). Corollary 5.3. In the notation introduced above, we have the following formula: (i) If a b ðmod 2Þ; then p1 ap ðF ; TÞ pk1 p12k ap ðF ; TÞ ¼
pk ð1 pk1 Þðp2k 1Þðp1k þ 1Þ 1 p32k þ
p22kþð32kÞðaþbÞ=2 ðpk1 1Þð1 p22k Þ : 1 p32k
(ii) If acb ðmod 2Þ; then p1 ap ðF ; TÞ pk1 p12k ap ðF ; TÞ ¼
pk ð1 pk1 Þðp2k 1Þðp1k þ 1Þ 1 p32k þ
p33kþð32kÞðaþb1Þ=2 Zð1 pk1 Þð1 p2k ZÞð1 þ p1k ZÞ : 1 p32k
6. Proof of main theorem In this section, we present a proof of Theorem 4.1. We keep the notation and the definitions from the previous sections. The proof is performed by comparing the Fourier coefficients of the both sides of the expression of Theorem 4.1. For this, we simply write their Fourier expansions as X X ð2Þ G˜ pþ2k1 ¼ aðTÞqT ; genus yð2Þ ðF Þ ¼ bðTÞqT ; ðk;
2
Þ
TASym2 ðZÞ
genus yð2Þ ðF Þ ¼
X
TASym2 ðZÞ
TASym2 ðZÞ
b ðTÞqT ;
ð2Þ
Fk ðZ; wp ; pÞ ¼
X
TASym2 ðZÞþ
The proof is reduced to showing the following two propositions.
cðTÞqT :
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Proposition 6.1. If rank To2; then aðTÞ ¼
Bk;wp ðbðTÞ pk1 b ðTÞÞ: 2kðpk1 1Þ
ð6:1Þ
Proposition 6.2. If TASym2 ðZÞ is positive, then the following identity holds: aðTÞ ¼ ak;p ðbðTÞ pk1 b ðTÞÞ þ bk;p cðTÞ;
ð6:2Þ
where ak;p ¼
Bk;wp ; 2kðpk1 1Þ
bk;p ¼ ð1Þkþ1
ð1 þ p1k Þð1 p2k Þ Bk;wp : 1 p32k 2k pk
Proof of Proposition 6.1. We note that formula (6.1) is equivalent to ð1Þ G˜ pþ2k1 ¼ ðk;
2
Þ
Bk;wp ðgenus yð1Þ ðF Þ pk1 genus yð1Þ ðF ÞÞ: 2kðpk1 1Þ
ð6:3Þ
ð1Þ
The space of Eisenstein series for G0 ðpÞ is two dimensional. Moreover, the righthand side of (6.3) has the value 0 in the cusp 0. Therefore it must be proportional to the canonical Eisenstein series for the cusp N whose Fourier expansion is well known to be equal to the left-hand side of (6.3). & Proof of Proposition 6.2. First we note that the Fourier coefficient aðTÞ is given as follows: aðTÞ ¼
2ð1 wT wp ðpÞpk2 ÞBk1;wT wp f ðTÞ2k3 ð1 p2k3 ÞB2k2 Y Fqð2Þ ðT; qk wp ðqÞÞ;
ð6:4Þ
q : prime
where f ðTÞ ¼ f ðTÞ pordp ð f ðTÞÞ (cf. Theorem 2.2). Next we calculate the right-hand side of (6.2). By Siegel’s main theorem, we can write the ‘‘genus theta part’’ as bðTÞ pk1 b ðTÞ ¼
Y qpN
aq ðF ; TÞ pk1
Y qpN
aq ðF ; TÞ:
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Using 3
aN ðF ; TÞ ¼ 22k p2k1 jD0 ðTÞjk2 f ðTÞ2k3 p1 fð2k 2Þ!g1 ; 3
aN ðF ; TÞ ¼ 22k p2k1 jD0 ðTÞjk2 f ðTÞ2k3 p12k fð2k 2Þ!g1 ; aq ðF ; TÞ ¼ aq ðF ; TÞ ðqapÞ;
we get bðTÞ pk1 b ðTÞ 3
¼ 22k p2k1 jD0 ðTÞjk2 f ðTÞ2k3 fð2k 2Þ!g1 Y aq ðF ; TÞðp1 ap ðF ; TÞ pk1 p12k ap ðF ; TÞÞ: qap
Let us now calculate each factor in the right-hand side of the above identity. We already know that Y
aq ðF ; TÞ ¼
qap
ð1 wT wp ðpÞp1k ÞLðk 1; wT wp Þ 1 1 p22k Lðk; wp Þzð2k 2Þ Y ð2Þ Fq ðT; qk wp ðqÞÞ: q : prime
We note that the last factor p1 aq ðF ; TÞ pk1 p12k ap ðF ; TÞ has already been calculated in Corollary 5.3. Consequently, we get the following expression: bðTÞ pk1 b ðTÞ ¼ A B ðC þ Ca;b Þ; 3
A ¼ 22k p2k1 jD0 ðTÞjk2 f ðTÞ2k3 fð2k 2Þ!g1 ; B¼
1 wT wp ðpÞp1k Lðk 1; wT wp Þ 1 p22k Lðk; wp Þzð2k 2Þ 22k
C¼
2k
ð1 p Þð1 p Þ ; p2k1 ð1 p32k Þ
Y q : prime
Fqð2Þ ðT; qk wp ðqÞÞ;
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Ca;b ¼
8 > > > > > > > >
p33kþð32kÞðaþb1Þ=2 Zð1 pk1 Þð1 p2k ZÞð1 þ p1k ZÞ > > > > > 1 p32k > > :
if a b
ðmod 2Þ;
if acb
ðmod 2Þ: ð6:5Þ
Finally we calculate the Fourier coefficient cðTÞ: By Proposition 2.3, we have 3 1 wT wp ðpÞp1k Lðk 1; wT wp Þ detðTÞk2 22k 1p Lðk; wp Þzð2k 2Þ Y ð2Þ k Fq ðT; q wp ðqÞÞ
cðTÞ ¼ x1 2;k
q : prime 3
¼ ð1Þk 22k p2k1 fð2k 2Þ!g1 jD0 ðTÞjk2 f ðTÞ2k3
1 wT wp ðpÞp1k Lðk 1; wT wp Þ 1 p22k Lðk; wp Þzð2k 2Þ
Y
Fqð2Þ ðT; qk wp ðqÞÞ:
q : prime
A simple calculation shows that bk;p cðTÞ ¼ ak;p A B C:
ð6:6Þ
Hence the right-hand side of (6.2) becomes ak;p ðbðTÞ pk1 b ðTÞÞ þ bk;p cðTÞ ¼ ak;p A B ðC þ Ca;b Þ þ bk;p cðTÞ ¼ ak;p A B Ca;b
ðby ð6:5ÞÞ
ðby ð6:6ÞÞ:
Therefore, the proof of (6.2), our objective, is reduced to showing the following identity: aðTÞ ¼ ak;p A B Ca;b :
ð6:7Þ
To prove this, we first assume that D0 ðTÞc0 ðmod pÞ: In this case, a b ðmod 2Þ holds. Hence, by (6.5), we get ak;p A B Ca;b ¼
2Bk1;wT wp f ðTÞ2k3 ð1 p2k3 ÞB2k2
¼ aðTÞ
ðby ð6:4ÞÞ:
Y q : prime
Fqð2Þ ðT; qk wp ðqÞÞ
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Next we assume that D0 ðTÞ 0 ðmod pÞ: This is equivalent to acb ðmod 2Þ: If we put D0 ðTÞ ¼ p vðvAZÞ; then we have 3
22k p2k1 ðp jvjÞk2 f ðTÞ2k3 ; A¼ ð2k 2Þ! 1
3
B ¼ 222k p12k pk2 jvj2k k ð2k 2Þ!
Ca;b ¼
Bk1;wT wp Bk;wp B2k2
Y
1 wT wp ðpÞp1k 1 p22k
Fqð2Þ ðT; qk wp ðqÞÞ;
q : prime
p33k pordp ð f ðTÞÞ ð32kÞ Zð1 pk1 Þð1 p2k ZÞð1 þ p1k ZÞ : 1 p32k
Noting wT wp ðpÞ ¼ Z; we have ak;p A B Ca;b ¼
2ð1 wT wp ðpÞpk2 ÞBk1;wT wp f ðTÞ2k3 ð1 p2k3 ÞB2k2
Y
Fqð2Þ ðT; qk wp ðqÞÞ
q : prime
¼ aðTÞ: This shows that identity (6.7) holds in both cases. This completes the proof of Proposition 6.2. & Acknowledgments We would like to express our gratitude to the referee for his/her valuable comments. References [1] A.N. Andrianov, Quadratic Forms and Hecke Operators, Springer, Berlin, Heidelberg, New York, 1987. [2] S. Bo¨cherer, C.-G. Schmidt, p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier 50 (2000) 1375–1443. [3] F. Hirzebruch, D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Inv. Math. 36 (1976) 57–113. [4] H. Katsurada, An explicit formula for Siegel series, Amer. J. Math. 121 (1999) 415–452. [5] N. Koblitz, p-adic congruence and modular forms of half integral weight, Math. Ann. 274 (1986) 199–220. [6] S. Nagaoka, A remark on Serre’s example of p-adic Eisenstein series, Math. Z. 235 (2000) 227–250. [7] S. Nagaoka, On p-adic Siegel–Eisenstein series (joint work with H. Katsurada), in: T. Ibukiyama, W. Kohnen (Eds.), Proceedings of Japanese–German Seminar Explicit Structure of Modular forms and Zeta Functions, Hakuba, Japan, March 2002, pp. 52–57.
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[8] J.-P. Serre, Formes modulaires et fonctions zeˆta p-adiques, Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268. [9] G. Shimura, On Eisenstein series, Duke Math. J. 50 (1983) 417–476. [10] C.L. Siegel, U¨ber die analytische Theorie der quadratischen Formen, Ann. of Math. 36 (1935) 527–606. [11] J. Sturm, The critical values of zeta functions associated to the symplectic group, Duke Math. J. 48 (1981) 327–350. [12] T. Yang, An explicit formula for local densities of quadratic forms, J. Number Theory 72 (1998) 309–356.
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 118–131
http://www.elsevier.com/locate/jnt
Explicit upper bounds for values at s ¼ 1 of Dirichlet L-series associated with primitive even characters Ste´phane R. Louboutin Institut de Mathe´matiques de Luminy, UPR 9016, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France Received 13 May 2002; revised 20 January 2003 Communicated by A. Granville
Abstract Let S be a given finite set of pairwise distinct rational primes. We give an explicit constant kS such that for any even primitive Dirichlet character w of conductor qw 41 we have ( ( ) ) Y 1 Y wðpÞ 1 Lð1; wÞp ðlog qw þ kS Þ þ oð1Þ; 1 1 pAS 2 pAS p p where oð1Þ is an explicit error term which tends rapidly to zero when qw goes to infinity. r 2003 Elsevier Inc. All rights reserved. MSC: primary 11M20 Keywords: Dirichlet characters; L-function; Gauss sums
1. Introduction Let S be a given finite set of pairwise distinct rational primes. In [Lou1], by using integral representations of Dirichlet L-functions, we proved that there exists a computable constant kS such that for any primitive Dirichlet character w of
E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00159-8
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conductor qw 41 we have ( ( ) ) 1 Y Y wðpÞ 1 1 1 Lð1; wÞp ðlog qw þ kS Þ þ oð1Þ; 2 pAS pAS p p where oð1Þ is an explicit error term which tends to zero when qw goes to infinity. However, our proof did not yield good values for these constants kS (e.g. see Remark Pn Pa 7 below). In [Lou6, Theorem 4], by using bounds on the characters sums a¼1 b¼1 wðbÞ; we gave in the special case of even characters a new proof of this result yielding better values for kS : According to this new proof, for primitive even Dirichlet characters we may choose kS ¼ k0 þ o log 4 2
o X i¼1
1 log 1 pi
þ2
X log p ; p1 pAS
where k0 ¼ 2g 1 ¼ 0:154431; where p1 ; y; po are the primes pi AS which do not divide qw (we may have o ¼ 0) and where g ¼ 0:577? denotes Euler’s constant. Here, by generalizing the method introduced in [Lou5], we improve upon this result (see Theorem 5 below for a more complicated but explicit result which implies the following one, and note also that [Lou6, Theorem 1] is a special case of the present Theorem 1 (for o ¼ 0)): Theorem 1. Let S be a given finite set of pairwise distinct rational primes, and set keven :¼ 2 þ g logð4pÞ ¼ 0:046191?: Then, for any even primitive Dirichlet character w of conductor qw 41 we have ( ) Y wðpÞ Lð1; wÞ 1 pAS p ( ) ! X log p 1 Y 1 p 1 log qw þ keven þ o log 4 þ 2 þ oð1Þ; 2 pAS p p1 pAS where oX0 is the number of primes pAS which do not divide qw ; and where oð1Þ is an explicit error term which tends rapidly to zero when qw goes to infinity. Moreover, if S ¼ | or if S ¼ f2g; then this error term oð1Þ is always less than or equal to zero, and if none of the prime in S divides qw then this error term oð1Þ is less than or equal to zero for qw large enough. Corollary 2. Let w be an even primitive Dirichlet character of conductor qw 41: Set 8 > < k1 :¼ 2 þ g log ð4pÞ ¼ 0:04619? k2 :¼ 2 þ g log ðpÞ ¼ 1:43248? > : k3 :¼ 2 þ g log ðp=4Þ ¼ 2:81878?:
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Then, 8 > < ðlog qw þ k1 Þ=2 jLð1; wÞjp ðlog qw þ k2 Þ=4 > : ðlog qw þ k3 Þ=ð2j2 wð2ÞjÞ
in all cases; if qw is even; if qw is odd:
ð1Þ
In particular, if w is quadratic then 8 > < ðlog qw þ k1 Þ=2 Lð1; wÞp ðlog qw þ k2 Þ=4 > : ðlog qw þ k3 Þ=6
if wð2Þ ¼ þ1; if wð2Þ ¼ 0; if wð2Þ ¼ 1:
We refer the reader to [Le,Lou6, Section 5; Lou7,Mos,MP; Ram,SSW, Corollary 2] for various applications of such explicit bounds for jLð1; wÞj: We also refer the reader to [Lou4,Ram] for slight improvements on the first two bounds in (1). Of course, by using Burgess’ results it is possible to obtain better asymptotic bounds for jLð1; wÞj: For example, it follows from [Toy] that for any non-trivial quadratic Dirichlet character w modulo q a cube-free positive integer we have (
Y pAS
wðpÞ 1 p
)
( ) 1 þ oð1Þ Y 1 1 Lð1; wÞp log q 4 p pAS
(see also [GS,Pin] for even better bounds in the case that S ¼ |). Moreover, using [Gros] it is possible to have explicit values of these error terms oð1Þ: However, we will show in Section 5 below, that even in the simplest case that S ¼ | and q is prime, the error terms oð1Þ in such bounds Lð1; wÞpð14 þ oð1ÞÞlog q are not that small for reasonable values of q; and such asymptotic bounds are worse than the bound jLð1; wÞjpðlog qw þ keven Þ=2 for reasonable values of qw (say qw p1016 ). Therefore, contrary to our present bounds (1) used for example in [Lou7], such asymptotic bounds are of no practical use when dealing with class number problems for number fields. Finally, we mention that at the moment we do not have a satisfactory approach to obtain similar results for odd primitive Dirichlet characters (however, see [Lou3] for the case that S ¼ |). Here again, as in [Lou1,Lou2], our proof of Theorem 1 stems from the use of yfunctions to obtain integral representations of Dirichlet L-series. However, as we will be using non-primitive Dirichlet L-series, these y-functions do not satisfy simple functional equations. This will make our proof more complicated than the one given in [Lou2]. Our present proof is a generalization of the proof given in [Lou5] of the previous Corollary.
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1.1. Notation From now on, we let S be a given finite set of pairwise distinct rational primes. We let w be a primitive even Dirichlet character modulo qw 41; we set Y pX1 ð2Þ d1 :¼ pAS and gcdðp;qw Þ¼1
(which is square-free and relatively prime with qw ), we let o :¼ oðd1 ÞX0 denote the number of distinct prime factors of d1 (i.e. the number of primes pAS which do not divide qw ), we set Y pX1 ð3Þ d2 :¼ pAS and pjqw
Q (which is a square-free divisor of qw ), we set d :¼ d1 d2 ¼ pAS X1 (which is a squarefree divisor of qc ), and we let c be the even Dirichlet character modulo qc ¼ d1 qw induced by w: Hence, ( ) Y wðpÞ 1 Lð1; wÞ ¼ Lð1; cÞ: ð4Þ p pAS Notice that c is not primitive for d1 41: We let m and f denote the Mo¨bius and Euler * totient functions. Whenever DX1 is a positive square-free integer, we set fðDÞ ¼ 1 if Q * D ¼ 1 and fðDÞ ¼ pjD ðp 2Þ if D41:
2. First bound for Lðs; vÞ We have ðqc =pÞs=2 Gðs=2ÞLðs; cÞ ¼
Z
N
yðx; cÞxs=2
0
dx x
ðRðsÞ41Þ;
where yðx; cÞ ¼
X
2
cðnÞepn x=qc ¼
nX1
1X 2 cðnÞepn x=qc 2 nAZ
ðx40Þ
ð5Þ
(for c is even). Hence, for any a40 (to be suitably chosen later on (see (14) below)) it holds that: ðqc =pÞs=2 Gðs=2ÞLðs; cÞ ¼
Z
N
1=a
yð1=x; cÞxs=2
dx þ x
Z a
N
yðx; cÞxs=2
dx x
ðRðsÞ41Þ: ð6Þ
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Lemma 3. 1. (See [MV, Lemma 5.4]). Set tðwÞ :¼
qw X
wðaÞe2pia=qw :
a¼1
Then, for any bAZ it holds that tb ðcÞ :¼
qc X
cðaÞe2piab=qc ¼ mðd1 Þwðd1 Þmðdb Þfðdb Þw% ðbÞt1 ðwÞ;
a¼1
where db :¼ gcdðb; d1 Þ: In particular, it holds that pffiffiffiffiffi jtb ðcÞ= qw j ¼
if gcdðb; qw Þ41; if gcdðb; qw Þ ¼ 1:
0 fðdb Þ
ð7Þ
2. Set yðxÞ ¼
X
2
epn x
ðx40Þ:
nX1
Then, jyðx; cÞjp
X
mðdÞyðd2 x=qc Þ
ðx40Þ
ð8Þ
djd
and jyð1=x; cÞjp
X pffiffiffiffiffiffiffiffiffiffi X * 1Þ x=d1 mðd2 Þyðd21 d22 x=qc Þ fðd d1 jd1
ðx40Þ:
d2 jd2
Proof. Since cðnÞ ¼ 0 for gcdðn; dÞ41; we have X jyðx; cÞjp expðpn2 x=qc Þ nX1 gcdðn;dÞ¼1
¼
X
X
mðdÞexpðpn2 x=qc Þ
nX1 djn and djd
¼
X djd
mðdÞ
X
expðpðmdÞ2 x=qc Þ;
mX1
and (8) follows. Now, set yðx; a; qÞ ¼
X bAZ
epðaþbqÞ
2
x=q
ðx40 and q40Þ:
ð9Þ
ARTICLE IN PRESS S.R. Louboutin / Journal of Number Theory 104 (2004) 118–131
123
The Poisson summation formula yields 1 X 2piab=q pb2 =qx e e yðx; a; qÞ ¼ pffiffiffiffiffiffi qx bAZ
ðx40Þ:
Using (5), we obtain: yðx; cÞ ¼
qc X 2 1X cða þ bqc Þepðaþbqc Þ x=qc 2 a¼1 bAZ
qc 1X 1 X 2 ¼ cðaÞyðx; a; qc Þ ¼ pffiffiffiffiffiffiffiffi tb ðcÞepb =qc x 2 a¼1 2 qc x bAZ
and yð1=x; cÞ ¼
pffiffiffiffiffiffiffiffiffiffi X tb ðcÞ pb2 x=q c x=d1 pffiffiffiffiffi e qw bX1
(for c is even). To get (9), we notice that X X X 2 * 1Þ mðd2 Þyðd21 d22 x=qc Þ ¼ ab epb x=qc fðd d1 jd1
bX1
d2 jd2
with ab ¼
X 0
¼@
X
* 1 Þmðd2 Þ fðd
d1 jd1 ;d2 jd2 and d1 d2 jb
X d1 jgcdðb;d1 Þ
10
* 1 Þ A@ fðd
X
1 mðd2 ÞA ¼
d2 jgcdðb;d2 Þ
0
if gcdðb; d2 Þ41;
fðgcdðb; d1 ÞÞ if gcdðb; d2 Þ ¼ 1
pffiffiffiffiffi (for gcdðd1 ; d2 Þ ¼ 1) and use (7) to obtain 0pjtb ðcÞ= qw jpab (for d2 divides qw ). & According to (8) and (9) of Lemma 3, the integral representation (6) is valid for all s in the complex plane. For s ¼ 1; using (4) we obtain: ( ) Y wðpÞ 1 Lð1; wÞpI1 þ I2 ; pAS p where 1 I1 :¼ pffiffiffiffiffi qc
Z
N
1=a
dx jyð1=x; cÞj pffiffiffi x x
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and 1 I2 :¼ pffiffiffiffiffi qc
Z
N
a
dx jyðx; cÞj pffiffiffi x
and a40 is to be suitably chosen in due course (see (14) below).
3. Proof of Theorem 1 To compute bounds on I1 and I2 we now use (8) and (9) of Lemma 3 and the following Lemma 4 which will enable us to get simple formulae: Lemma 4. Let DX1 be a positive square-free rational integer. Let tX0 denote its number of prime divisors. It holds that X 1 if D ¼ 1; mðdÞ ¼ eD :¼ 0 if D41; djD X
mðdÞlog d ¼ ZD :¼
djD
log p
if D ¼ p is prime;
0
otherwise;
X mðdÞ Y 1 fðDÞ ¼ ; 1 ¼ d p D djD pjD X mðdÞ fðDÞ X log p log d ¼ ; d D pjD p 1 djD X fðdÞ Y * 1 fðDÞ t ; ¼2 1 ¼ 2t p D d djD pjD X djD
0
1 X log p * A: fðdÞlog d ¼ fðDÞ@log D p1 pjD
Proof. Use induction on the number t of distinct prime factors of D:
&
1. Let us first give an upper bound on I2 : Using (8) and the functional equation yð1=xÞ ¼
pffiffiffi pffiffiffi xyðxÞ þ ð x 1Þ=2
ðx40Þ
ð10Þ
ARTICLE IN PRESS S.R. Louboutin / Journal of Number Theory 104 (2004) 118–131
125
(see [Lan, Chapter XIII]) and setting Z keven :¼ 2 1
N
dx yðxÞ pffiffiffi þ 2 x
Z
N
yðxÞ 1
dx ¼ 2 þ g log ð4pÞ x
ð11Þ
(see [Lou2, (3) and Lemme p. 12]), we obtain: Z N 1 dx I2 ¼ pffiffiffiffiffi jyðx; cÞj pffiffiffi qc a x Z N X 1 dx p mðdÞ pffiffiffiffiffi yðd2 x=qc Þ pffiffiffi qc a x djd Z N X mðdÞ dx ¼ yðxÞ pffiffiffi d ad2 =qc x djd ! Z qc =ad2 X mðdÞ Z N dx dx yðxÞ pffiffiffi þ yð1=xÞ pffiffiffi ¼ d x x x 1 1 djd ! Z Z N qc =ad2 pffiffiffi X mðdÞ keven x1 dx pffiffiffi dx yðxÞ þ ¼ d x 2 2x x qc =ad2 1 djd 0 sffiffiffiffiffiffiffiffiffi1 X mðdÞ keven Z N dx 1 d1 qw ad2 A @ ¼ : yðxÞ þ log 1þ 2 d x 2 2 d1 qw ad d1 qw =ad2 djd Hence, using Lemma 4, we obtain 0 1 X log p fðdÞ@ A þ R2 ; log ðd1 qw =aÞ þ keven 2 þ 2 I2 p 2d p1 pjd where R2 ¼
rffiffiffiffiffiffiffiffiffi X mðdÞ Z N dx a yðxÞ þ ed 2 d x d qw 1 d1 qw =ad djd
is an explicit error term which tends to zero when qw goes to infinity. 2. In the same way, using (9) and the functional equation (10), we obtain: 1 ILL1 ¼ pffiffiffiffiffi qc
Z
N 1=a
dx jyð1=x; cÞj pffiffiffi x x
ð12Þ
ARTICLE IN PRESS S.R. Louboutin / Journal of Number Theory 104 (2004) 118–131
126
Z N X 1 X* dx p pffiffiffiffiffiffiffiffiffiffi mðd2 Þ yðd21 d22 x=qc Þ fðd1 Þ x d1 qc d1 jd1 1=a d2 jd2 Z N X X 1 dx * 1Þ ¼ pffiffiffiffiffiffiffiffiffiffi mðd2 Þ yðxÞ fðd 2 2 x d1 qc d1 jd1 d1 d2 =aqc d2 jd2 ! Z aqc =d21 d22 Z N X 1 X * dx dx ¼ pffiffiffiffiffiffiffiffiffiffi mðd2 Þ yðxÞ þ yð1=xÞ fðd1 Þ x x d1 qc d1 jd1 1 1 d2 jd2 ! Z N Z aqc =d21 d22 pffiffiffi X x1 1 X* keven dx dx mðd2 Þ yðxÞ pffiffiffi þ ¼ pffiffiffiffiffiffiffiffiffiffi fðd1 Þ 2x 2 x d1 qc d jd aqc =d21 d22 1 d jd 1
1
2
2
Z N X 1 X* keven dx ¼ pffiffiffiffiffi mðd2 Þ yðxÞ pffiffiffi þ fðd1 Þ 2 2 d1 qw d jd 2 x ad1 qw =d1 d2 d jd 1
1
2
2
!! sffiffiffiffiffiffiffiffiffiffiffiffi ad1 qw 1 ad1 qw : 1 log 2 d21 d22 d21 d22
Using Lemma 4, we obtain I1 p2o
fðdÞ d
rffiffiffiffiffi a þ R1 d1
ð13Þ
where !! Z N X 1 X * keven dx 1 ad1 qw R1 ¼ pffiffiffiffiffi mðd2 Þ yðxÞ pffiffiffi 1 log fðd1 Þ d1 qw d jd 2 2 x d21 d22 ad1 qw =d21 d22 d2 jd2 1 1 Z N X 1 X * dx ¼ pffiffiffiffiffi mðd2 Þ yðxÞ pffiffiffi fðd1 Þ 2 2 d1 qw d jd x ad1 qw =d1 d2 d2 jd2 1 1 0 1 X log p Zd2 ed2 fðd1 Þ@ Aþ p þ pffiffiffiffiffi keven 2 log ðad1 qw Þ þ 2 log d1 2 ffiffiffiffiffi fðd1 Þ 2 qw d1 d1 qw p1 pjd 1
is an explicit error term which tends to zero when qw goes to infinity. 3. Hence, using (12) and (13), we finally obtain ( ) Y wðpÞ Lð1; wÞ 1 pAS p ( ) ! rffiffiffiffiffi X log p 1 Y 1 d1 qw a oþ1 p 1 þ keven 2 þ 2 log þ R; þ2 2 pAS p d1 p1 a pAS where R ¼ R1 þ R2 is an explicit error term which tends to zero when qw goes to pffiffiffiffiffiffiffiffiffiffi infinity. To get the term log a þ 2oþ1 a=d1 as small as possible, we now choose a ¼ d1 =4o and obtain the following explicit result which proves Theorem 1:
ð14Þ
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Theorem 5. It holds that ( ( ! ) ) 1 Y Y X log p wðpÞ 1 Lð1; wÞp log ð4o qw Þ þ keven þ 2 þ R; 1 1 2 pAS pAS p p p1 pAS where R is the following explicit error term which tends to zero when qw goes to infinity: X mðdÞ Z N * 1 Þ X mðd2 Þ Z N 1 X fðd dx 2 dx o yð4 qw x=d Þ o yðd12 qw x=4o d21 d22 Þ pffiffiffi R¼ d x 2 d d x 1 d jd 2 1 1 djd d jd 1
0
1
2
2
1 X log p Z 2 fðd1 Þ ed ed2 fðd1 Þ@ A þ pdffiffiffiffi ffi þ pffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffi keven 2 log ðqw =4o Þ 2 : ffi o qw d1 p1 4 qw 2 qw d1 pjd1
If d2 ¼ 1 or if d2 is prime, then Rp0 for qw large enough. If d2 41 is not prime, then R¼
X mðdÞ Z N * 1 Þ X mðd2 Þ Z N dx 1 X fðd dx yð4o qw x=d2 Þ o yðd12 qw x=4o d21 d22 Þ pffiffiffi d x 2 d d x 1 2 1 1 djd d jd d jd 1
1
2
2
tends rapidly to zero when qw goes to infinity, for R ¼ Od ðq1w ecpqw Þ with c ¼ minð4o =d 2 ; 1=4o d22 Þ: Proof. Use Z Z N dx yðAxÞ x p 1
N 1
yðAxÞpdxffiffixp
Z 1
N
yðAxÞdx ¼
X nX1
1 pn2 A p pA e p e : pn2 A 6A
&
4. Some explicit special cases 1. Assume that d1 ¼ 1 and d ¼ d2 X1: Then o ¼ 0 and X mðdÞ Z N pffiffiffi dx ed Zd R¼ ð1 þ xÞyðqw x=d2 Þ þ pffiffiffiffiffiðkeven log qw Þ þ pffiffiffiffi ffi: d 1 x 2 qw qw djd In particular, for d ¼ 1 we have ed ¼ 1; Zd ¼ 0 and qw X54ekeven yields Rp0; which proves the first bound in (1) of Corollary 2. In the same way, if d ¼ 2 we have ed ¼ 0; Zd ¼ log 2 and qw X54ekeven yields Z Z N pffiffiffi 1 N dx log 2 log 2 Rp ð1 þ xÞyð5x=4Þ pffiffiffi p yðxÞ dx pffiffiffi 2 1 x 5 5 1 pn2 X log 2 e ¼ pffiffiffi þ p0; pn2 5 nX1 which proves the second bound in (1) of Corollary 2.
ARTICLE IN PRESS 128
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2. Assume that d ¼ d1 X1 and d2 ¼ 1: Then Z N X fðdÞ X mðdÞ Z N * dx dx R¼ yð4o qw x=d2 Þ 2o yðd 2 qw x=4o d2 Þ pffiffiffi d 1 x d 1 x djd djd 0 1 X log p ed fðdÞ A: ffi þ pffiffiffiffiffi@keven 2 log ðqw =4o Þ 2 þ pffiffiffiffiffiffiffiffiffi p1 4o qw 2d qw pjd In particular, for d ¼ 2 we have ed ¼ 0 and o ¼ 1; and qw X5 yields Z 1 N dx 1 yðqw xÞ þ pffiffiffiffiffiðkeven 2 log qw Þ Rp 2 1 x 4 qw Z 1 N 1 p yðqw xÞ þ pffiffiffiffiffiðkeven 2 log qw Þ 2 1 4 qw 2 1 1 X epqw n ¼ pffiffiffiffiffiðkeven 2 log qw Þ þ p0; 4 qw 2qw nX1 pn2 which proves the third bound in (1) of Corollary 2. 3. Assume that d1 ¼ 3 and d2 ¼ 2: Then o ¼ 1; d ¼ 6; ed ¼ 0; ed2 ¼ 1 and Zd2 ¼ log 2; and qw X5 yields Z Z Z 1 N dx 1 N dx 1 N dx yðqw xÞ þ yð4qw x=9Þ þ yðqw x=16Þ pffiffiffi Rp 2 1 x 3 1 x 3 1 x 1 þ pffiffiffiffiffiðkeven 2 log qw log 3Þ 3 qw Z 7 N 1 p yðqw x=16Þdx þ pffiffiffiffiffiðkeven 2 log qw log 3Þ 6 1 3 qw 2 1 56 X epqw n =16 p0: ¼ pffiffiffiffiffiðkeven 2 log qw log 3Þ þ 3 qw 3qw nX1 pn2 Hence, we obtain: Corollary 6. Set k :¼ 2 þ g logðp=12Þ ¼ 3:91739?: Let w be an even primitive Dirichlet character of conductor qw 41; and assume that wð2Þ ¼ 0 but wð3Þ ¼ 1: Then, jLð1; wÞjpðlog qw þ kÞ=8: Remark 7. This result should be compared with the previously known two bounds on jLð1; wÞj for even primitive Dirichlet characters of conductors qw 41 and such that wð2Þ ¼ 0 but wð3Þ ¼ 1: the first one quoted in [Le, Lemma 3] (whose proof stems from the use of [Lou1] (see the proof of [Lou6, (1)])) according to which jLð1; wÞjpðlog qw þ k0 Þ=8 where k0 ¼ logð216Þ þ 8 ¼ 13:37527?; and the second one given in [Lou6, (11)] according to which jLð1; wÞjpðlog qw þ k00 Þ=8 where k00 ¼ 6:
ARTICLE IN PRESS S.R. Louboutin / Journal of Number Theory 104 (2004) 118–131
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5. Better asymptotic bounds Throughout this section, we let Pw denote a primitive Dirichlet character modulo pX8731 a prime. We set Sw ðnÞ ¼ nk¼1 wðkÞ: Then, for any arbitrary positive integer r it holds that ( jSw ðnÞjp
ðtrivial boundÞ;
n 11=ðrþ1Þ 1=4r
35n
p
ðlog pÞ
1=ð2rþ2Þ
ðBurgess’ bound ðsee½Gros; Theorem 1; p: 1177 ÞÞ:
Hence, for X X1 real, it holds that X S ðnÞ w jLð1; wÞj ¼ p nX1 nðn þ 1Þ
X 1pnpX þ1
1 nþ1
!
X
þ
n4X þ1
! jSw ðnÞj : nðn þ 1Þ
Since X 1pnpX þ1
1 2 p 1 þ g þ log X þ nþ1 X
(where g ¼ 0:577215? denotes Euler constant), and since X jSw ðnÞj Z N p1=4r ðlog pÞ1=ð2rþ2Þ 1=ð2rþ2Þ 11=ðrþ1Þ 1=4r p 35n p ðlog pÞ dn ¼ 35ðr þ 1Þ ; n2 X 1=ðrþ1Þ X n4X þ1 we obtain
jLð1; wÞjp 1 þ g þ log X þ
2 p1=4r ðlog pÞ1=ð2rþ2Þ þ 35ðr þ 1Þ : X X 1=ðrþ1Þ
ð15Þ
Since the Burgess bound is better than the trivial bound for nXXp;r :¼ ð35Þrþ1 pðrþ1Þ=4r
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi log pX1225 p1=4 log p;
we choose X ¼ Xp;r for which 35ðr þ 1Þp1=4r ðlog pÞ1=ð2rþ2Þ X 1=ðrþ1Þ ¼ r þ 1: Using (15) with these values of X ; we obtain jLð1; wÞjplog Xp;r þ r þ g þ
2 ¼ Xp;r
1 1 þ þ or ð1Þ log p: 4 4r
ARTICLE IN PRESS S.R. Louboutin / Journal of Number Theory 104 (2004) 118–131
130
It remains to choose the integer rX1 to obtain the best possible bound. Set c1 ¼ log ð35Þ þ 1; c2 ¼ logð35Þ þ g and 1 1 1 þ hðrÞ :¼ log Xp;r þ r þ g ¼ log p þ log log p þ c1 r þ c2 : 4 4r 2 Then hðrÞphðrp Þ ¼ 14 log p þ
qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p c1 log p þ 2 log log p þ c2 where r ¼ rp :¼ log 4c1 ; for
which 2=Xp;rp ¼ oð1Þ: Choosing r equal to the greatest integer less than or equal to rp ; we finally obtain: Theorem 8. Set c1 ¼ logð35Þ þ 1 ¼ 4:555348? and c2 ¼ logð35Þ þ g ¼ 4:132563?: If w is a primitive Dirichlet character of prime conductor pX8731; then pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 jLð1; wÞjp log p þ c1 log p þ log log p þ c2 þ oð1Þ 4 2
ð16Þ
where the error term oð1Þ is explicit and tends to zero as p goes to infinity. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Now, since in the range 9731ppp5 1051 the main term 14 log p þ c1 log p þ 1 2 log log p þ c2 of this bound (16) is greater than the worst of our explicit bounds ðlog p þ keven Þ=2 on jLð1; wÞj; this asymptotic bound (16) is useless when it comes to practical application of bounds for Lð1; wÞ (as in [Le,MP,Mos,SSW]).
References [Gros] [GS] [Lan] [Le] [Lou1]
[Lou2] [Lou3] [Lou4] [Lou5] [Lou6] [Lou7] [MV]
E. Grosswald, On Burgess’ bound for primitive roots modulo primes, Amer. J. Math. 103 (1981) 1171–1183. A. Granville, K. Soundararajan, Upper bounds for jLð1; wÞj; Quart. J. Math. 53 (2002) 265–284. S. Lang, Algebraic Number Theory, Graduate Texts in Mathematics, Vol. 110, 2nd Edition, Springer, New York, 1994. M. Le, Upper bounds for class numbers of real quadratic fields, Acta Arith. 68 (1994) 141–144. S. Louboutin, Majoration au point 1 des fonctions L associe´es aux caracte`res de Dirichlet primitifs, ou au caracte`re d’une extension quadratique d’un corps quadratique imaginaire principal, J. Reine Angew. Math. 419 (1991) 213–219. S. Louboutin, Majorations explicites de jLð1; wÞj; C. R. Acad. Sci. Paris 316 (1993) 11–14. S. Louboutin, Majorations explicites de jLð1; wÞj (suite), C. R. Acad. Sci. Paris 323 (1996) 443–446. S. Louboutin, Majorations explicites de jLð1; wÞj (troisie`me partie), C. R. Acad. Sci. Paris 332 (2001) 95–98. S. Louboutin, Majorations explicites de jLð1; wÞj (quatrie`me partie), C. R. Acad. Sci. Paris 334 (2002) 625–628. S. Louboutin, Explicit upper bounds for jLð1; wÞj for primitive even Dirichlet characters, Acta Arith. 101 (2002) 1–18. S. Louboutin, Explicit lower bounds for residues at s ¼ 1 of Dedekind zeta functions and relative class numbers of CM-fields, Trans. Amer. Math. Soc. 355 (2003) 3079–3098. H.L. Montgomery, R.C. Vaughan, The exceptional set in Goldbach’s problem, Acta Arith. 27 (1975) 353–370.
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C. Moser, Nombre de classes d’une extension cyclique re´elle de Q de degre´ 4 ou 6 et de conducteur premier, Math. Nachr. 102 (1981) 45–52. C. Moser, J.-J. Payan, Majoration du nombre de classes d’un corps cubique de conducteur premier, J. Math. Soc. Japan 33 (1981) 701–706. J. Pintz, Elementary methods in the theory of L-functions VII. Upper bound for Lð1; wÞ; Acta Arith. 32 (1977) 397–406. O. Ramare´, Approximate formulae for Lð1; wÞ; Acta Arith. 100 (2001) 245–256. R.G. Stanton, C. Sudler, H.C. Williams, An upper bound for the period of the simple continued pffiffiffiffi fraction for D; Pacific J. Math. 67 (1976) 525–536. M. Toyoizumi, Remarks on the upper bound for Lð1; wÞ; Acta Arith. 55 (1990) 137–140.
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 132–155
http://www.elsevier.com/locate/jnt
Irrationality results for values of generalized Tschakaloff series II Masaaki Amoua,1 and Masanori Katsuradab,,2 b
a Department of Mathematics, Gunma University, Tenjin-cho, Kiryu 376-8515, Japan Department of Mathematics, Hiyoshi Campus, Keio University, Hiyoshi, Yokohama 223-8521, Japan
Received 12 August 2002; revised 21 April 2003 Communicated by D. Goss
Abstract The study of irrationality properties of values of the generalized Tschakaloff series f ðxÞ defined by (1.2) below was initiated by Duverney (Portugal. Math. 53(2) (1996) 229; Period. Math. Hungar. 35 (1997) 149), and continued by the authors (J. Number Theory 77 (1999) 155). The present paper proceeds to extend our previous result (Amou and Katsurada, 1999, Theorem). The irrationality of f ðaÞ for any aAQ\f0g is proved in a quantitative form under fairly general growth conditions on the coefficients of f ðxÞ (Theorem 1), while the same result is shown in a certain ‘limiting’ situation of Theorem 1, at the cost of loosing a quantitative aspect (Theorem 2). The linear independence of certain values of a system of f ðxÞ is also obtained (Theorem 3). The key idea in proving our previous result is a Mahler’s transcendence method, due to Loxton and van der Poorten (in: A. Baker, D.W. Masser (Eds.), Transcendence Theory: Advances and Applications, Academic Press, San Diego, 1977, pp. 211–226), applied to an appropriate sequence of functions (see (2.4) and (2.5)). In order to establish Theorems 1 and 2, this method is enhanced by a certain technique which allows us to improve zero estimates for the remainder terms of Pade´-type approximations (see Lemmas 3 and 4). r 2003 Elsevier Inc. All rights reserved. MSC: primary 11J72; secondary 11J82 Keywords: Irrationality; Irrationality measure; q-Difference equation; Siegel’s lemma; Pade´ approximation
Corresponding author. E-mail addresses: [email protected] (M. Amou), [email protected] (M. Katsurada). 1 Supported in part by Grant-in-Aid for Scientific Research (No. 13640007), The Ministry of Education, Science, Sports, Culture of Japan. 2 Supported in part by Grant-in-Aid for Scientific Research (No. 13640041), The Ministry of Education, Science, Sports, Culture of Japan. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00143-4
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1. Introduction The arithmetical study of values of the entire function Tq ðxÞ ¼
N X
xn
n¼0
qnðnþ1Þ=2
;
where q is a complex parameter, jqj41; was initiated by Tschakaloff [Ts] in 1921. He modified the method of Perron [Pe] (of which origin dates back to Hermite [He]) so as to be applied to the q-difference equation Tq ðqxÞ ¼ xTq ðxÞ þ 1;
ð1:1Þ
in order to construct rational approximations of Tq ðxÞ; and then proved the linear independence over Q of the values of Tq ðxÞ at rational points under a certain assumption on qAQ: Subsequent research have been made in various aspects; we refer the reader to the introductions of [AK,Du2,VW] for the details. After his work [Du1,Du2] on the values of certain generalizations of Tq ðxÞ; Professor Daniel Duverney kindly suggested, in personal correspondence with the first author (dated June 28, 1997), several problems of investigating arithmetical properties of values of the generalized Tschakaloff series f ðxÞ ¼
N X sn u0 u1 ?un xn ; t v v ?vn qnðnþ1Þ=2 n¼0 n 0 1
ð1:2Þ
where q is an integer, jqjX2; and fsn gN n¼0 ;
ftn gN n¼0 ;
fun gN n¼0 ;
fvn gN n¼0
ð1:3Þ
are nonzero integer sequences. A partial solution of one of the problems above was given in the authors’ previous paper [AK], in which it was proved, by means of a refinement of the method developed in [Du2], a quantitative irrationality result for f ðxÞ at rational points under certain conditions on (1.3) (see Remark of Theorem 1). In fact, the particular case of (1.2) in the form Hq ðxÞ ¼
N X
xn
n¼0
n!qnðnþ1Þ=2
was introduced and studied already in 1991 by Haas [Ha], who proved the linear independence over Q of certain values of Hq ðxÞ and its derivatives in a quantitative form. The present paper proceeds further with our study on the values of the generalized Tschakaloff series (1.2), for which we shall prove three main results (Theorems 1, 2 and 3 below). It is the first aim of the present paper to improve our previous result [AK, Theorem], even under more general conditions on (1.3). Let l:c:m:ðt0 ; y; tn Þ denote the (positive) least common multiple of t0 ; y; tn : We shall prove
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Theorem 1. Let q be an integer, jqjX2; and let f ðxÞ be given by (1.2). Suppose that the nonzero integer sequences in (1.3) satisfy jsn j; jtn j; jvn jpexpðc0 nd Þ;
jun jpexpðc0 nZ Þ;
l:c:m:ðt0 ; t1 ; y; tn Þpexpðc0 n1þZ Þ
ð1:4Þ ð1:5Þ
for all nX1; where c0 is a positive constant and 0odo1;
ð1:6Þ
1 0pZomin ; 1 d : 2
ð1:7Þ
Then, for any aAQ\f0g; f ðaÞ is irrational. Furthermore, let m ¼ maxðd; ZÞ;
ð1:8Þ
and n ¼ max 2;
1 : 1 m
ð1:9Þ
Then the lower estimate f ðaÞ M 4expf cðlog NÞ2n=ð1þnÞ g N
ð1:10Þ
holds for any integers M and N with NX2; where c is a positive constant depending only on f and a: Remark. Our previous result [AK, Theorem] asserts that, instead of (1.10), f ðaÞ M 4expf cðlog NÞ8=5 g N by assuming jsn j; jtn j; jun j; jvn jpexpðc0 nd Þ for all nX1 with 0odo1=4; Theorem 1 improves the exponent 8/5 to 4/3 under the assumption above with 0odo1=2: In [Du2, Theorem 2], the irrationality of the values of f ðxÞ at rational points was proved when sn ¼ tn ¼ un ¼ 1 ðnX0Þ; by assuming certain divisibility properties of fvn g and a growth condition such as jvn j ¼ eoðnÞ ðn-NÞ: (The result above is proved in a more general setting.) This suggests that the assertion of Theorem 1 remains
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valid under the ‘limiting’ situation d-1–0 of (1.4), (1.5), and (1.7). We can in fact prove, at the cost of loosing a quantitative aspect, Theorem 2. Let q be an integer, jqjX2; and let f ðxÞ be given by (1.2). Suppose that the nonzero integer sequences in (1.3) satisfy jsn j; jtn j; jvn jpexpfoðnÞg;
jun jpexp c0 ;
ð1:11Þ
l:c:m:ðt0 ; t1 ; y; tn Þpexpðc0 nÞ
ð1:12Þ
for all nX1; where c0 is a positive constant. Then, for any aAQ\f0g; f ðaÞ is irrational. Remarks. (1) The proof of Theorem 2 is not a straightforward modification of that of Theorem 1. Hence it is impossible to obtain the corresponding quantitative assertion by simply letting m-1–0 (and so n-N) in (1.10). (2) If one considers the ‘boundary’ case d ¼ 1 of Theorem 1, the range of Z in (1.7) becomes empty; even in this case the assertion of Theorem 1 fails for the function f ðxÞ ¼ 1 þ Tq ðxÞ Tq ðqxÞ ¼ 1 þ
N X
ð1 qn Þ
n¼1
xn qnðnþ1Þ=2
;
since f ð1Þ ¼ 0 by (1.1). One of the main ideas in proving [AK, Theorem] was a refinement of Mahler’s transcendence method, due to Loxton and van der Poorten [LP], applied to the sequence of functions which satisfy a chain of functional equations (see (2.4) and (2.5) below). In order to establish Theorems 1 and 2, our previous frame of the proof is enhanced by a new technique which allows us to improve zero estimates for the remainder terms of Pade´-type approximations (see the proofs of Lemmas 3 and 4). It is the second aim of the present paper to study linear independence of the values N of a system of generalized Tschakaloff series. Let fsn; j gN n¼0 and ftn; j gn¼0 ð j ¼ 1; y; mÞ be sequences of nonzero integers, and let gj ðxÞ be the functions defined by gj ðxÞ ¼
N X sn; j u0 u1 ?un xn t v v ?vn qnðnþ1Þ=2 n¼0 n; j 0 1
ð1:13Þ
for j ¼ 1; y; m: Theorem 3. Let m and q be integers with mX2 and jqjX2; and let gj ðxÞ ð1pjpmÞ be given by (1.13). Suppose that the nonzero integer sequences in (1.13) satisfy jsn; j j; jtn; j j; jvn jpc0 ng ;
jun jpexp c0 ;
l:c:m:ðt0; j ; t1; j ; y; tn; j Þpexpðc0 nÞ
ð1:14Þ ð1:15Þ
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for all nX1 and j ¼ 1; y; m; where c0 and g are positive constants. Let a1 ; y; am be nonzero rational numbers such that ai ajqjk for any i; jAf1; y; mg with iaj; and any kAZ: ð1:16Þ a j Then the numbers 1, gj ðaj Þ ð1pjpmÞ are linearly independent over the rationals. Furthermore, the lower estimate m X Hj gj ðaj Þ4expf cðlog HÞ4m=ð2mþ1Þ g ð1:17Þ H0 þ j¼1 holds for any ðH0 ; y; Hm ÞAZmþ1 with H :¼ maxðjH1 j; y; jHm jÞX2; where c is a positive constant depending only on the gj ’s and aj ’s. Remark. When m ¼ 1; the proof of Theorem 3 is reduced to that of [AK, Theorem], which gives (1.17) with the exponent 8=5 instead of 4m=ð2m þ 1Þ: For simplicity, we shall prove Theorem 3 supposing mX2: Let jðxÞ be a function holomorphic near x ¼ 0; its hth derivative is denoted by jðhÞ ðxÞ ðh ¼ 0; 1; yÞ; and its kth primitive jð kÞ ðxÞ ðk ¼ 0; 1; yÞ is defined inductively by Z x jð kÞ ðxÞ ¼ jð kþ1Þ ðxÞ dx ðkX1Þ; jð0Þ ðxÞ ¼ jðxÞ: 0
Then from Theorem 3 we can show the following corollary. Corollary. Let q be a positive integer with jqjX2; and f ðxÞ given by (1.2). Let c and m be integers with cX0 and mX2: Suppose that the nonzero integer sequences in (1.3) satisfy jsn j; jtn j; jvn jpc0 ng ;
jun jpexp c0 ;
l:c:m:ðt0 ; t1 ; y; tn Þpexpðc0 nÞ
ð1:18Þ ð1:19Þ
for all nX1: Then, for any a1 ; y; am AQ\f0g satisfying (1.16), the numbers 1 and f ðiÞ ðaj Þ ð cpipc; 1pjpmÞ are linearly independent over the rationals. The assertion of this corollary is in particular valid for Tq ðxÞ and Hq ðxÞ: Remarks. (1) It seems difficult to prove Theorem 3 under a more general assumption that ai aqk aj
for any i; jAf1; y; mg with iaj; and any kAZ;
ð1:20Þ
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instead of (1.16); this is in fact best possible in the case of Tschakaloff series, since, if (1.20) fails, the numbers 1 and Tq ðaj Þ ð1pjpmÞ become linearly dependent by (1.1). (2) The following results (relevant to higher derivatives and primitives) are to be compared with our corollary. The linear independence of the values of Tq ðxÞ; together with its primitives and derivatives of any order, at a single nonzero rational point was first established by Duverney [Du2, The´ore`me 1], while the main result of Haas [Ha,Satz] asserts the lower estimate for the linear form K0 þ Pc Pm ðiÞ i¼ c j¼1 Ki; j Hq ðaj Þ with K0 ; Ki; j AZ; by assuming (1.20) and certain divisibility properties of aj ð1pjpmÞ; q and K0;1 : Theorems 1 and 2 are proved in the next section. The last section is devoted to showing Theorem 3 and its corollary. Because the basic frame of the proofs are unaltered from our previous work, we reuse the notations introduced in [AK, Section 2]. The authors would like to thank the referee for valuable comments and refinements of the first version of the present paper.
2. Irrationality We divide this section into five subsections; the first four subsections are devoted to the proof of Theorem 1, while Theorem 2 is proved in the last subsection. Throughout the following sections, the letters r and h are assigned for integer parameters, a for nonzero rational number, and ci ði ¼ 1; y; 32Þ for positive constants depending at most on f and a; but not depending on r and h: The notation j5c means that jjjpCc holds for some constant C (not depending on r and h), and further j^c that both j5c and c5j hold. 2.1. Main proposition (1) Theorem 1 is a direct conclusion of the following proposition, the proof of which will begin at Section 2.3, while the next subsection is prepared for describing its outline. Proposition 1. For any rXc1 ; there exist integers Mr and Nr such that expðc2 r2n ÞpjNr jpexpðc3 r2n Þ
ð2:1Þ
expð c4 r1þn ÞpjNr f ðaÞ Mr jpexpð c5 r1þn Þ;
ð2:2Þ
and
where n is defined by (1.9) with (1.8).
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The deduction of Theorem 1 is as follows. Let M and N be any integers with NX2: Then, by the second inequality in (2.2), f ðaÞ M X Mr M f ðaÞ Mr N N N N r
r
1 1 expð c5 r1þn Þ X NjNr j jNr j for any Mr =Nr aM=N with rXc1 ; and hence we have f ðaÞ M X1 1 ; N 2 NjNr j
ð2:3Þ
provided that Mr =Nr aM=N and Noð1=2Þ expðc5 r1þn Þ: In view of (2.1) and (2.3) we may take c1 so large that the estimates Mr pexpð c7 r2n Þ expð c6 r2n Þp f ðaÞ Nr hold for any rXc1 : Hence there is an integer r0 with ror0 5r such that Mr0 =Nr0 aMr =Nr : We can therefore choose an integer r satisfying the conditions above with r^ðlog NÞ1=ð1þnÞ : This, together with (2.1) and (2.3), concludes the assertion of Theorem 1. 2.2. Outline of the proof The sequence of entire functions fh ðxÞ ¼
N X snþh Uh ðnÞ xn t V ðnÞ qnðnþ1Þ=2 n¼0 nþh h
ðhX0Þ;
ð2:4Þ
where Uh ðnÞ ¼ uh u1þh ?unþh ;
Vh ðnÞ ¼ vh v1þh ?vnþh ;
plays a fundamental roˆle throughout the following. Note that f0 ðxÞ ¼ f ðxÞ; and fh ðxÞ satisfies the chain of functional equations uh s h fh ðqxÞ ¼ þ xfhþ1 ðxÞ ðhX0Þ: ð2:5Þ vh t h In order to prove Proposition 1 we first construct Pade´-type approximations of the form Qr;h ðxÞfh ðxÞ Pr;h ðxÞ ¼ xsðr;hÞ Tr;h ðxÞ;
ð2:6Þ
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where Pr;h ðxÞ and Qr;h ðxÞ are polynomials of degree at most 2r with integer coefficients, sðr; hÞ is an integer at least 3r; and Tr;h ðxÞ is an entire function satisfying Tr;h ð0Þa0: The construction of (2.6) is carried out by applying Siegel’s lemma, which also gives upper estimates for the absolute values of the coefficients of Pr;h ðxÞ and Qr;h ðxÞ: The repeated use of (2.5) yields the relation xh fh ðq h xÞ ¼ Gh f ðxÞ Fh ðxÞ ðhX0Þ;
ð2:7Þ
where Fh ðxÞ ¼
h X
xh i qið2h iþ1Þ=2
i¼1
sh i Vh iþ1 ði 2Þ th i Uh iþ1 ði 2Þ
ð2:8Þ
and Gh ¼ qhðhþ1Þ=2
V0 ðh 1Þ U0 ðh 1Þ
ð2:9Þ
(see [AK, Lemma 4]). Writing a ¼ A=B with relatively prime integers A and B; we find from (2.7) (with x ¼ a) that Ah dh fh ðq h aÞ ¼ nh f ðaÞ mh
ð2:10Þ
with the integers dh ¼ l:c:m:ðt0 ; y; th 1 ÞU0 ðh 1Þ;
mh ¼ Bh dh Fh ðaÞ;
nh ¼ Bh dh Gh :
ð2:11Þ
We next substitute x ¼ q h a in (2.6), multiply both sides by ðqh BÞ2r Ah dh ; and then use (2.10) to obtain Nr;h f ðaÞ Mr;h ¼ ðqh BÞ2r Ah dh ðq h aÞsðr;hÞ Tr;h ðq h aÞ
ð2:12Þ
with the integers Mr;h ¼ ðqh BÞ2r fQr;h ðq h aÞmh þ Ah dh Pr;h ðq h aÞg
ð2:13Þ
Nr;h ¼ ðqh BÞ2r Qr;h ðq h aÞnh :
ð2:14Þ
and
The desired rational approximations (see Proposition 1) are obtained by (2.12) with a sequence of h’s which are suitably chosen with h^rn (see Lemma 3 in Section 2.4). The approximation (2.12) is also utilized to prove Theorem 2. In this case, h is chosen with h^maxðr2 ; cðrÞÞ for each r; where cðrÞ is defined by (2.31) and (2.32) (see Lemma 4 in Section 2.6).
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2.3. Construction of auxiliary functions Lemma 1. Let r and h be arbitrary. Then for the function fh ðxÞ there exist auxiliary polynomials Pr;h ðxÞ ¼
2r X
ai xi ;
Qr;h ðxÞ ¼
i¼0
2r X
bi x i ;
ð2:15Þ
i¼0
not both zero, with integer coefficients ai and bi ; such that 2
m
max ðjai j; jbi jÞpec8 ðr þrh Þ ;
ð2:16Þ
sðr; hÞX3r:
ð2:17Þ
0pip2r
and sðr; hÞ in (2.6) satisfies
Moreover, if we suppose sðr; hÞp4r þ 1; then we have the following estimates for Tr;h ðxÞ in (2.6): þh2Z Þ
1 for jxjp 2
2
m
Þ
for jxjpe c11 ðr þrh
jTr;h ðxÞjpec9 ðr þrh
ð2:18Þ
and 2
jTr;h ðxÞjXe c10 ðr þrh
d
2
m
þh2Z Þ
:
ð2:19Þ
Remark. The temporary assumption for sðr; hÞ above is shown to be valid (on some sequence of h’s) in Lemma 3 below. Proof. Let Pr;h ðxÞ and Qr;h ðxÞ be given in the form (2.15). Then Qr;h ðxÞfh ðxÞ Pr;h ðxÞ ¼
N X
Ln xn ;
ð2:20Þ
n¼0
where Ln ¼
minð2r;nÞ X i¼0
bi
sn iþh Uh ðn iÞ ðn iÞðn iþ1Þ=2 q an tn iþh Vh ðn iÞ
ðn ¼ 0; 1; 2; yÞ;
ð2:21Þ
with the convention that an ¼ 0 for n42r: Hence to satisfy (2.17) we solve a system of 3r linear homogeneous equations Ln ¼ 0 ðn ¼ 0; 1; y; 3r 1Þ with 4r þ 2 unknowns ai and bi ði ¼ 0; y; 2rÞ: By setting D ¼ th t1þh t3r 1þh Vh ð3r 1Þqð3r 1Þð3rÞ=2 ;
ð2:22Þ
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the system can be written as DLn ¼
minð2r;nÞ X
Dn;i bi Dan ¼ 0
ðn ¼ 0; 1; y; 3r 1Þ
ð2:23Þ
i¼0
with the integer coefficients D and Dn;i ; where Dn;i ¼ D
sn iþh Uh ðn iÞ ðn iÞðn iþ1Þ=2 q : tn iþh Vh ðn iÞ
ð2:24Þ
Using (1.4) and noting (1.6), we obtain d
jDjpe6rc0 ð3r 1þhÞ jqj9r
2
=2
pec12 ðr
2
þrhd Þ
and for any n and i with 0pipnp3r 1 d
Z
2
jDn;i jpjDjec0 ð3r 1þhÞ þ3rc0 ð3r 1þhÞ pec13 ðr þrh
m
Þ
by (1.8). Siegel’s lemma (cf. [Ba, p. 13, Lemma 1]) therefore ensures the existence of the required solution. We now suppose s :¼ sðr; hÞp4r þ 1 to prove (2.18) and (2.19). From (2.6) and (2.20) we have N X Tr;h ðxÞ ¼ Lnþs xn : ð2:25Þ n¼0
The expression (2.21), together with (1.4) and (2.16), gives the estimates jLnþs jp
2r X
jbi jjsnþs iþh jjUh ðn þ s iÞjjqj ðnþs iÞðnþs iþ1Þ=2
i¼0 2
m
d
Z
p ð2r þ 1Þec8 ðr þrh Þ ec0 ðnþsþhÞ ec0 ðnþsþ1ÞðnþsþhÞ jqj ðnþs 2rÞ 2
p ec14 ðr þrh
m
2
=2
þnhZ Þ c15 n2
for all nX0; by using (1.6)–(1.8) and s5r: Here the exponent of the last upper bound (as a function of n) is asymptotically equal to c15 n2 as n-N; and it attains its maximum at n ¼ c14 hZ =2c15 with the value c14 ðr2 þ rhm Þ þ ðc14 hZ Þ2 =4c15 ^r2 þ rhm þ h2Z : Hence the estimate 1 2 m 2Z jLnþs jp ec16 ðr þrh þh Þ ð2:26Þ 2 follows for all nX0: From this and (2.25) we obtain (2.18). We next write Tr;h ð0Þð¼ Ls Þ ¼ Js =Ks ; where Js and Ks are relatively prime integers with Ks 40: Then from (2.21) we have Ks pjts 2rþh ?tsþh jjVh ðsÞjjqjsðsþ1Þ=2 ;
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and hence by (1.4) d 1 2 d Ks pec0 ðsþ2rþ2ÞðhþsÞ jqjsðsþ1Þ=2 p ec17 ðr þrh Þ ; 2
where the last inequality follows by (1.6) and s5r: We use (2.25), together with this estimate and (2.26), to obtain for jxjp1=2 jTr;h ðxÞjX jLs j jxj
N X
jLnþ1þs jjxjn
n¼0
X 2e
c17 ðr2 þrhd Þ
2
jxjec16 ðr þrh
m
þh2Z Þ
:
Hence (2.19) holds with some sufficiently large c11 : &
2.4. Estimation of sðr; hÞ Let Pr;h ðxÞ and Qr;h ðxÞ be polynomials satisfying the assertion of Lemma 1. The following crude bound for sðr; hÞ is used to find a sharp upper bound for sðr; hÞ on a suitable sequence of h’s. Lemma 2. For any r and h we have sðr; hÞpc18 ðr2 þ rhm Þ: Proof. Our task is to look for an integer n ðX3rÞ as small as possible for which Ln given by (2.21) does not vanish. Since Qr;h ðxÞ is neither the null polynomial nor a monomial by (2.17), there exist integers k and c with koc such that bk bc a0 and bi ¼ 0 for all iok and i4c: Then using (2.21) with nX3r; we have sn cþh Uh ðn cÞ ðn cÞðn cþ1Þ=2 jqj jLn jX jbc j tn cþh Vh ðn cÞ c 1 X sn iþh Uh ðn iÞ ðn iÞðn iþ1Þ=2 jqj jbi j t V ðn iÞ n iþh h i¼k Uh ðn cÞ ðn cÞðn cþ1Þ=2 sn cþh 0 jqj X t Ln ; Vh ðn cÞ n cþh where L0n
¼
c k X j¼1
sn cþjþh Un cþ1þh ð j 1Þ fðn cþ1Þþ?þðn cþjÞg jqj jbc j j : tn cþjþh Vn cþ1þh ð j 1Þ
It follows from (1.4), (1.6)–(1.8), (2.16), and cp2r the upper estimates 2
m
d
Z
2
jL0n jp2rec8 ðr þrh Þ ec0 ðnþhÞ e2rc0 ðnþhÞ jqj ðn 2rþ1Þ pec19 ðr þrh
m
þrnZ Þ c20 n
;
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d
while the lower estimate jsn cþh =tn cþh j4e c0 ðnþhÞ follows by (1.4). Comparing the resulting bounds, we obtain Ln a0 if nbmaxðr2 ; rhm ; rnZ Þ: Since Zo1=2 by (1.7), the condition nbrnZ is superseded by nbr2 ; and hence the assertion follows. & For a function fðxÞ holomorphic near x ¼ 0; its order of the zeros at x ¼ 0 is denoted by ord fðxÞ: Lemma 3. For any r and any aXc21 there exists an h in the range ½arn php2½arn
ð2:27Þ
such that sðr; hÞp4r þ 1 holds, where n is defined by (1.9). Proof. Let Rr;h ðxÞ be the remainder of the approximation (2.20), namely, Rr;h ðxÞ :¼ Qr;h ðxÞfh ðxÞ Pr;h ðxÞ ¼ xsðr;hÞ Tr;h ðxÞ: Then it follows from (2.7) that xh Rr;h ðq h xÞ ¼ Qr;h ðq h xÞfGh f ðxÞ Fh ðxÞg xh Pr;h ðq h xÞ ¼ Qr;h ðq h xÞGh f ðxÞ fQr;h ðq h xÞFh ðxÞ þ xh Pr;h ðq h xÞg: For any integers h and k with 0ohok; we define the determinant Dðh; kÞ by Q ðq h xÞG Qr;h ðq h xÞFh ðxÞ þ xh Pr;h ðq h xÞ h r;h Dðh; kÞ :¼ Qr;k ðq k xÞGk Qr;k ðq k xÞFk ðxÞ þ xk Pr;k ðq k xÞ Q ðq h xÞG xh Rr;h ðq h xÞ h r;h ¼ : Qr;k ðq k xÞGk xk Rr;k ðq k xÞ We first prove that sðr; hÞp4r þ 1 holds for an h in the range Hphp2H; upon assuming DðH; 2HÞa0 for some positive integer H: (Such an H is specified in the last step of the proof.) Let H be fixed as above. Then we have Dðh; h þ 1Þa0 for some h with Hphp2H 1: In fact, if this is not the case, the linear space spanned over QðxÞ by the nonzero vectors ðQr;h ðq h xÞBh ; Qr;h ðq h xÞAh ðxÞ þ xh Pr;h ðq h xÞÞ ðHphp2HÞ is one-dimensional. But this contradicts that DðH; 2HÞa0: We can therefore choose such an h: Then, from the definitions of Dðh; kÞ we have ord Dðh; h þ 1ÞXminðsðr; hÞ þ h; sðr; h þ 1Þ þ h þ 1Þ; and ord Dðh; h þ 1Þpdeg Dðh; h þ 1Þph þ 4r þ 1:
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A comparison of these two bounds gives minðsðr; hÞ; sðr; h þ 1Þ þ 1Þp4r þ 1; and hence h or h þ 1 has the desired property. We remark that the estimate for sðr; hÞ in Lemma 2 is irrelevant to the argument up to the present stage. We next look for a positive integer H as small as possible such that DðH; 2HÞa0 holds. Since ord Rr;2H ðq 2H xÞX3r and ord Qr;2H ðq 2H xÞp2r; the difference ord fx2H Rr;2H ðq 2H xÞQr;H ðq H xÞg ord fxH Rr;H ðq H xÞQr;2H ðq 2H xÞg is greater than H sðr; HÞ: Hence DðH; 2HÞa0 holds if HXsðr; HÞ: It follows from Lemma 2 that the last condition is fulfilled for any H satisfying c18 ðr2 þ rH m ÞpH: We can therefore find c21 such that, for any aXc21 ; DðH; 2HÞa0 holds with H ¼ ½arn by (1.9). This proves the lemma. &
2.5. Conclusion of the proof of Theorem 1 We now proceed to the proof of Proposition 1, while preserving the notations in Section 2.2. Let Pr;h ðxÞ and Qr;h ðxÞ be polynomials satisfying the assertion of Lemma 1, and let h be in the range (2.27) for any r with a :¼ c22 Xc21 ; so that the estimates 3rpsðr; hÞp4r þ 1
ð2:28Þ
are now valid. Setting a ¼ A=B as before, we consider (2.12) with (2.13) and (2.14). Our task is to estimate jNr;h f ðaÞ Mr;h j from above and below. Note that 1þZ
1pjdh jpec23 h
ð2:29Þ
holds by the definition of dh in (2.11) with (1.4) and (1.5), and that 2
jqjh =2 pjnh jpjqjh
2
=2 c24 h1þZ
ð2:30Þ
e
holds by the definition of nh in (2.11) with (2.9), (1.4), and (1.5). Since h^rn by (2.27), we have hbmaxðr2 ; rhm ; h2Z Þ; and hence we may suppose jq h ajpminð1=2; e c11 ðr
2
þrhm þh2Z Þ
Þ
(see (2.18) and (2.19)), by taking c21 (in Lemma 3) sufficiently large. Then it follows from (2.12), (2.18), (2.28), and (2.29) that 1þZ
2
jNr;h f ðaÞ Mr;h jp jqj hr jAjhþ4rþ1 ec23 h ec9 ðr þrh 2
p jqj hr ec25 ðr þrh
m
þh1þZ þh2Z Þ
m
þh2Z Þ
pjqj hr=2
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if hbr; where the last condition is, by nX2 (see (1.9)), fulfilled by taking c21 large enough. On the other hand, from (2.12), (2.19), (2.28), and (2.29) we obtain 2
jNr;h f ðaÞ Mr;h jX jqj hð2rþ1Þ jBj ð2rþ1Þ e c10 ðr þrh 2
d
Þ
d
X jqj hð2rþ1Þ e c26 ðr þrh Þ Xjqj 3hr if hbr; where the last condition is again fulfilled by taking c21 large enough. We combine the results above to conclude that, for any r; there is an h ¼ hðrÞ in the range (2.27) (with a ¼ c21 ) such that e 3hr pjNr;h f ðaÞ Mr;h jpe hr=2 holds, where Nr;h a0 since Mr;h is an integer. Denoting Mr ¼ Mr;h ; Nr ¼ Nr;h with these Mr;h ; Nr;h we obtain (2.1) and (2.2), and this completes the proof of Proposition 1. Theorem 1 is thus established. 2.6. Proof of Theorem 2 The first condition in (1.11) is restated as that jsn j; jtn j; jvn jpexpfjðnÞng
ð2:31Þ
for all nX1; where jðnÞ is a positive function satisfying jðnÞ ¼ oð1Þ as n-N: For the proof of Theorem 2 we define an associated function cðrÞ by jðkÞpEðrÞ=r
for all kXcðrÞ;
ð2:32Þ
where EðrÞ is any positive function satisfying EðrÞ ¼ oðrÞ as r-N: The preliminary results for the proof of Theorem 2 are collected in the following lemma. Lemma 4. For any large rb1 there exist an h in the range ½c27 maxðr2 ; cðrÞÞ php2½c27 maxðr2 ; cðrÞÞ ;
ð2:33Þ
and auxiliary polynomials Pr;h ðxÞ and Qr;h ðxÞ for fh ðxÞ (as in (2.6)), not both zero, with integer coefficients ai and bi ; such that 2
max ðjai j; jbi jÞpec28 ðr þEðrÞhÞ
ð2:34Þ
3rpsðr; hÞp4r þ 1;
ð2:35Þ
0pip2r
and
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where sðr; hÞ is defined by (2.6). Moreover, we have 0ojTr;h ðq h aÞjpec29 ðr
2
þEðrÞhÞ
ð2:36Þ
:
Proof. We follow the argument given in the previous subsections with suitable modifications. Let r and h be arbitrary with hXcðrÞ: We solve a system of 3r linear homogeneous equations with 4r þ 2 unknowns, which has the same form as (2.23). In the present case, the estimate for jDj (see (2.22)) is, by (2.31) and (2.32), ! 3r 1þh X jDjp exp 2 jðkÞk jqjð3r 1Þð3rÞ=2 k¼h
pe
6EðrÞð3r 1þhÞ
jqj9r
2
=2
pec30 ðr
2
þEðrÞhÞ
:
Next if we estimate jDn;i j (see (2.24)) by using the second inequality in (1.11), (2.31), and (2.32), the extra factor eð3r 1þhÞEðrÞ=rþ3c0 r appears but is absorbed into the resulting upper bound for jDj above. Siegel’s lemma therefore ensures the existence of a solution of this system such that (2.34) and the first inequality in (2.35) are valid. Also the upper estimate sðr; hÞpc31 ðr2 þ EðrÞhÞ
ð2:37Þ
is obtained similar to the proof of Lemma 2. We can further show that (2.36) holds if 2
jq h ajpminð1=2; e c32 ðr þEðrÞhÞ Þ;
ð2:38Þ
similar to (2.18) and (2.19). It remains to show that we can choose an h in the range (2.33) for which the second inequality in (2.35) and condition (2.38) are valid. From the proof of Lemma 3 (except the last step) we have sðr; hÞp4r þ 1 for some h with Hphp2H; where H is any integer satisfying HXsðr; HÞ: Here the last condition and (2.38) with HXcðrÞ are, in view of (2.37) and rb1; fulfilled by taking H ¼ ½c27 maxðr2 ; cðrÞÞ : This completes the proof. & We can now conclude the proof of Theorem 2. Let a ¼ A=B for integers A and B with B40: We suppose that f ðaÞ ¼ A0 =B0 for some integers A0 and B0 with B0 40; to obtain the contradiction. As was shown in Section 2.2, relation (2.12) holds with (2.13) and (2.14). Then multiplying B0 both sides of (2.12), we have B0 fNr;h f ðaÞ Mr;h g ¼ B0 ðqh BÞ2r Ah dh ðq h aÞs Tr;h ðq h aÞ: The left-hand side of this equality is an integer since Mr;h ; Nr;h and B0 f ðaÞ integers, while the right-hand side does not vanish by the first inequality in Contrary to this fact, the right-hand side is less than 1 in absolute value sufficiently large r: In fact, from (1.11), (1.12), the second inequality in
are all (2.36). for all (2.36),
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and (2.38), it does not exceed jB0 ðqh BÞ2r dh ðq h aÞs Tr;h ðq h aÞjpB0 B2r jajs jAjh e2c0 h jqj ðs 2rÞh ec29 ðr
2
þEðrÞhÞ
;
which with s^r (see (2.35)) implies the claim above for all large rb1 (with an h in the range (2.33)). This contradiction terminates the proof of Theorem 2.
3. Linear independence Throughout this section, let mX2 be an integer, let aj ð1pjpmÞ be nonzero rational numbers, and let ci ði ¼ 33; y; 49Þ denote positive constants depending at most on the gj ðxÞ’s and aj ’s. This section, divided into three subsections, is aimed at proving Theorem 3. The main proposition (Proposition 2), which directly concludes Theorem 3, is stated in the next subsection, and its proof is given in Sections 3.2 and 3.3. The last subsection is devoted to showing corollary. 3.1. Main proposition (2) We write aj ¼ Aj =Bj ð1pjpmÞ with relatively prime integers Aj and Bj ; and define g˜ j ðxÞ ¼
N s u u ?u Anþ1 X j;n 0 1 n j n¼0
xn
qnðnþ1Þ=2 tj;n v0 v1 ?vn Bnþ1 j
ð1pjpmÞ:
For the proof of Theorem 3 we introduce the sequence of functions g˜ h; j ðxÞ ðhX0Þ defined by g˜ h; j ðxÞ ¼
N X snþh; j Uh; j ðnÞ xn ; t V ðnÞ qnðnþ1Þ=2 n¼0 nþh; j h; j
where ¼ Uh ðnÞAnþ1 ; Uh; j ðnÞ ¼ uh u1þh ?unþh Anþ1 j j ¼ Vh ðnÞBnþ1 Vh; j ðnÞ ¼ vh v1þh ?vnþh Bnþ1 j j with the notations in (2.5). Note that g˜ 0; j ðxÞ ¼ g˜ j ðxÞ: In view of the relations g˜ j ðxÞ ¼ aj gj ðaj xÞ ð1pjpmÞ; it is sufficient for establishing Theorem 3 to prove the linear independence over Q of the numbers 1 and g˜ j ð1Þ for j ¼ 1; 2; y; m: Proposition 2. For any rXc33 there exist mðm þ 1Þ linear forms Li; j ¼ Ni g˜ j ð1Þ Mi; j
ð0pipm; 1pjpmÞ
ð3:1Þ
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with integer coefficients Ni and Mi; j ; such that jNi jpec34 r jLi; j jpe c35 r
4m
2mþ1
ð0pipmÞ;
ð3:2Þ
ð0pipm; 1pjpmÞ
ð3:3Þ
M0;m y M1;m a0: y y y Mm;m
ð3:4Þ
and N0 N 1 D :¼ y N
m
M0;1 M1;1 y Mm;1
y
The deduction of Theorem 3 from this proposition is as follows. Let ðH0 ; y; Hm ÞAZmþ1 be arbitrary with H :¼ maxðjH1 j; y; jHm jÞX2; and let L ¼ H0 þ
m X
Hj g˜ j ð1Þ
j¼1
be a linear form. Then for any iAf0; 1; 2; y; mg; by (3.1), Ni L ¼ L i þ
m X
Hj Li; j ;
L i ¼ H0 Ni þ
j¼1
m X
Hj Mi; j :
j¼1
It follows from (3.4) that Li a0 for some iAf0; 1; y; mg: For such an i we have jNi LjX1 mH max jLi; j j; 1pjpm
2mþ1
and hence, by (3.3), jNi LjX1=2 holds for all H satisfying Hpec35 r (3.2) implies, by taking r as r^ðlog HÞ1=ð2mþ1Þ ; that
=2m: This with
4m=ð2mþ1Þ 1 : jLjX jNi j 1 4e cðlog HÞ 2
Assertion (1.17) of Theorem 3 therefore follows. 3.2. Construction of simultaneous approximations The purpose of this subsection is to construct successive simultaneous approximations for the numbers g˜ j ð1Þ with 1pjpm (Lemma 6), which is fundamental in proving the linear independence of 1 and g˜ j ð1Þ ð1pjpmÞ: The
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basic frame of the construction is a natural extension of that given in Section 2.3, so the details will be omitted. Lemma 5. Let r and h be arbitrary. Then for the functions g˜ h; j ðxÞ ð1pjpmÞ there exist auxiliary polynomials Pr;h; j ðxÞ ¼
mr X
ai; j xi ;
Qr;h ðxÞ ¼
i¼0
mr X
bi xi ;
ð3:5Þ
i¼0
not all zero, with integer coefficients ai; j and bi ; such that 2
max ðjai; j j; jbi jÞpec36 ðr þr log hÞ
ð3:6Þ
ðm þ 1Þrpsðr; h; jÞpc37 ðr2 þ r log hÞ;
ð3:7Þ
0pipmr 1pjpm
and
where sðr; h; jÞ is defined by Qr;h ðxÞg˜ h; j ðxÞ Pr;h; j ðxÞ ¼ xsðr;h; jÞ Tr;h; j ðxÞ;
Tr;h; j ð0Þa0:
ð3:8Þ
1 2
ð3:9Þ
Moreover we have the estimates 2
jTr;h; j ðxÞjpec38 ðr þr log hÞ
for jxjp
and 2
2
jTr;h; j ðxÞjXe c39 ðr þr log hÞ
for jxjpe c40 ðr
2
þr log hÞ2
:
ð3:10Þ
Proof. We construct Pade´-type approximations (of the second kind) for the functions g˜ h; j ðxÞ; and then deduce the assertions of the lemma, without the upper estimate for sðr; h; jÞ in (3.7), similarly to the proof of Lemma 2. & Lemma 6. For any r and any hbr4 ; there exist integers Mr;h; j and Nr;h such that 2
2
ec41 h pjNr;h jpec42 h ;
ð3:11Þ
Lr;h; j ¼ Nr;h g˜ j ð1Þ Mr;h; j
ð3:12Þ
and the approximation forms
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satisfy 2
e c43 ðr þr log hÞh pjLr;h; j jpe c44 rh
ð3:13Þ
for j ¼ 1; y; m: Proof. Let Pr;h; j ðxÞ for 1pjpm and Qr;h; j ðxÞ be polynomials satisfying the assertion of Lemma 5 with hbr4 : Then the desired approximation forms are given by Lr;h; j ¼
m Y
th 1;i Bhi U0 ðh 1Þðaj q sðr;h; jÞþmr Þh Tr;h; j ðq h Þ;
ð3:14Þ
i¼1
where th 1; j ¼ l:c:m:ðt0; j ; y; th 1; j Þ: In fact, by using the functional relation between g˜ h; j ðq h xÞ and g˜ j ðxÞ as in (2.10), we can express Lr;h; j in the form (3.12) with some integers Mr;h; j and Nr;h : Estimates (3.13) therefore follow from (1.14), (1.15), (3.7), (3.9), and (3.10), and this with (3.12) implies Nr;h a0 since Mr;h; j is an integer. Then estimates (3.11) follow from (1.14), (1.15), and (3.6). Lemma 6 is proved. & In order to prove Proposition 2, a set of m þ 1 linearly independent (i.e. satisfying (3.4)) approximation forms Lr;hi ; j ðh0 oh1 o?ohm Þ should be selected. This is achieved by choosing suitable hi ’s with hi ^r2m ; which also gives (3.2) and (3.3) by (3.11) and (3.13), respectively. The abbreviations Ni ¼ Nr;hi ;
Mi; j ¼ Mr;hi ; j ;
Li; j ¼ Lr;hi ; j
and Ti; j ¼ Tr;hi ; j
ð3:15Þ
will be used hereafter, and hence with these hi ’s (3.12) is rewritten as Li; j ¼ Ni g˜ j ð1Þ Mi; j for 0pipm and 1pjpm: 3.3. Nonvanishing of the determinant We preserve the notations introduced in the previous subsections. In this subsection we show the nonvanishing of the determinant D (see Lemma 9), formed by a set of suitably selected simultaneous approximations of 1 and g˜ j ð1Þ with 1pjpm (see Lemma 7). A key to showing this nonvanishing is Lemma 8, which isolates the diagonal term (as a dominant part) from other terms of D: Lemma 7. For any given real number a40; there is a constant b40; depending only on m; c37 (see (3.7)), c40 (see (3.10)) and a; for which the following four properties are valid. There exist positive integers h0 ; h1 ; y; hm such that (i) hi Xc40 ðr2 þ r log hi Þ2 =logjqj for 0pipm; (ii) br2m phi p2br2m for 0pipm;
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(iii) hi Xahiþ1 for 0pipm 1; (iv) sðr; h0 ; jÞ ¼ sðr; h1 ; jÞ ¼ ? ¼ sðr; hm ; jÞ for 1pjpm: Remark. The number a will be chosen so that the assertions of Lemmas 8 and 9 hold. Proof. The number of m-tuples of indices ðsðr; h; 1Þ; y; sðr; h; mÞÞ with h varying in the range br2m php2br2m is not greater than X ðrÞ :¼ fc37 ðr2 þ r logð2br2m Þgm : Dirichlet’s box principle therefore yields, if ½br2m =a 4ðm þ 1ÞX ðrÞ; the existence of positive integers hi ð0pipmÞ satisfying properties (i)–(iv) above. & Let hi ð0pipmÞ be positive integers having properties (i)–(iv) of Lemma 7, and put sj :¼ sðr; h0 ; jÞ mr ¼ ? ¼ sðr; hm ; jÞ mr
ð3:16Þ
for j ¼ 1; y; m: Noting (1.16) we may assume without loss of generality that ja1 q s1 j4ja2 q s2 j4?4jam q sm j by a suitable reordering of a1 ; y; am : From (3.1) the determinant D in (3.4) is rewritten as N0 N 1 D¼ y N m
L0;1 L1;1 y Lm;1
L0;m ? L1;m ; y y ? Lm;m ?
and assertions (3.2) and (3.3) of Proposition 2 hold by Lemma 6. Our task is therefore to show the nonvanishing of the determinant on the right-hand side. Lemma 8. Let Sm denote the mth permutation group, and let i be its identity element. Then for any pASm \fig; we have L1;1 L2;2 ?Lm;m c h Xe 45 m : L 1;pð1Þ L2;pð2Þ ?Lm;pðmÞ
Proof. From (3.14), noting (3.15) and (3.16), we have Li; j ¼
m Y k¼1
thi 1;k Bhki U0 ðhi 1Þðaj q sj Þhi Ti; j ðq hi Þ:
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We now suppose that pð jÞ ¼ j for all j ¼ 1; 2; y; k 1; but pðkÞak for some kAf1; 2; y; mg: Then T ?T sk hk sm hm ða q Þ ?ða q Þ k m k;k m;m ¼ ; m;pðmÞ ðapðkÞ q spðkÞ Þhk ?ðapðmÞ q spðmÞ Þhm Tk;pðkÞ ?Tm;pðmÞ
L1;1 ?Lm;m L ?L 1;pð1Þ
where we write Ti; j ¼ Ti; j ðq hi Þ for brevity. It is easy to see that the first absolute value on the right-hand side attains its minimum when pðkÞ ¼ k þ 1; pðk þ 1Þ ¼ k and pð jÞ ¼ j for all jXk þ 2; and this implies that L1;1 ?Lm;m L ?L 1;pð1Þ
m;pðmÞ
ak q sk hk hkþ1 Tk;k ?Tm;m X a q skþ1 T ?T kþ1
pðkÞ;k
pðmÞ;m
:
Since jak q sk =akþ1 q skþ1 j is not smaller than ai ai inf jqjn ; jqjn 41; 1pi; jpm 1; iaj; nAZ ; aj aj which is expressed as ec46 by (1.16), and the lower estimates Tk;k ?Tm;m T ?T
2 2 m Y 2 e c39 ðr þr log hi Þ 2 X Xe c47 ðr þr log h0 Þ c38 ðr2 þr log hi Þ e m;pðmÞ i¼k
k;pðkÞ
follow by (3.9), (3.10) (see (i) of Lemma 7), and (iii) of Lemma 7. L1;1 ?Lm;m L ?L 1;pð1Þ
m;pðmÞ
c ðh h Þ c ðr2 þr log h Þ2 0 Xe 46 k kþ1 47 :
Then the assertion holds by taking að51Þ in Lemma 7 large enough.
&
We can now prove the nonvanishing of D: Lemma 9. We have Da0: Proof. Let Di be the ði; 0Þth cofactor of D: Then it follows from Lemma 8 and the first inequality in (3.13) that jD0 jX jL1;1 ?Lm;m j
X
jL1;pð1Þ ?Lm;pðmÞ j
pASm \fig
X jL1;1 ?Lm;m j ðm! 1Þe c45 hm jL1;1 ?Lm;m j
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153
1 X jL1;1 ?Lm;m j 2 1 2 2 X e c43 fðr þr log h1 Þh1 þ?þðr þlog hm Þhm g 2 2
X e c48 ðr þr log h1 Þh1 by (iii) of Lemma 7, while the upper estimate jDi jpm!e c44 mrhm holds for 1pipm by the second inequality in (3.13). We therefore obtain, from the resulting bounds above and Lemma 6, m X jNi jjDi j jDjX jN0 jjD0 j i¼1
Xe
c41 h20 c48 ðr2 þr log h1 Þh1
2
mec42 h1 m!e c44 mrhm
2
4 ec49 h0 ; provided that að51Þ in Lemma 7 is taken sufficiently large. Lemma 9 is proved. & The proof of Proposition 3 is now complete. Theorem 3 is thus established. 3.4. Proof of Corollary Let f ðxÞ; q; c; m and aj ð1pjpmÞ be as in the Corollary, and write aj ¼ Aj =Bj with relatively prime integers. We shall prove the nonvanishing of the linear form L ¼ K0 þ
c m X X
Ki; j f ðiÞ ðaj Þ
ð3:17Þ
i¼ c j¼1
for any K0 ; Ki; j AZ with Ki; j being not all zero. We may assume without loss of P n generality that ðK c; j ; y; Kc; j Þa0 for all j ¼ 1; y; m: Let f ðxÞ ¼ N n¼0 cn x be the Maclaurin series for f ðxÞ in (1.2). Then from the definitions we have f ðhÞ ðxÞ ¼
N X
nðn 1Þ?ðn h þ 1Þcn xn h
n¼h
and f ð kÞ ðxÞ ¼
N X n¼0
cn xnþk ðn þ 1Þðn þ 2Þ?ðn þ kÞ
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for h; k ¼ 0; 1; y: These expressions are combined to give xi f ðiÞ ðxÞ ¼
N X n¼0
n! cn x n ðn iÞ!
ð3:18Þ
for any integer i; with the convention that 1=ðn iÞ! ¼ 0 if 0pnoi: Defining the entire functions gj ðxÞ ¼
c X
i ðiÞ Ki; j a i j x f ðxÞ ð1pjpmÞ;
i¼ c
we see from (3.17) that L ¼ K0 þ
m X
gj ðaj Þ:
j¼1
It follows from (3.18) that the Maclaurin series for gj ðxÞ is given by gj ðxÞ ¼ PN n n¼0 Cn; j x ; where Cn; j ¼ cn
c X i¼ c
¼
n!cn dn; j n! Ki; j a i j ¼ ðn iÞ! ðn þ cÞ!Acj Bcj
dn; j sn u0 u1 ?un c c ðn þ 1Þ?ðn þ cÞAj Bj tn v0 v1 ?vn
with some integer dn; j ðnX0Þ: Note that each dn; j ð1pjpmÞ is a nonzero polynomial in n of degree at most c with integer coefficients. Then setting sn; j ¼ dn; j sn ;
tn; j ¼ ðn þ 1Þ?ðn þ cÞAcj Bcj tn
for nX0 and 1pjpm; we see that each gj ðxÞ (except the finitely many terms with dn; j ¼ 0) satisfies the assumptions of Theorem 3. It is therefore applied (with the excluded terms being suitably supplemented) for gj ðxÞ ð1pjpmÞ to show that L þ L0 eQ with some L0 AQ; where L0 comes from the supplemented terms above. This immediately implies the nonvanishing of L: The Corollary is thus proved. &
References [AK] M. Amou, M. Katsurada, Irrationality results for values of generalized Tschakaloff series, J. Number Theory 77 (1999) 155–169. [Ba] A. Baker, Transcendental Number Theory, Cambridge University Press, Cambridge, 1979. [Du1] D. Duverney, A criterion of irrationality, Portugal. Math. 53 (2) (1996) 229–237. [Du2] D. Duverney, Sur les proprie´te´s arithme´tiques de la fonction de Tschakaloff, Period. Math. Hungar. 35 (1997) 149–157. [Ha] M. Haas, U¨ber die lineare Unabha¨ngigkeit von Werten einer speziellen Reihe, Arch. Math. 56 (1991) 148–162.
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[He] Ch. Hermite, Sur la fonction exponentielle, Oeuvrest. III, Gauthier-Villars, Paris, 1917, pp. 150–181. [LP] J.H. Loxton, A.J. van der Poorten, Transcendence and algebraic independence by a method of Mahler, in: A. Baker, D.W. Masser (Eds.), Transcendence Theory: Advances and Applications, Academic Press, San Diego, 1977, pp. 211–226. [Pe] O. Perron, Irrationalzahlen, 2. Aufl, Chelsea, New York, 1951. P n nðn 1Þ=2 I, Math. Ann. [Ts] L. Tschakaloff, Arithmetische Eigenschaften der unendlichen Reihe N n¼0 x a 80 (1921) 62–74; II, Math. Ann. 84 (1921) 100–114. [VW] K. Va¨a¨na¨nen, R. Wallisser, A linear independence measure for certain p-adic numbers, J. Number Theory 39 (1991) 225–236.
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Journal of Number Theory 104 (2004) 156–161
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On the general Kloosterman sum and its fourth power mean$ Zhang Wenpeng Research Center for Basic Science, Xi’an Jiaotong University Xi’an Shaanxi, PR China Received 5 October 2002; revised 18 April 2003 Communicated by D. Goss
Abstract The main purpose of this paper is to study the asymptotic property of the fourth power mean of the general Kloosterman sums, and give an interesting calculating formula. r 2003 Elsevier Inc. All rights reserved. Keywords: General Kloosterman sums; Fourth power mean; Asymptotic formula
1. Introduction Let qX3 be a positive integer. For any integers m and n; we define the general Kloosterman sums Sðm; n; w; qÞ as follows:
Sðm; n; w; qÞ ¼
q X 0 a¼1
wðaÞe
ma þ na% ; q
P where 0 denotes the summation over all a such that ða; qÞ ¼ 1; aa% 1 mod q; w denotes a Dirichlet character mod q and eðyÞ ¼ e2piy : This summation is very important, because it is a generalization of the classical Kloosterman sums Sðm; n; w0 ; qÞ ¼ Sðm; n; qÞ; where w0 is the principal character mod q: The various properties of Sðm; n; qÞ were investigated by many authors. $
This work is supported by N.S.F.(10271093) and P.N.S.F. (2002A11) of PR China. E-mail address: [email protected].
0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00154-9
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Perhaps the most famous property of Sðm; n; qÞ is the estimate (see [4,2]): 1
1
jSðm; n; qÞjpdðqÞq2 ðm; n; qÞ2 ;
ð1Þ
where dðqÞ is the divisor function, ðm; n; qÞ denotes the greatest common divisor of m; n and q: If q be a prime p; then Chowla [3] and Malyshev [7] also proved a similar result for Sðm; n; w; pÞ: On the other hand, Kloosterman [6] studied the fourth power mean of Sða; 1; pÞ; and proved the identity p1 X
S 4 ða; 1; pÞ ¼ 2p3 3p2 p 1:
a¼1
This identity can also be found in Iwaniec [5]. Salie´ [8] and Davenport (independently) obtained the estimate p1 X
S6 ða; 1; pÞ5p4 :
a¼1
The main purpose of this paper is to study the asymptotic properties of the 2kth power mean X
q X
jSðm; n; w; qÞj2k ;
ð2Þ
w mod q m¼1
and give an exact formula for (2) with k ¼ 2: That is, we shall prove the following: Theorem. Let qX3 be an integer. Then for any fixed integer n with ðn; qÞ ¼ 1; we have the identity X
q X
jSðm; n; w; qÞj4 ¼ f2 ðqÞq2 dðqÞ
pa jjq
w mod q m¼1
where fðqÞ is the Euler function, and paþ1 wq:
Y 1
Q
pa jjq
2 pa1 1 a 4pa1 þ ; a þ 1 pa ðp 1Þ ða þ 1Þpa
denotes the product over all p such that pa jq
In view of Salie´ and Davenport’s results on moments of Kloosterman sum which only involve single average, is it really necessary here to use the double average over m and w rather than average over m alone? This is a unsolved problem. For general integer kX3 and ðn; qÞ ¼ 1; whether there exists an exact formula for (2) is also an open problem.
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2. Some lemmas To complete the proof of the theorem, we need the following Lemmas. Lemma 1. Let p be a prime, a be any positive integer. Then we have the identity a
a
p p X 0 X0
1 ¼ ða þ 1Þpa 2
a¼1 b¼1 pa jða1Þðb1Þ
pa1 1 þ a 4pa1 : p1
Proof. If a ¼ 1; then we have p p X 0 X0
1¼
p1 X p1 X
1 ¼ 2ðp 1Þ 1 ¼ 2p 3:
a¼1 b¼1 pjða1Þðb1Þ
a¼1 b¼1 pjða1Þðb1Þ
So Lemma 1 holds for a ¼ 1: Now we suppose aX2: From the properties of the Mo¨bius function (see [1]) we immediately get a
a
p p X 0 X0
a
1¼
p X
a
p X
X
mðdÞ
a¼1 b¼1 djða;pa Þ pjða1Þ ðb1Þ
a¼1 b¼1 pjða1Þ ðb1Þ
¼
X
mðdÞ
djpa a
¼
a1
12
a¼1 b¼1 pa jða1Þðb1Þ
p X
1
a¼1 a
a
1 pX 1 a1 pX X i¼1
¼ 2pa 1 þ
1
a¼1 b¼1 pa jðda1Þðhb1Þ
a
p p X X
¼ 2pa 1 þ
a
p =d X p =h X
mðhÞ
hjpa
mðhÞ
hjðb;pa Þ a
X
X
1 2pa1
a¼1 b¼1 pi ja pai jb
a1 X
ðpai 1Þðpi 1Þ 2pa1
i¼1
¼ ða þ 1Þpa 2 This completes the proof of Lemma 1.
pa1 1 þ a 4pa1 : p1
&
Lemma 2. Let n; k1 ; k2 are integers with ðn; k1 k2 Þ ¼ ðk1 ; k2 Þ ¼ 1: Then for any character w mod k1 k2 ; there exist integers n1 and n2 with ðn1 ; k1 Þ ¼ ðn2 ; k2 Þ ¼ 1 such
ARTICLE IN PRESS Z. Wenpeng / Journal of Number Theory 104 (2004) 156–161
159
that n n1 k22 þ n2 k12 ðmod k1 k2 Þ; and that for these integers we have jSðm; n; w; k1 k2 Þj ¼ jSðm; n1 ; w1 ; k1 Þj jSðm; n2 ; w2 ; k2 Þj; where w ¼ w1 w2 with w1 mod k1 and w2 mod k2 : Proof. Since ðk1 ; k2 Þ ¼ 1; we have ðk12 ; k22 Þ ¼ 1: Therefore there exists integer n1 such that n1 k22 n mod k12 ; and n2 such that n2 k12 n mod k22 : These imply that k12 j n1 k22 þ n2 k12 n and k22 j n1 k22 þ n2 k12 n: That implies, n n1 k22 þ n2 k12 ðmod k1 k2 Þ: It is clear that ðk1 ; n1 Þ ¼ ðk1 ; n1 k22 Þ ¼ ðk1 ; nÞ ¼ 1 and ðk2 ; n2 Þ ¼ 1: Let w ¼ w1 w2 with w1 mod k1 and w2 mod k2 : Then for integers k1 ; k2 ; n1 and n2 ; we have k1 k2 X 0 X0
mðak2 þ bk1 Þ þ ðn1 k22 þ n2 k12 Þak2 þ bk1 k1 k2 a¼1 b¼1 k1 k2 X 0 ma þ n1 a% X 0 mb þ n2 b% ¼ w1 ðk2 Þw2 ðk1 Þ w1 ðaÞe w1 ðbÞe k2 k1 a¼1 b¼1
Sðm; n; w; k1 k2 Þ ¼
w1 w2 ðak2 þ bk1 Þe
¼ w1 ðk2 Þw2 ðk1 ÞSðm; n1 ; w1 ; k1 ÞSðm; n2 ; w2 ; k2 Þ: Now Lemma 2 follows from jw1 ðk2 Þw2 ðk1 Þj ¼ 1:
&
Lemma 3. Let p be a prime, and a a positive integer. Then for any integers u and v with ðuv; pÞ ¼ 1; we have X
a
p X
jSðmu; v; w; pa Þj4 ¼ f2 ðpa Þp2a ða þ 1Þ 1
w mod pa m¼1
2 pa1 1 a 4pa1 þ : a þ 1 pa ðp 1Þ ða þ 1Þpa
Proof. From the properties of characters we have jSðm; n; w; qÞj2 ¼
q q X 0 X0 a¼1
¼
q X 0 a¼1
b¼1
% wðabÞe
% mða bÞ þ nða% bÞ q
q X % a% 1Þ 0 mbða 1Þ þ nbð wðaÞ e : q b¼1
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Now by the orthogonality relation for characters we immediately get 2 q q X % a% 1Þ 0 X0 mbða 1Þ þ nbð jSðm; n; w; qÞj ¼ fðqÞ e : b¼1 q a¼1
X
4
w mod q
ð3Þ
Note that the trigonometric identity q X q; sa e ¼ q 0; a¼1
if qjs; if q[s:
From (3), Lemma 1 we have a
X w mod
p X pa
jSðmu; v; w; pa Þj4
m¼1
pa 2 pa pa X X X % 0 0 mubða 1Þ þ v bð a 1Þ % ¼ fðpa Þ e a p a¼1 m¼1 b¼1 pa pa pa pa X muðb1 b2 Þða 1Þ þ vðb1 b2 Þða% 1Þ 0 X0 X0 X ¼ fðp Þ e pa a¼1 b ¼1 b ¼1 m¼1 a
1
2
¼ fðpa Þ
pa pa X0 X0 X mub2 ðb1 1Þða 1Þ þ vb2 ðb1 1Þða% 1Þ 0 X e pa a¼1 b ¼1 b ¼1 m¼1
¼ fðpa Þ
pa pa X0 X0 X mðb1 1Þða 1Þ þ vb2 b1 að1 0 X % b1 Þð1 aÞ e pa a¼1 b ¼1 b ¼1 m¼1
pa
pa
1
pa
1
pa
¼ f2 ðpa Þpa
2
pa
2
pa
X0 X0
1
a¼1 b¼1 pa jða1Þðb1Þ
pa1 1 a a1 þ a 4p ¼ f ðp Þp ða þ 1Þp 2 p1 2 pa1 1 a 4pa1 2 a 2a þ ¼ f ðp Þp ða þ 1Þ 1 : a þ 1 pa ðp 1Þ ða þ 1Þpa 2
a
a
This proves Lemma 3.
&
3. Proof of the theorem From the Lemmas 2 and 3 in the above section, we can complete the proof of the theorem. In fact for any integer n with ðn; qÞ ¼ 1; let q has the prime power
ARTICLE IN PRESS Z. Wenpeng / Journal of Number Theory 104 (2004) 156–161
decomposition q ¼
Qr
i¼1
pai i ; m ¼
Pr
mi q i¼1 pai : i
161
It is clear that if mi ði ¼ 1; 2; y; rÞ pass
through a complete residue system modulo pai i ; then m pass through a complete residue system modulo q: Note that Sðm; ni ; wi ; pai i Þ ¼ Sðmi q=pai i ; ni ; wi ; pai i Þ (where ni as the definition of Lemma 2), ðq=pai i ; pai i Þ ¼ 1 and dðpai i Þ ¼ ai þ 1: From Lemmas 2 and 3 we immediately obtain the identity X
q X
jSðm; n; w; qÞj4
w mod q m¼1
¼
2
r Y
4
i¼1 r Y
"
wi mod
3
ai
pi X
X a pi i
jSðmi q=pai i ; ni ; wi ; pai i Þj4 5
mi ¼1
2 pai i 1 1 ai 4pai i 1 þ ¼ þ 1Þ 1 ai ai þ 1 pi ðpi 1Þ ðai þ 1Þpai i i¼1 Y 2 pa1 1 a 4pa1 þ ¼ f2 ðqÞq2 dðqÞ 1 : a þ 1 pa ðp 1Þ ða þ 1Þpa pa jjq
!#
i f2 ðpai i Þp2a i ðai
This completes the proof of the Theorem.
&
Acknowledgments The author expresses his gratitude to the referee for very helpful and detailed comments.
References [1] T.M. Apostol, Introduction to Analytic Number Theory, Springer, New York, 1976. [2] J.H.H. Chalk, R.A. Smith, On Bombieri’s estimate for exponential sums, Acta Arith. 18 (1971) 191–212. [3] S. Chowla, On Kloosterman’s sum, Norkse Vid. Selbsk. Fak. Frondheim 40 (1967) 70–72. [4] T. Estermann, On Kloostermann’s sum, Mathematica 8 (1961) 83–86. [5] H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics 17 (1997) 61–63. [6] H.D. Kloosterman, On the representation of numbers in the form ax2 þ by2 þ cz2 þ dt2 ; Acta Math. 49 (1926) 407–464. [7] A.V. Malyshev, A generalization of Kloosterman sums and their estimates, Vestnik Leningrad Univ. 15 (1960) 59–75 (in Russian). [8] H. Salie´, Uber die Kloostermanschen Summen Sðu; v; qÞ; Math. Z. 34 (1931) 91–109.
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 162–169
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Progression-free sets in finite abelian groups Vsevolod F. Lev Department of Mathematics, University of Haifa at Oranim, Tivon 36006, Israel Received 11 November 2002; revised 23 May 2003 Communicated by D. Goss
Abstract Let G be a finite abelian group. Write 2G :¼ f2g: gAGg and denote by rkð2GÞ the rank of the group 2G: Extending a result of Meshulam, we prove the following. Suppose that ADG is free of ‘‘true’’ arithmetic progressions; that is, a1 þ a3 ¼ 2a2 with a1 ; a2 ; a3 AA implies that a1 ¼ a3 : Then jAjo2jGj=rkð2GÞ: When G is of odd order this reduces to the original result of Meshulam. As a corollary, we generalize a result of Alon and show that if an integer kX2 and a real e40 are fixed, j2Gj is large enough, and a subset ADG satisfies jAjXð1=k þ eÞjGj; then there exists A0 DA such that 1pjA0 jpk and the elements of A0 add up to zero. When G is of odd order or cyclic this reduces to the original result of Alon. r 2003 Elsevier Inc. All rights reserved.
1. Motivation and background Since the fundamental paper of Roth [R53] one of the central problems in combinatorial number theory is to estimate the maximum possible cardinality of an integer set AD½1; m ; free of three-term arithmetic progressions; here m is a growing to infinity integer parameter. Equivalently, one can consider progression-free subsets of Zm ; the cyclic group of order m: If DðZm Þ denotes the maximum possible cardinality of such a subset, then the result of Roth can be formulated as DðZm Þ ¼ Oðm=ln ln mÞ; and further refinements due to Heath-Brown [H87], Szemere´di [S90], and Bourgain [B99] as DðZm Þ ¼ Oðm=lna mÞ with an absolute constant a40: A natural generalization is to replace Zm with an arbitrary finite abelian group G and estimate DðGÞ; the maximum possible cardinality of a subset ADG; free of E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00148-3
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three-term arithmetic progressions. For abelian groups G of odd order, Brown and Buhler in [BB82] and independently Frankl et al. in [FGR87] proved that DðGÞ ¼ oðjGjÞ as jGj-N: A quantitative estimate is due to Meshulam [M95] who has shown that DðGÞp2jGj=rkðGÞ for any non-trivial abelian group G of odd order; here and below throughout the paper rkðGÞ denotes the rank of G: As Meshulam shows, along with the result of Heath-Brown and Szemere´di this implies that DðGÞ ¼ OðjGj=lnb jGjÞ for some absolute constant b40 and any non-trivial abelian group G of odd order. Suppose now that G is either abelian of odd order or finite cyclic. Using the result of Frankl et al. in the first case, and the result of Roth is the second case, Alon proved in [A87] the following. Suppose that an integer kX2 and a real e40 are fixed. Then provided that jGj is large enough, for any subset ADG satisfying jAj4ð1=k þ eÞjGj there exists A0 DA such that 1pjA0 jpk and the elements of A0 add up to zero.
2. Notation and summary of results Our goal is to extend the results of Meshulam and Alon onto arbitrary finite abelian groups, not necessarily of odd order. First we discuss a question which we have not addressed yet, namely: what is a three-term arithmetic progression and when we say that a set is free of such progressions? The answer is quite straightforward for groups of odd order but somewhat subtler for even order groups: is the set f1; 6gDZ10 free of arithmetic progressions? Let G be an abelian group. We say that a three-element sequence P ¼ ða1 ; a2 ; a3 Þ of elements of G is an arithmetic progression if a1 þ a3 ¼ 2a2 ; and not all elements of P are equal to each other. If, moreover, all elements of P are pairwise distinct (equivalently, if a1 aa3 ) then we say that P is a true arithmetic progression. Clearly, if G has no elements of even order then any arithmetic progression is true and the notions of ‘‘arithmetic progression’’ and ‘‘true arithmetic progression’’ coincide. By PF[G] we denote the family of all subsets of G; free of true arithmetic progressions; that is, all subsets ADG such that a1 þ a3 ¼ 2a2 with a1 ; a2 ; a3 AA implies that a1 ¼ a3 : Assuming that G is finite, we let DðGÞ :¼ maxfjAj: AAPF½G g; the maximum possible cardinality of a subset of G; free of true arithmetic progressions. One last bit of notation before we can formulate our results. For an abelian group G we write 2G :¼ f2g: gAGg: Clearly, 2G is a subgroup of G; and if G is finite of odd order then 2G ¼ G: Furthermore, 2G is the trivial group if and only if G is an elementary abelian 2-group.
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Theorem 1. For any finite abelian group G such that 2G is non-trivial we have DðGÞo
2jGj : rkð2GÞ
Corollary 1. There exist absolute real constants C and b40 with the property that for any finite abelian group G such that 2G is non-trivial we have DðGÞoC
jGj ðln j2GjÞb
:
When G is of odd order Theorem 1 and Corollary 1 reduce to [M95, Theorem 1.2] and [M95, Corollary 1.3], respectively. We prove Theorem 1 in the next section using a version of Meshulam’s argument (which itself originates from Roth’s method) and employing some additional ideas. Corollary 1 can be derived from Theorem 1 and the result of Heath-Brown and Szemere´di in the same way as [M95, Corollary 1.3] is derived from [M95, Theorem 1.2]; the only difference is that instead of a ‘‘large’’ cyclic subgroup HpG one should consider a ‘‘large’’ cyclic subgroup Hp2G: We refer the reader to [M95] for the details. Theorem 2. Let G be a finite abelian group. Then for any integer kX2 and any subset ADG satisfying jAjXð1=kÞjGj þ 4kDðGÞ there exists A0 DA such that 1pjA0 jpk and the elements of A0 add up to zero. Corollary 2. For any integer kX2 and real e40 there exists an integer n0 ðk; eÞ with the following property. Let G be a finite abelian group such that j2Gj4n0 ðk; eÞ and suppose that a subset ADG satisfies jAjXð1=k þ eÞjGj: Then there exists A0 DA such that 1pjA0 jpk and the elements of A0 add up to zero. When G is a finite cyclic group, Corollary 2 reduces to [A87, Theorem 1.1] and Theorem 2 is analogous to [A87, Proposition 2.5]. The proof of Theorem 2 can be conducted following the proof of [A87, Proposition 2.5], except that the threeuniform hypergraph considered by Alon is to be replaced by the graph in which two vertices are adjacent if and only if the corresponding elements are the endterms of a true three-term arithmetic progression. For completeness and reader’s convenience we outline the proof at the end of the next section; details can be retrieved from [A87]. Corollary 2 is immediate from Theorem 2 and Corollary 1.
3. Proof of Theorems 1 and 2 Proof of Theorem 1. Using induction on the integer nX1; we show that if G is a finite abelian group such that rkð2GÞXn; then DðGÞo2jGj=n: The case n ¼ 1 is trivial and we assume that nX2: Suppose that AAPF½G ; we want to show that jAjo2jGj=n:
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Consider the automorphism j A AutðGÞ defined by jðgÞ ¼ 2g ðgAGÞ: The image of j is 2G and the kernel of j is G0 :¼ fgAG: 2g ¼ 0g: Thus G=G0 D2G and writing s :¼ jG0 j and t :¼ j2Gj we have st ¼ jGj: The set A is a union of t disjoint subsets contained in a G0 -coset each. Let n1 ; y; nt denote the cardinalities of these subsets so that n1 þ ? þ nt ¼ jAj and 0pni pjG0 ði ¼ 1; y; tÞ: By Aˆ we denote the Fourier coefficients of the characteristic function of A; that is, for a character wAGˆ we write ˆ AðwÞ :¼
X
w% ðaÞ;
aAA
where w% is the character conjugate to w: The cartesian square and cartesian cube of A are denoted by A2 and A3 ; respectively. Since AAPF½G using orthogonality relations we obtain X
2 ˆ 2 ˆ ðAðwÞÞ Aðw% Þ ¼ jGj#fða1 ; a2 ; a3 ÞAA3 : a1 þ a3 ¼ 2a2 g
wAGˆ
¼ jGj#fða1 ; a2 ÞAA2 : 2a1 ¼ 2a2 g ¼ jGj#fða1 ; a2 ÞAA2 : a2 a1 AG0 g ¼ jGjðn21 þ ? þ n2t Þ p jGjðn1 þ ? þ nt Þ max ni 1pipt
p jGjjAjjG0 j: We single out those summands corresponding to real characters w: Notice that w is ˆ w% 2 Þ ¼ jAj and we get real if and only if w2 ¼ w0 ; the principle character. In this case Að X
X
2 ˆ 2 ˆ ðAðwÞÞ Aðw% ÞX
ˆ w2 aw wAG: 0
2 ˆ 2 ˆ ðAðwÞÞ Aðw% Þ jGjjAjjG0 j
ˆ w2 ¼w wAG: 0
X
¼ jAj
2 ˆ ðAðwÞÞ
jGjjAjjG0 j
ˆ w2 ¼w wAG: 0
¼ jAjS jGjjAjjG0 j; where S ¼
P
ˆ w2 ¼w wAG: 0
2 ˆ ðAðwÞÞ : Set
M :¼
max
ˆ w2 aw wAG: 0
ˆ w% 2 Þj: jAð
ð1Þ
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ˆ Since AðwÞ is real for any real character w; from (1) and the Parseval identity it follows that X
jAjS jGjjAjjG0 jp M
2 ˆ jAðwÞj
ˆ w2 aw wAG: 0
0 ¼ M@
X
1
2 ˆ jAðwÞj
SA
wAGˆ
¼ ðjAjjGj SÞM:
ð2Þ
ˆ w% 2 Þj ¼ M and let W oG be the kernel of Fix now a non-real character w such that jAð 2 2 w ; that is, W ¼ fgAG: w ðgÞ ¼ 1g: We have ˆ w% 2 Þ ¼ Að
X
w2 ðaÞ
aAA
0 1 X 1 X@ 2 ¼ w ðgÞ 1A jW j gAG aAA-ðgþW Þ 1 X 2 w ðgÞjðA gÞ-W j jW j gAG 1 X ¼ ð w2 ðgÞÞðDðW Þ jðA gÞ-W jÞ: jW j gAG
¼
As ððA gÞ-W ÞAPF½W for any gAG; all differences DðW Þ jðA gÞ-W j are non-negative and writing for brevity k :¼ DðW Þ=jW j we obtain X ˆ w% 2 Þjp 1 M ¼ jAð ðDðW Þ jðA gÞ-W jÞ ¼ kjGj jAj: jW j gAG Comparing with (2) we conclude that jAjS jGjjAjjG0 jpðjAjjGj SÞðkjGj jAjÞ; after simplifications, kS þ jAj2 pjAjjG0 j þ kjAjjGj:
ð3Þ
To estimate S we represent A as a union of s disjoint subsets contained in a 2G-coset each. (Recall that ½G : 2G ¼ jG0 j ¼ s:) We denote the cardinalities of these subsets by m1 ; y; ms so that m1 þ ? þ ms ¼ jAj and 0pmi pj2Gj ði ¼ 1; y; sÞ: Using
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orthogonality relations once again we get S¼
X wAGˆ
! 1 X 2 2 ˆ w ðgÞ jAðwÞj jGj gAG
1 X X 2 2 ˆ w ðgÞjAðwÞj jGj gAG wAGˆ X ¼ #fða1 ; a2 ÞAA2 : a1 a2 ¼ 2gg ¼
gAG
¼ jG0 j#fða1 ; a2 ÞAA2 : a2 a1 A2Gg ¼ jG0 jðm21 þ ? þ m2s Þ X jG0 jðm1 þ ? þ ms Þ2 =s ¼ jAj2 : Combined with (3) this yields jAj jG0 jpkðjGj jAjÞ:
ð4Þ
We now use the induction hypothesis as applied to W to estimate k ¼ DðW Þ=jW j: Since the quotient group G=W is isomorphic to the image of w2 ; which is a finite subgroup of the multiplicative group of the field of complex numbers, we conclude that G=W is cyclic or trivial. It follows that 2G=2W is cyclic or trivial, too, whence rkð2W ÞXrkð2GÞ 1Xn 1 and therefore ko2=ðn 1Þ: Substituting to (4) we get jAj jG0 jo2ðjGj jAjÞ=ðn 1Þ which gives jAjo
2=ðn 1Þ þ jG0 j=jGj jGj: 2=ðn 1Þ þ 1
ð5Þ
Since the quotient jG0 j=jGj can be rather large, at this point we have to take into account the algebraic structure of G: Write r :¼ rkðGÞ and G ¼ "ri¼1 Gi ; where the direct summands are cyclic subgroups of G the orders of which di :¼ jGi j ði ¼ 1; y; rÞ satisfy 2pd1 j?jdr : Set j :¼ maxfiA½1; n : di ¼ 2g (with the understanding that max Ø ¼ 0) and put G 0 :¼
j M i¼1
Gi ;
G 00 :¼
r M i¼jþ1
Gi :
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Consequently, we have G ¼ G 0 "G 00 ; 2G 0 ¼ f0g; and rkðG 00 Þ ¼ rkð2G00 Þ ¼ rkð2GÞXn:
ð6Þ
We distinguish two cases. First we assume that G satisfies rkð2GÞ ¼ rkðGÞ: In this case G 0 is the trivial group and r Y Y jG0 j 1 2 1 1 2 ¼ : ð7Þ p p no jGj d d 2 2 nðn
1Þ iA½1;r : d is odd i iA½1;r : d is even i i
i
From (5) and (7) we obtain jAjo
2=ðn 1Þ þ 2=nðn 1Þ 2 jGj ¼ jGj; 2=ðn 1Þ þ 1 n
as required. It remains to consider the case when rkð2GÞorkðGÞ; that is, the group G0 is nontrivial. To this end we observe that DðGÞpjG 0 jDðG 00 Þ: Indeed, if BDG satisfies jBj4jG 0 jDðG00 Þ; then there exists a coset g þ G 00 (with some gAG) such that jB-ðg þ G 00 Þj4DðG 00 Þ; by the definition of DðG 00 Þ; the set B gDG 00 contains a true arithmetic progression, and so does B: Now by (6) and the argument above, as applied to the group G00 ; we have DðGÞpjG0 jDðG 00 ÞojG 0 j
2jG00 j 2jGj ¼ : n n
This completes the proof of Theorem 1. &
Proof of Theorem 2 (Sketch). Suppose that G; k; and A are as in the theorem. We can assume that kX3 as otherwise the assertion follows readily from the pigeonhole principle. Define A0 to be the set of all those elements of A which are the midterms of at least 2ðk 1Þ true three-term arithmetic progressions with the elements in A: A0 :¼ faAA: #fða1 ; a2 ÞAA2 : a1 þ a2 ¼ 2a; a1 aa2 gX2ðk 1Þg: Write A00 :¼ A\A0 and consider the graph G ¼ ðA00 ; EÞ on the vertex set A00 in which two (distinct) vertices a1 ; a2 AA00 are adjacent if and only if there exists aAA00 such that a1 þ a2 ¼ 2a: The number of edges of G is jEjpðk 2ÞjA00 j (as any aAA00 satisfies at most 2ðk 2Þ equalities of the form a1 þ a2 ¼ 2a with a1 ; a2 AA00 ; a1 aa2 ), hence by a simple probabilistic argument there is an independent vertex set BDA00 of cardinality jBjXjA00 j2 =ð4jEjÞXjA00 j=ð4ðk 2ÞÞ: By the construction we have BAPF½G whence jBjpDðGÞ; therefore jA00 jp4ðk 2ÞDðGÞ
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and consequently jA0 j4jGj=k: By [A87, Corollary 2.3] there exists an integer hA½1; k and a sequence ða1 ; y; ah Þ of not necessarily distinct elements of A0 such that a1 þ ? þ ah ¼ 0: By the choice of the set A0 ; for any iA½1; h there are at least 2ðk 1Þ ð1Þ ð2Þ ð1Þ ð2Þ ð1Þ ð2Þ representations 2ai ¼ ai þ ai with ai ; ai AA and ai aai ; and it is not difficult to conclude that the sum a1 þ ? þ ah can be re-written as a sum of h pairwise distinct elements of A; see the proof of [A87, Proposition 2.5] for the explanation. Thus we have found hA½1; k pairwise distinct elements of A with zero sum. &
References [A87] N. Alon, Subset sums, J. Number Theory 27 (1987) 196–205. [B99] J. Bourgain, On triples in arithmetic progression, Geom. Funct. Anal. 9 (1999) 968–984. [BB82] T.C. Brown, J.C. Buhler, A density version of a geometric Ramsey theorem, J. Combin. Theory Ser. A 32 (1982) 20–34. [FGR87] P. Frankl, G. Graham, V. Ro¨dl, On subsets of abelian groups with no 3-term arithmetic progression, J. Combin. Theory Ser. A 45 (1987) 157–161. [H87] R. Heath-Brown, Integer sets containing no arithmetic progressions, J. London Math. Soc. 35 (1987) 385–394. [M95] R. Meshulam, On subsets of finite abelian groups with no 3-term arithmetic progressions, J. Combin. Theory Ser. A 71 (1995) 168–172. [R53] K.F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953) 104–109. [S90] E. Szemere´di, Integer sets containing no arithmetic progressions, Acta Math. Hungar. 56 (1990) 155–158.
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Heights on a subvariety of an abelian variety Takashi Ichikawa Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan Received 9 January 2003; revised 25 May 2003 Communicated by S.-W. Zhang
Abstract Extending Ullmo–Zhang’s result on the Bogomolov conjecture, we give conditions that a closed subvariety of an abelian variety A defined over a number field is isomorphic to an abelian variety in terms of the value distribution of a Neron–Tate height function on the subvariety. As a corollary of the result, we prove the Bogomolov conjecture which claims that if an irreducible curve X in A is not isomorphic to an elliptic curve, then for the pseudodistance defined by the Neron–Tate height, the distribution of algebraic points on X is uniformly discrete. These results can be extended in the case where base fields are finitely generated over Q via Moriwaki’s height theory. r 2003 Elsevier Inc. All rights reserved. Keywords: Abelian varieties; Neron–Tate height; Arakelov theory
1. Introduction % of Q; and let hˆL be the Let A be an abelian variety over the algebraic closure Q % associated with a symmetric ample line bundle Neron–Tate height function on AðQÞ % let rðxÞ be the L on A: For a closed subvariety X of A and for each xAX ðQÞ; ˆ % hL ðy xÞorg is a finite set, and let RðX Þ maximal real number r such that fyAX ðQÞj % be the infimum of rðxÞ ðxAX ðQÞÞ: The positivity of RðX Þ means that for the pseudodistance defined by hˆL ; the distribution of algebraic points on X is uniformly discrete. In [1], Bogomolov conjectured that for proper and smooth algebraic curves C of genus 41 embedded in those Jacobian varieties JacðCÞ; RðCÞ are positive and expressed as a certain height function on the moduli space of curves C (see [9] for the E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.07.002
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% has height review of recent progress in this conjecture). Since an element of AðQÞ 0 if and only if it is torsion, if X contains a translation of a nontrivial abelian % satisfy that rðxÞ ¼ 0; and hence RðX Þ ¼ 0: subvariety of A; then some xAX ðQÞ On the other hand, if X does not satisfy the above condition, then each rðxÞ is seen to be positive from the result of Ullmo [6] and Zhang [8] proved by using Arakelov theory which claims that any irreducible closed subvariety of A having Zariski dense small points must be the translation of an abelian subvariety by a torsion point. In this paper, extending Ullmo–Zhang’s result we give conditions that a subvariety of A is isomorphic to an abelian variety in terms of the value distribution of hˆL : % let Theorem. Let X be an irreducible closed subvariety of A, and for each xAX ðQÞ; r0 ðxÞ be the maximal real number r such that % hˆL ðy xÞorg fyAX ðQÞj is not Zariski dense in X. Then the following four conditions are equivalent: % is equal to 0. (1) The infimum of r0 ðxÞ ðxAX ðQÞÞ (2) X is isomorphic to an abelian variety. (3) X is represented as a translation A1 þ a of an abelian subvariety A1 of A, where % % gives the minimum height (for hˆL ) on X ðQÞ: aAX ðQÞ % satisfies that r0 ðxÞ ¼ 0: (4) Any xAX ðQÞ Note that rðxÞ ¼ r0 ðxÞ if dimðX Þ ¼ 1; hence as a consequence of (1) ) (2) one can prove the Bogomolov conjecture on the positivity of RðX Þ: Corollary. Let X be an irreducible curve in A which is not isomorphic to an elliptic curve. Then RðX Þ is positive. In the above theorem, assertion (4) ) (1) is trivial and (3) ) (4) follows from that the set of torsion points on A1 is Zariski dense in A1 : Hence to show the theorem, it is enough to prove (1) ) (2) and (2) ) (3). We prove (1) ) (2) by using Ullmo– Zhang’s result and a theory of Albanese varieties, and prove (2) ) (3) by decomposing (up to isogeny) an abelian variety to a product of abelian subvarieties which are orthogonal to each other for the bilinear form associated with a given Neron–Tate height function. In [4], Moriwaki extended a theory of height functions for algebraic varieties defined over a finitely generated field K over Q; and for an abelian variety A with symmetric ample line bundle L which are defined over K; he constructed the % % with respect to a polarization B% of K: From canonical height function hˆBL on AðKÞ results in [4] containing a generalization of Ullmo–Zhang’s result for hˆBL% ; one can see that the above theorem and corollary are extended for hˆBL% :
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2. Proof of (1) ) (2) From the argument in [8, p. 164], it is enough to show that X becomes a single point under the assumption that faAAjX þ a ¼ X g ¼ f0g: By this assumption, there is a positive integer n such that An {ða1 ; y; an Þ/ða1 a2 ; y; an1 an ÞAAn1 gives a birational morphism j : X n -jðX n Þ: Let Ln1 be a symmetric ample line bundle on An1 defined by the product of the pullbacks of L via the projections An1 -A to the ith components ði ¼ 1; y; n 1Þ: Then % hˆLn1 ðða1 ; y; an1 ÞÞ ¼ hˆL ða1 Þ þ ? þ hˆL ðan1 Þðai AAðQÞÞ: % such that By condition (1), for any e40; there is an element xe of X ðQÞ % hˆL ðx xe Þoeg Se ¼ fxAX ðQÞj is Zariski dense in X : Then we have hˆLn1 ðjðx1 ; y; xn ÞÞ ¼
n1 X
hˆL ððxi xe Þ ðxiþ1 xe ÞÞo4ðn 1Þe
i¼1
for any ðx1 ; y; xn ÞASen ; and jðSen Þ is Zariski dense in jðX n Þ: Therefore, the irreducible closed subvariety jðX n Þ of An1 has Zariski dense small points, and hence by Zhang [8, Corollary 3] (see [4, Theorem 8.1] in the case where base fields are finitely generated over QÞ; jðX n Þ becomes an abelian subvariety of An1 because 0AjðX n Þ: By Hironaka’s theorem [2], one can take a resolution p : Y -X of X : Then j gives a birational morphism Y n -jðX n Þ; and hence the Albanese variety AlbðY n Þ of Y n is isomorphic to the abelian variety jðX n Þ: Fix a point y0 on Y ; and let a : Y -AlbðY Þ;
an : Y n -AlbðY n Þ
be the Albanese maps such that aðy0 Þ ¼ 0; an ððy0 ; y; y0 ÞÞ ¼ 0 respectively. For each i ¼ 1; y; n; let pi : Y n -Y denote the projection to the ith component, and let ii : Y -Y n be the morphism such that for any yAY ; the jth component of ii ðyÞ is y if j ¼ i; and is y0 if jai; respectively. Then by the property of Albanese varieties, there exist homomorphisms bi : AlbðY n Þ-AlbðY Þ;
gi : AlbðY Þ-AlbðY n Þ
such that a3pi ¼ bi 3an ; an 3ii ¼ gi 3a; respectively. Hence we have a3pi 3ij ¼ bi 3an 3ij ¼ bi 3gj 3a;
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which implies that bi 3gi is the identity map and that bi 3gj ðiajÞ is the 0-map because aðY Þ generates AlbðY Þ: Thus the map g : AlbðY Þn -AlbðY n Þ defined by gððz1 ; y; zn ÞÞ ¼ g1 ðz1 Þ þ ? þ gn ðzn Þððz1 ; y; zn ÞAAlbðY Þn Þ is an injective homomorphism between abelian varieties. Therefore, we have n dimðAlbðY ÞÞpdimðAlbðY n ÞÞ ¼ dimðjðX n ÞÞ ¼ n dimðX Þ: Denote the inclusion X +A by i; and represent i3p : Y -A as t3a þ a0 ; where t is a homomorphism AlbðY Þ-A and a0 AA: Then X ¼ pðY ÞCtðAlbðY ÞÞ þ a0 ; and hence by the above inequality, we have X ¼ tðAlbðY ÞÞ þ a0 : Since each fiber of the birational map j is stable under translations by the diagonal ða; y; aÞ for aAtðAlbðY ÞÞ; we conclude that tðAlbðY ÞÞ ¼ f0g: This completes the proof. Remark. From this proof and the theorem of successive minima [7, Theorem 1.10], one can see that if dimðX ÞX1 and the morphism j : X n -jðX n Þ given by jððx1 ; y; xn ÞÞ ¼ ðx1 x2 ; y; xn1 xn Þ ðxi AX Þ becomes a birational morphism, then 1 n ˆ % hðjðX ÞÞ40; inffr0 ðxÞjxAX ðQÞgX 4ðn 1Þ n ˆ where hðjðX ÞÞ denotes the height of jðX n Þ for the line bundle Ln1 with admissible metric giving the Neron–Tate height. If X is a proper and smooth curve of genus 42 which is not hyperelliptic, then the morphism j : X 2 -jðX 2 ÞCJacðX Þ is a birational morphism, and hence
1ˆ 2 ÞÞ40: RðX ÞX hðjðX 4
3. Proof of (2) ) (3) First, we show the following lemma which claims, together with the reducibility of % can be decomposed (up to isogeny) abelian varieties, that any abelian variety over Q to a product of abelian subvarieties which are orthogonal to each other for the bilinear form associated with a given Neron–Tate height function. Let A1 ; A2 be % let M be a symmetric ample line bundle on A1 A2 ; and abelian varieties over Q; ˆ % associated with M: denote by hM the Neron–Tate height function on ðA1 A2 ÞðQÞ By the construction of hˆM ; there is a symmetric bilinear form % ðA1 A2 ÞðQÞ-R % lˆM : ðA1 A2 ÞðQÞ
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% such that hˆM ðaÞ ¼ lˆM ða; aÞ ðaAðA1 A2 ÞðQÞÞ: Then we have: Lemma. There exists an abelian subvariety A0 of A1 A2 such that the homomorphism f : A1 A0 -A1 A2 given by f ðða1 ; a0 ÞÞ ¼ ða1 ; 0Þ þ a0 is an isogeny, and that % % and iA0 ðA0 ðQÞÞ are orthogonal for lˆf ðMÞ ; where iA1 : A1 -A1 A0 ; iA1 ðA1 ðQÞÞ 0 0 iA0 : A -A1 A are the closed immersions given by iA1 ða1 Þ ¼ ða1 ; 0Þ; iA0 ða0 Þ ¼ ð0; a0 Þ; respectively. Proof. For i ¼ 1; 2; define the closed immersion ii : Ai -A1 A2 by i1 ða1 Þ ¼ ða1 ; 0Þ; i2 ða2 Þ ¼ ð0; a2 Þ; and the projection pi : A1 A2 -Ai by pi ðða1 ; a2 ÞÞ ¼ ai : Let P be the Poincare´ line bundle on A1 Aˆ 1 ; where Aˆ 1 is the dual abelian variety of A1 : Then by the universality of P (cf. [5, p. 125]), there is a unique homomorphism jM : A2 -Aˆ 1 such that ð1A1 jM Þ ðPÞDM#p 1 ði 1 ðMÞÞ1 #p 2 ði 2 ðMÞÞ1 : Denote by m; pi ði ¼ 1; 2Þ the homomorphisms A1 A1 -A1 defined as the multiplication and the projection to the ith component respectively. Since M1 ¼ i 1 ðMÞ is an ample line bundle on A1 ; there is a unique isogeny fM1 : A1 -Aˆ 1 such that ð1A1 fM1 Þ ðPÞDm ðM1 Þ#p 1 ðM1 Þ1 #p 2 ðM1 Þ1 : Let d be the degree of fM1 ; and take the isogeny f0M1 : Aˆ 1 -A1 satisfying that fM1 3f0M1 ¼ ½d; where ½d denotes the homomorphism defined as the multiplication by d: Then F ¼ f0M1 3jM is a homomorphism A2 -A1 satisfying that fM1 3F ¼ ½d3jM ¼ jM 3½d: Hence we have ð1A1 FÞ ðm ðM1 Þ#p 1 ðM1 Þ1 #p 2 ðM1 Þ1 Þ Dð1A1 ½dÞ ðM#p 1 ði 1 ðMÞÞ1 #p 2 ði 2 ðMÞÞ1 Þ: It is known (see [4, Section 3] in the case where base fields are finitely generated over % the associated QÞ that for an abelian variety with symmetric line bundle over Q; Neron–Tate height function becomes the unique quadratic form which is, modulo a bounded function, the height function defined using Arakelov theory for a model of the abelian variety with symmetric line bundle. Therefore, the Neron–Tate height functions associated with symmetric line bundles are functorial for taking these pullbacks, and are additive for taking these tensor products. Hence for any
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% by the above isomorphism we have ða1 ; a2 ÞAðA1 A2 ÞðQÞ; 2 lˆM ðða1 ; 0Þ; ð0; d a2 ÞÞ ¼ hˆM ðða1 ; d a2 ÞÞ hˆM ðða1 ; 0ÞÞ hˆM ðð0; d a2 ÞÞ ¼ hˆN ðða1 ; d a2 ÞÞ ¼ hˆM1 ða1 þ Fða2 ÞÞ hˆM1 ða1 Þ hˆM1 ðFða2 ÞÞ ¼ 2 lˆM1 ða1 ; Fða2 ÞÞ; where N ¼ M#p 1 ði 1 ðMÞÞ1 #p 2 ði 2 ðMÞÞ1 : Let Z : A2 -A1 A2 be the homomorphism with finite kernel given by ZðaÞ ¼ ðFðaÞ; d aÞ; and let A0 be an abelian subvariety of A1 A2 defined as the image of Z: Then by the above equality, A0 satisfies the condition in lemma. & % Using this lemma, we prove (2) ) (3). Denote by A1 an abelian variety over Q which is isomorphic to X : Then the Albanese variety of X becomes A1 ; and hence one can identify A1 with an abelian subvariety of A; and X is represented as a translation of A1 : Then by Poincare´’s complete reducibility theorem, there is an abelian subvariety A2 of A such that the homomorphism f : A1 A2 -A given by fðða1 ; a2 ÞÞ ¼ a1 þ a2 is an isogeny. Hence one can take a positive integer j and an isogeny c : A-A1 A2 satisfying that c3f ¼ ½ j: Put M ¼ f ðLÞ; Z ¼ cðX Þ: Let A0 CA1 A2 ; f : A1 A0 -A1 A2 be as in lemma, and take a positive integer k and an isogeny g : A1 A2 -A1 A0 satisfying that g3 f ¼ ½k: Since gði1 ðA1 ÞÞ ¼ iA1 ðA1 Þ; we have gðZÞ þ iA1 ðA1 Þ ¼ gðZÞ; and hence the image of gðZÞ by the projection A1 A0 -A0 consists of a single point b which satisfies that gðZÞ ¼ iA1 ðA1 Þ þ iA0 ðbÞ % iA0 ðbÞ are orthogonal for lˆf ðMÞ : Therefore, we have and that iA1 ðA1 ðQÞÞ; % and hence if we take an element a of hˆf ðMÞ ðiA0 ðbÞÞ ¼ inffhˆf ðMÞ ðgðzÞÞjzAZðQÞg; % such that ðg3cÞðaÞ ¼ iA0 ðbÞ; then X ¼ A1 þ a and X ðQÞ % ð jkÞ2 hˆL ðaÞ ¼ ð jkÞ2 inffhˆL ðxÞjxAX ðQÞg: This completes the proof of (2) ) (3), and hence of theorem.
Acknowledgments The author wishes to thank Professor N. Terai for his valuable comments in reading [3] which helped the author to learn Arakelov theory.
References [1] F.A. Bogomolov, Points of finite order on an abelian variety, Math. USSR-Izv. 17 (1981) 55–72.
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[2] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79 (1964) 109–326. [3] S. Kawaguchi, A. Moriwaki, I. Yamaki, Introduction to Arakelov geometry, in: Proceedings of the Symposium on Algebraic Geometry in East Asia, World Scientific, Singapore, pp. 1–74. [4] A. Moriwaki, Arithmetic height functions over finitely generated fields, Invent. Math. 140 (2000) 101–142. [5] D. Mumford, Abelian Varieties, Oxford University Press, Oxford, 1974. [6] E. Ullmo, Positivite´ et discre´tion des points alge´briques des courbes, Ann. of Math. 147 (1998) 167–179. [7] S. Zhang, Small points and adelic metrics, J. Algebraic Geom. 4 (1995) 281–300. [8] S. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. 147 (1998) 159–165. [9] S. Zhang, Small points and Arakelov theory, in: Proceedings of the ICM, Vol. II, Berlin, 1998, Doc. Math. II (1998) 217–225.
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Kuznietsov trace formula and weighted distribution of Hecke eigenvalues Charles C.C. Li Mathematics Department, UCLA, Box 941555, Los Angeles, CA 90095, USA Received 20 January 2003; revised 17 April 2003 Communicated by D. Goss
Abstract Let ðX ; mÞ be a measurable topological space. Let S1 ; S2 ; y be a family of finite subsets of X : Suppose each xASi has a weight wix ARþ assigned to it. We say fSi g is fwi g-distributed with respect to the measure m if for any continuous function f on X ; we have P w f ðxÞ R xASi ix limi-N P w ¼ X f ðxÞ dmðxÞ: xASi
ix
Let SðN; kÞ be the space of modular cusp forms over G0 ðNÞ of weight k and let EðN; kÞCSðN; kÞ be a basis which consists of Hecke eigenforms. Let ar ðhÞ be the rth Fourier coefficient of h: Let xhp be the eigenvalue of h relative to the normalized Hecke operator Tp0 : Let jj jj be the Petersson norm on SðN; kÞ: In this paper we will show that for any even integer 2 4pr
e kX3; fxhp : hAEðN; kÞg; p[N is fjar ðhÞj jjhjj2
g-distributed with respect to a polynomial times the
Sato–Tate measure when N-N: r 2003 Elsevier Inc. All rights reserved.
1. Introduction Let SðN; kÞ be the space of modular cusp forms on G0 ðNÞ of weight k and for p[N; let Tp be the Hecke operator on SðN; kÞ as defined in [Se2]. We shall consider Tp0 ¼ p ðk 1Þ=2 Tp : Denote the eigenvalue of a Hecke eigenform h relative to Tp0 by xhp : The Ramanujan–Petersson conjecture (Deligne’s Theorem) asserts that jxhp jp2 for ð p; NÞ ¼ 1: Furthermore, it is conjectured that the set fxhp : ðp; NÞ ¼ 1g is E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00149-5
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equidistributed with respect to the Sato–Tate measure (refer to [Se1, Chapter 1]) dmN ðxÞ ¼
( qffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 p 1 4 dx 0
when xA½ 2; 2 ; otherwise:
Let EðN; kÞ be a basis of SðN; kÞ consisting of Hecke eigenforms. Serre [Se2] considered the distribution of fxhp : hAEðN; kÞg for fixed p: He used the Selberg trace formula for the Hecke operators and showed that when N-N; the set fxhp g is equidistributed as dmp ðxÞ ¼
pþ1 ðp1=2
þ p 1=2 Þ2 x2
dmN ðxÞ:
ð1Þ
In this paper, we established a weighted distribution for fxhp : hAEðN; kÞg for fixed p: Let ðX ; mÞ be a measurable topological space. Let S1 ; S2 ; y; Si ; y be a family of finite subsets of X : Suppose each xASi has a weight wix ARþ assigned to it. Let dx be the Dirac measure at x: Define P wix dx : dmi ¼ PxASi xASi wix We say fSi g is fwi g-distributed with respect to measure dm if lim dmi ¼ lim
i-N
i-N
P wix dx PxASi ¼ dm: xASi wix
This means for any continuous function f on X ; we have P Z xASi wix f ðxÞ P f ðxÞ dmi ðxÞ ¼ lim ¼ f ðxÞ dmðxÞ: i-N X X xASi wix
Z lim
i-N
When wi;x ¼ 1; the definition is the same as the definition of equidistribution given in [Se2, Section 1]. In this paper, we will use Kuznietsov trace formula to obtain a certain weighted distributions. Suppose hASðN; kÞ is a Hecke eigenform with Fourier expansion hðzÞ ¼
N X
ar e2rpiz ; Re z40:
r¼1
Write ar ðhÞ ¼ ar : Let jj jj be the Petersson norm on SðN; kÞ [Ge, p. 24 (2.6)]. We can assume k is even because SðN; kÞ is empty when k is odd. Define
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polynomials Xn by Xn ð2 cos fÞ ¼
sinðn þ 1Þf : sin f
Let r be a positive integer. Let p be a fixed prime. Let rp ¼ ordp r: Then we have Theorem 1.1. Let k be an even number X3: Consider the family of sets SN ¼ 2 4pr
e fðxhp Þ : hAEðN; kÞg; p[N with weight wrh ¼ jar ðhÞj jjhjj2
assigned to xhp : Then the family
of sets fSN : p[Ng is fwrh g-distributed with respect to X X2i ðxÞdmN ðxÞ 0piprp
when N-N: The proof will be given at the end of this paper. A more general result is given in Theorem 5.7. Corollary 1.2. Let k; SN ; wrh be as above. If p[r; then the family of sets fSN : p[Ng; N-N is fwrh g-distributed with respect to the Sato–Tate measure. Proof. If p[r; rp ¼ ordp r ¼ 0 and X0 ðxÞ ¼ 1: The corollary follows easily. & The technique used here can be generalized to other groups (example G ¼ GSpð2kÞ), refer to [Li] for the generalizations.
2. Construction of test functions
1 Let G ¼ GL2 : The unipotent group is N ¼ CG: Write e ¼ : Let 1 1 Z be the center of G; let M be the diagonal subgroup of G: Denote U% ¼ U=Z for any cos y sin y subset U of GL2 : Define KN ¼ sin y cos y : When poN; define Kp ¼ GL2 ðZp Þ: a b Q Define K0 ðNÞp ¼ c d AKp : c 0ðmod NÞ ; K0 ðNÞ ¼ poN K0 ðNÞp : Let A be the adeles of Q: Let Afin be the finite component of A: Let L2 be the Hilbert space of continuous functions j on ZðAÞGðQÞ\GðAÞ such R that ZðAÞGðQÞ\GðAÞ jjðgÞj2 dgoN: The subset of cuspidal functions in L2 is denoted 1
by L20 : Let R be the right regular representation of GðAÞ on L2 : There is an embedding SðN; kÞ-L20 [Ge, p. 42]. The map is denoted by f /jf : Denote the image of the map by AðN; kÞ: Suppose ðp; V Þ is a representation of a measurable topological group G and f is a R continuous function on G: Define pð f Þv ¼ G f ðgÞpðgÞv dg:
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We are going to construct a function f ¼ fN ffin on GðAÞ ¼ GðRÞ GðAfin Þ: The main property of this function is given in Proposition 2.1. The results quoted below are well known. The function fN ¼ fk is a function defined on GL2 ðRÞ: It is the conjugate of a normalized matrix coefficient. Explicitly it is defined by fk ðgÞ ¼ dpk /pðgÞv0 ; v0 S where pk is the discrete series representation of lowest weight k; v0 is the lowest weight unit vector and dpk is the formal degree. Explicitly we can take
fk ðgÞ ¼
8 < k 1 :
2k ðdet gÞk=2 4p ððaþdÞþiðb cÞÞk
0
a b c d if det go0:
if g ¼
and det g40;
ð2Þ
Refer to [Va, p. 192] or [KL] for the details. Define cðNÞ ¼ ½G0 ð1Þ : G0 ðNÞ : We take measðKp Þ ¼ 1: One can easily show that 1 1 measðK0 ðNÞÞ ¼ ½K0 ð1Þ:K ¼ cðNÞ : 0 ðNÞ If R be a ring, let M2 ðRÞ be all the 2 2 matrix over R: Now we define Mðn; NÞ ¼
g¼
Y Y a b Zp : det ðgÞAn AM2 Z p c d
and c 0 ðmod NÞ :
Define ( f n ðgÞ ¼
1 meas ðK0 ðNÞÞ
0
¼ cðNÞ
if g ¼ zm; zAZðAfin Þ; mAMðn; NÞ; otherwise:
Proposition 2.1. Suppose f ¼ fk f n ; Rð f Þ vanishes on AðN; kÞ> : On AðN; kÞ it acts by Rð f Þjh ¼ j k n
2 1 Tn h
:
Proof. The idea of the proof can be found in [Ge, Lemma 3.7] and also [Ro, Lemma 2.12]. A complete proof can be found in [KL]. & Corollary 2.2. Let hAEðN; kÞ; f ¼ fk f n then 0 Rð f Þjh ¼ n1=2 @
Y
1 Xnp ðxhp ÞAjh :
pjn
Q np Q Proof. Recall that n ¼ p : We have Tn0 ¼ p Tp0 np : By Serre [Se2, Sections 2 and 3, Lemma 1], Tp0 np ¼ Xnp ðTp0 Þ: The corollary follows easily from the previous proposition. &
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3. Kuznietsov trace formula Let f be a continuous function on GðAÞ: The kernel of Rð f Þ is defined as X f ðx 1 gyÞ: Kðx; yÞ ¼
ð3Þ
gAGðQÞ
Another way to express the kernel is Kðx; yÞ ¼
X
Rð f ÞfðxÞfðyÞ:
f
Here f runs through an orthonormal basis of L2 : When f ¼ fk f n ; Rð f Þ annihilates AðN; kÞ> : We can sum over an orthonormal basis of AðN; kÞ: An orthonormal basis can be taken as fjjjjh ðxÞjj : hAEðN; kÞg: It is easy to show that h
Kðx; yÞ ¼
X hAEðN;kÞ
Rð f Þjh ðxÞ jh ðyÞ : jjjh jj jjjh jj
ð4Þ
Use y to denote a character on Q\A: We can decompose y into yN yfin : Here yN (resp. yfin ) is the infinite (resp. finite) component of y: There exists rAQ; such that rAZþ throughout the whole paper. Under this yN ðxÞ ¼ e2pirx : We assume Q assumption yfin is trivial on p Zp : The character y can also be regarded as a character on NðQÞ\NðAÞ: Q rp We factorize r into p : Normalize measure on Qp by taking measðZp Þ ¼ 1: Define measure on A by using the product measure. We can show that measðQ\AÞ ¼ 1: Measure on NðAÞ is defined by identifying A with NðAÞ: From (3), the kernel Kðx; yÞ is invariant under left multiplication of elements in NðQÞ NðQÞ: Kuznietsov trace formula is the equality obtained by expanding the following integral using the two formulas (3) and (4) Z Z KTFð f Þ ¼ Kðn1 ; n2 Þyðn 1 ð5Þ 1 n2 Þ dn1 dn2 : NðQÞ\NðAÞ
NðQÞ\NðAÞ
The integral is convergent because Q\A is compact. The expression obtained using formula (3) is called the geometric side. Using formula (4) we obtain the spectral side. Proposition 3.1. When f ¼ fk f n ; then KTFð f Þ is equal to 0 1 X Y @ n1=2 Xnp ðxhp ÞAwrh ; hAEðN;kÞ
pjn
ARTICLE IN PRESS C.C. Li / Journal of Number Theory 104 (2004) 177–192
182
recall that wrh ¼
jar ðhÞj2 e 4pr jjhjj2
:
Proof. Using (4) and Corollary 2.2 0
X
Kðx; yÞ ¼ n1=2
@
hAEðN;kÞ
Y
1 Xnp ðxhp ÞA
pjn
jh ðxÞjh ðyÞ jjhjj2
:
Thus the spectral side of (5) is 0
X
n1=2
@
hAEðN;kÞ
Y pjn
1
2 Z 1
1 Xnp ðxhp ÞA j ðnÞyðn Þdn : h 2 jjhjj NðQÞ\NðAÞ
From [Ge, Chapter 3, Lemma 3.6], Z
jh ðnÞyðn 1 Þ dn ¼ NðQÞ\NðAÞ
The proposition follows easily.
ar e 2pr
if rAZþ ;
0
otherwise:
&
Let dAG: We define Nd ¼ fðn1 ; n2 ÞAN N : n 1 1 dn2 Bdg; here g1 Bg2 if g1 ¼ zg2 for some z in the center. Denote the image of d in NðQÞ\GðQÞ=NðQÞ by ½d : Proposition 3.2. KTFð f Þ ¼
Z
X ½d ANðQÞ\GðQÞ=NðQÞ
Nd ðQÞ\NðAÞNðAÞ
1 f ðn 1 1 dn2 Þyðn1 n2 Þ dn1 dn2 :
Proof. The geometric side of KTFð f Þ Z Z X
1 ¼ f ðn 1 1 gn2 Þyðn1 n2 Þ dn1 dn2 NðQÞ\NðAÞ
NðQÞ\NðAÞ
gAGðQÞ
ARTICLE IN PRESS C.C. Li / Journal of Number Theory 104 (2004) 177–192
Z
X
¼
½d ANðQÞ\GðQÞ=NðQÞ
¼
XZ
Z
NðQÞ\NðAÞ
Z
X
NðQÞ\NðAÞ
X
½d
183
1 f ðn 1 1 gn2 Þyðn1 n2 Þ dn1 dn2 :
gANðQÞdNðQÞ
1
1 f ðn 1 1 m1 dm2 n2 Þyðn1 n2 Þ dn1 dn2 :
ðm1 ;m2 ÞANd ðQÞ\NðQÞNðQÞ
1 Replace n1 by m 1 1 n1 ; n2 by m2 n2 : Since y is trivial on m1 ; m2 ANðQÞ; KTFð f Þ is equal to Z X
1 f ðn 1 & 1 dn2 Þyðn1 n2 Þ dn1 dn2 : ½d ANðQÞ\GðQÞ=NðQÞ
Nd ðQÞ\NðAÞNðAÞ
Denote Id ð f Þ ¼
Z Nd ðQÞ\NðAÞNðAÞ
1 f ðn 1 1 dn2 Þyðn1 n2 Þ dn1 dn2 :
An element dAGðQÞ is said to be admissible if the map Nd ðAÞ-C : ðn1 ; n2 Þ/yðn 1 1 n2 Þ is trivial. Lemma 3.3. If d is not admissible, then Id ð f Þ ¼ 0: Proof. Assume that d is not admissible. Let ðn1 ; n2 ÞANd ðAÞ such that yðn 1 1 n2 Þa1: Replace n1 ; n2 by n1 n1 ; n2 n2 ; respectively. Z
1
1 1 f ðn 1 Id ð f Þ ¼ 1 n1 dn2 n2 Þyðn1 n1 n2 n2 Þ dn1 dn2 : Nd ðQÞ\NðAÞNðAÞ
Thus Id ð f Þ ¼ yðn 1 1 n2 ÞId ) Id ¼ 0: Theorem 3.4. Z KTFð f Þ ¼ NðAÞ
f ðnÞyðnÞ dn þ
&
X mAQ
Z NðAÞ
Z NðAÞ
0 m 1 f n 1 1 1 0 n2 yðn1 n2 Þ dn1 dn2 :
Proof. By the Bruhat decomposition G ¼ NM,NM 01 10 N: Thus a representative g 0
set of NðQÞ\GðQÞ=NðQÞ is 0 1 : gAQ , 01 m0 : mAQ : From Theorem 3.2, the P geometric side ¼ ½d Id : By Lemma 3.3, Id ð f Þ ¼ 0 unless d is admissible.
g 0 1 t 1 gt When d ¼ 0 1 ; simple calculation shows that Nd ¼ : ; 1 1 If d is admissible, we have yððg 1ÞtÞ ¼ 1 for any tAA: Because y is non-trivial, this
ARTICLE IN PRESS C.C. Li / Journal of Number Theory 104 (2004) 177–192
184
cannot happen unless g ¼ 1: Thus Ie ð f Þ ¼ ¼
Z f Z
Nd ðQÞ\NðAÞNðAÞ
diagonal\NðAÞNðAÞ
n 1 1
1 0
0 n2 yðn 1 1 n2 Þ dn1 dn2 1
1 f ðn 1 1 n2 Þyðn1 n2 Þ dn2 dn1 :
Letting m1 ¼ n1 ; m2 ¼ n 1 1 n2 ; the diagonal becomes ðm1 ; eÞ: The integral becomes Z
f ðm2 Þyðm2 Þ dm2 dm1 ¼
NðQÞe\NðAÞNðAÞ
Z
f ðm2 Þyðm2 Þdm2 : NðAÞ
When d ¼ 01 m0 ; simple calculation shows that Nd ðQÞ ¼ fðe; eÞg: Thus d is admissible. We have Id ð f Þ ¼
Z NðAÞNðAÞ
0 f n 1 1 1
m 0
n2 yðn 1 1 n2 Þ dn1 dn2 :
We can prove the theorem by summing up all the terms.
&
4. Evaluation of Integrals Lemma 4.1. For uAQ; yfin ðuÞ ¼ yN ð uÞ ¼ e 2pru Proof. We have 1 ¼ yðuÞ ¼ yN ðuÞyfin ðuÞ: The lemma follows easily.
&
The following lemmas show us how to evaluate f n : Suppose R is a ring, we denote R ¼ fx2 : xAR g:
2
Q Lemma 4.2. Suppose g ¼ ac db AGðAfin Þ and det ðgÞAn Zp : Then gA supp f n if and Q only if gAM2 ð Zp Þ and c 0 mod N:
AZðAfin Þ; mAMðn; NÞ: Taking the determinant z Q on both sides, we see that z is in Zp : Thus z can be absorbed into m; so gAMðn; NÞ: It is easy to see that gAMðn; NÞ if and only if g satisfies the conditions in the lemma. &
Proof. Write g ¼ zm; z ¼
Lemma 4.3. Suppose g ¼ Q nA 2 Zp : fin
z
a b n only if det gA c d AGðAfin Þ; then g is in supp f
ARTICLE IN PRESS C.C. Li / Journal of Number Theory 104 (2004) 177–192
Q
zAA fin ;
2
Z p :
185
z
Under this assumption, say det g ¼ nz u for uA Let z ¼ : Let z Q n 11 m12 m ¼ z 1 g ¼ m if and only if mAM2 ð Zp Þ and m21 m21 m22 : Then gA supp f 0 mod N: z Proof. If g is in supp f n ; then g ¼ zm with z ¼ AZðAfin Þ and mAMðn; NÞ: z Q Thus det g ¼ z2 det mAnA 2 Zp : This proves the first part. fin z 2 Suppose det g ¼ nz u as stated in the lemma. Let z ¼ : Obviously z gA supp f n if and only if z 1 gA supp f n : One can easily show that det z 1 g ¼ Q nuAn Z p : The lemma follows easily from the above lemma. & From now on we call the z ¼
z
appearing in the previous lemma a
z
z-part of g. Define TrueðstatementÞ ¼
1 if statement is true; 0 otherwise:
Proposition 4.4. Z
f n ðnfin Þyfin ðnfin Þ dnfin NðAfin Þ
¼ cðNÞn1=2 Trueðn1=2 AZ
and n1=2 jrÞ:
ð6Þ
Proof. Write nfin ¼ 10 1t : Its determinant is 1. From Lemma 4.3, nfin A supp f n only
2
if 1AnQ 2 p Zp for all primes p: Thus np is even for all p: As a result nA7Q : We can assume n is positive. Write n ¼ n02 ; n0 AZþ : 0 1 0 0 Now a z-part of nfin can be z ¼ n0 n00 : Let m ¼ z 1 nfin ¼ n0 nn0t : The lower left entry of m is 0, which is divisible by N: By Lemma 4.3, nfin is in supp f n if and only if Q 0 0 n nt Zp Þ: Or equivalently 0 n0 AM2 ð t0 ¼ n0 tA
Y
Zp :
Thus (6) is equal to Z
Z cðNÞ tAn0 1
Q
Zp
yfin ðtÞ dt ¼ cðNÞ Q
Zp
0 0 t t yfin 0 d 0 : n n
ARTICLE IN PRESS C.C. Li / Journal of Number Theory 104 (2004) 177–192
186
Q
Because yfin is trivial on
Zp ; integral (6) is equal to 0 X Z t y cðNÞn0 dt0 fin Q n0 0 sþn
sAZ=n0 Z
X
¼ cðNÞn0
sAZ=n0 Z
Zp
sr Y Zp ¼ cðNÞ e 0 meas s þ n0 n
The result follows easily.
X sAZ=n0 Z
e
sr : n0
&
Next, we evaluate Z Afin Afin
0 m f n n 1 n2 yfin ðn 1 1 1 Þyfin ðn2 Þ dn1 dn2 : 1 0
ð7Þ
Define X
Klu ðn; yN Þ ¼
yN
s1 ;s2 AZ=uZ;s1 s2 n ðmod uÞ
s s 1 2 yN : u u
Proposition 4.5. Integral (7) a0 only if m ¼ 7un2 for some integer u 0 ðmod NÞ: Under this assumption, integral (7) is equal to cðNÞ Klu ð8n; yN Þ: Proof. Let ni ¼
1
ti ; i ¼ 1; 2: 1 n 1 1
0
m
1
0
n2 ¼
t1
m t1 t2
1
t2
:
ð8Þ
0 m
2 n Notice det n 1 1 1 0 n2 ¼ m: From Lemma 4.3, (8) A supp f only if mAnQp Zp for all p: Thus we have ordp ðmÞ np ðmod 2Þ for all p: As a result mA7nQ2 : Let m ¼ z 2
as the z-part. Write m ¼ 7nz for some zAQ : We can take z ¼ z
1
0 m n z 1 n 1 1 1 0 n2 ¼ z 1 : By Lemma 4.3, (8) is in supp f only if z AZp for all p: Hence z ¼ 1=u for some uAZþ : Let m ¼ 7un2 ; uAZ: m¼
ut1 u
7n ðut1 Þðut2 Þ u
ut2
! :
Write t01 ¼ ut1 ; t02 ¼ ut2 : By Lemma 4.3 again, (8) is in supp f n if and only if Y t01 ; t02 A Zp ; u 0 ðmod NÞ; t01 t02 7n ðmod uÞ:
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187
Integral (7) becomes Z f t1 ;t2
¼ ¼ ¼ ¼
n
t1
m t1 t2
1
t2
yfin ð t1 Þyfin ðt2 Þ dt1 dt2
0 0 t01 t2 t1 t02 d f
y y d fin fin Q 00 Q u u u u u t02 t01 ;t02 A Zp ;t1 t2 A7nþu Zp 0 0 0 Z X t t t t0 yfin 1 yfin 2 d 1 d 2 cðNÞ Q Q u u u u ðs1 þu Zp Þðs2 þu Zp Þ s1 ;s2 AZ=uZ;s1 s2 ¼7n Y 2 s s X 1 2 cðNÞ yN yN meas Zp u u s1 ;s2 AZ=uZ;s1 s2 ¼7n s s X 1 2 cðNÞ yN & yN ¼ cðNÞ Klu ð8n; yN Þ: u u s ;s AZ=uZ;s s ¼8n Z
n
1
2
t01
1 2
Proposition 4.6. When kX3; (
Z
fk ðnÞyN ðnÞ dn ¼
NðRÞ
Proof. Write n ¼
1
e 4pr ð4prÞk 1 ðk 2Þ!
if r40;
0
otherwise:
ð9Þ
t : By (2), the integral in (9) is equal to 1 k 1 4p
Z
2k
N
N
ð2 þ itÞ
k
e2pirt dt:
When r40; use the x-axis and the upper semi-circle as the contour. We can get the result easily by evaluating the residue of the integrand at t ¼ 2i: When ro0; use the x-axis and the lower semi-circle as the contour. The result follows easily. & Proposition 4.7. When kX3; Z NðRÞNðRÞ
0 fk n 1 1 1
m 0
n2 yN ðn 1 1 n2 Þ dn1 dn2
ð10Þ
is non-zero only if r; m are all positive. Under this condition, the integral is equal to 1 e 4pr ð4piÞk rk 1 pffiffiffiffiffiffiffi ð mÞ2 Jk 1 ð4pr mÞ; 2ðk 2Þ!
here Jk is the Bessel J function.
ARTICLE IN PRESS 188
C.C. Li / Journal of Number Theory 104 (2004) 177–192
m
1 Proof. When m40; det n1 n2 ¼ mo0: The value of fk at it is 0. So we 1 1 ti can assume mo0: Write ni ¼ ; i ¼ 1; 2: By (2), (10)¼ 1 k 1 4p
Z
N
N
Z
N
2k ð mÞk=2 e2pirðt2 t1 Þ
N
ð t1 þ t2 þ ið 1 þ m t1 t2 ÞÞk
dt1 dt2 :
First assume r40; evaluate the residue of the integrand at t2 ¼ i þ t1mþi; (10) becomes k 1 4p
Z
N
N
2k ð mÞk=2 ð2pirÞk 1 e ðk 1Þ! ð1 it1 Þk 2pi
m 2pirðiþt þi t1 Þ 1
dt1 :
Use x-axis and lower semi-circle as the contour, the integral can be calculated by evaluating the residue of the integrand at t1 ¼ i: Notice that the path is counterclockwise. Refer to [Wa, Chapter 2.1], we have 1
1
e2xðt tÞ ¼
N X
tn Jn ðxÞ:
N
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi Let x ¼ 4pr m; t ¼ it1 þ i= m be the residue theorem, (10) becomes k 1 2pi 2k ð mÞk=2 ð2pirÞk 1 4pr pffiffiffiffiffiffiffi ð iÞk 1 e ð 2piÞ J ð4pr
mÞ : k 1 k 1 4p ð iÞk ðk 1Þ! ð mÞ 2 We can get the result easily. When ro0; use the axis and upper semi-circle as the contour. It is easy to show that (10) is 0. & Theorem 4.8. Let k be an even number X3: Let n; N; r be any positive integers. Q Factorize n into p pnp : Assume GCDðN; nÞ ¼ 1: Define yN ðxÞ ¼ e2pirx ; then we have 0 1 X Y jar ðhÞj2 e 4pr @ Xnp ðxhp ÞA jjhjj2 hAEðN;kÞ pjn ¼ Trueðn1=2 AZ; n1=2 jrÞ
e 4pr ð4prÞk 1 cðNÞ ðk 2Þ!
N X e 4pr ð4piÞk rk 1 1 4pn1=2 r Jk 1 þ cðNÞ KlvN ðn; yN Þ: Nv Nv 2ðk 2Þ! v¼1 Proof. The spectral side is obtained by Proposition 3.1.
ð11Þ
ð12Þ
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189
The geometric side is obtained by Theorem 3.4. The integral Ie ð f Þ is the product of (6) and (9). Using the results in Proposition 4.6 and Corollary 4.4, we can get Ie ð f Þ:
Let d ¼ 01 m0 : Then Id ð f Þ is product of (7) and (10). By Proposition 4.5 ð7Þa0 only if m ¼ 7n and Nju: Write u ¼ Nv: By Proposition 4.7, ð10Þa0 only if mo0: Thus u2 ( ) !
n 0 ðNvÞ P 2 d¼ : vAZþ : d Id ð f Þ is a sum over 1 0 Multiply the results we obtained in Propositions 4.5 and 4.7. Summing up all the terms in the geometric side. We can obtain the formula by equating the spectral side and geometric side and then dividing both sides by n1=2 : &
5. Weighted distribution Lemma 5.1. jKlu ðn; yN Þjpun: Proof. Obviously jKlu ðn; yN Þjpjfs1 s2 n ðmod uÞgj: It suffices to prove jfs1 s2 n ðmod uÞgjpun for u ¼ pup and n ¼ pnp : If np Xup ; jfs1 s2 n ðmod uÞgjpu2 pun: Assume np oup : jfs1 s2 n ðmod uÞgj ¼
u X
jft : st n ðmod uÞgj
s¼1
¼
u X
gcdðs; uÞ ¼
psp jfs : ordp ðsÞ ¼ sp ; 1pspugj
sp ¼0
s¼1;gcdðs;uÞjn
p
np X
np X
psp
sp ¼0
pu p pðnp þ 1Þpup pnu ps p
From [Iw, equation (5.16)], Jk ðxÞpminfxk ; x 1=2 g:
Proposition 5.2. Let k be an even integer X3; then 1 cðNÞ
X
Y
hAEðN;kÞ
pjn
Xnp ðxhp Þwrh
&
ARTICLE IN PRESS C.C. Li / Journal of Number Theory 104 (2004) 177–192
190
¼ Trueðn
1=2
AZ; n
1=2
ðkþ1Þ=2 e 4pr ð4prÞk 1 n þO jrÞ : ðk 2Þ! N k 1
Here the constant in the O-notation depends on y only. Proof. Form the inequalities given above, (12) is {
k 1 N X e 4pr ð4pÞk rk 1 1 4pn1=2 r cðNÞ ðNvnÞ Nv Nv 2ðk 2Þ! v¼1
{r
cðNÞ ðkþ1Þ=2 X 1 cðNÞ n {r k 1 nðkþ1Þ=2 : k 1 N k 1 v N v
The proposition follows easily from Theorem 4.8.
&
Corollary 5.3. The average of weight 1 cðNÞ
X
wrh ¼
hAEðN;kÞ
e 4pr ð4prÞk 1 1 þO : N k 1 ðk 2Þ!
Proof. Put n ¼ 1 in the above proposition. & Let p1 ; p2 ; ypc be distinct primes. For any prime p; let Ip be an interval in R containing all the possible values of xhp for any Hecke eigenform h: By Deligne’s result, we can take Ip ¼ ½ 2; 2 ; but we do not need this strong result. We only need the fact that Ip is a finite interval. Refer to [Ro, Proposition 2.9] for the proof of this fact. Denote the set of real valued continuous functions on I ¼ Ip1 Ip2 ? Ipc by CðIp1 Ip2 ? Ipc Þ: Define a topological structure on it by using the LN norm jj f jjN ¼ maxfj f ðxÞjg: Let k be an even integer X3; define a functional FN on CðIp1 Ip2 ? Ipc Þ by P FN f ¼
h h r hAEðN;kÞ f ðxp1 ; y; xpc Þwh P r hAEðN;kÞ wh
Proposition 5.4. lim
N-N;ðN;p1 ;y;pc Þ¼1
FN Xnp1 ? Xnpc ¼
c Y i¼1
Trueð2jnpi ; npi =2prpi Þ
ARTICLE IN PRESS C.C. Li / Journal of Number Theory 104 (2004) 177–192 np
191
np
Proof. Let n ¼ p1 1 ?pc c : From Corollaries 5.2 and 5.3, Qc
i¼1
FN f ¼
Trueð2jnpi ; npi =2prpi Þe e 4pr ð4prÞk 1 ðk 2Þ!
ð4prÞk 1 ðk 2Þ!
4pr
ðkþ1Þ=2
þ OðnN k 1 Þ
1 þ OðN k 1 Þ
Letting N-N; the proposition follows easily.
:
&
Proposition 5.5. Z
Xn ðxÞXm ðxÞ dmN ðxÞ ¼ dnm :
R
Proof. A proof can be found in [Se2, Section 2.2]. p dnm ¼ 2
Z
p
sin ny 2 sin my sin y dy: sin y sin y
0
ny Recall Xn ðxÞ ¼ sin sin y ; x ¼ 2 cos y: Make a substitution x ¼ 2 cos y; we have
p dnm ¼ 2 The proposition follows easily.
Z
2
Xn ðxÞXm ðxÞ
2
sin y dx: 2
&
Theorem 5.6. Define measure X
dmi ðxÞ ¼
X2n0 ðxÞ dmN ðxÞ:
0pn0 prpi
Also define Fð f Þ ¼
Z
Z ?
Ip1
I pc
f ðx1 ; yxc Þdm1 ðx1 Þ?dmc ðxc Þ:
Then for any f ACðIp1 Ip2 ? Ipc Þ; lim FN ð f Þ ¼ Fð f Þ:
N-N
Proof. By the previous proposition Z Ipi
Xnpi ðxÞ dmi ðxÞ ¼ Trueð2jnpi ; npi =2prpi Þ:
ð13Þ
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Take the product over i ¼ 1; yc and by Proposition 5.4, we have FðXnp1 ? Xnpc Þ ¼
lim
N-N;ðN;p1 ;y;pc Þ¼1
FN ðXnp1 ? Xnpc Þ:
One can easily show that jFN ð f Þjpjj f jjN : Thus FN is a continuous linear functional. Because deg Xn ¼ n; the linear span of fXn g consists of all the one variable polynomials. Thus the linear span of fXnp1 ? Xnpc g consists of all the possible polynomials. The theorem follows by the fact that polynomials are dense in CðIp1 Ip2 ? Ipc Þ: & Theorem 5.7. Let k be an even integer X3: Consider the family of sets SN ¼ fðxhp1 ; y; xhpc Þ : hAEðN; kÞg; ðN; p1 ?pc Þ ¼ 1 with weight fwrh g assigned to ðxhp1 ; y; xhpc Þ: Then the family of sets fSN ; p[Ng is fwrh g-distributed with respect to the measure dm1 ?dmc when N-N: Here mi is given by Eq. (13). The measure mi has the following properties: (a) it is supported on ½ 2; 2 ; (b) it is a polynomial times the Sato–Tate measure on ½ 2; 2 ; (c) it depends only on ordp ðrÞ: Proof. Follows easily from the previous theorem. Proof of 1.1. Take c ¼ 1 and p1 ¼ p:
&
&
References [Ge] [Iw] [KL] [Li] [Ro]
[Se1] [Se2] [Va] [Wa]
S. Gelbart, Automorphic Forms on Adele Groups, Annals of Mathematics Studies, No. 83, Princeton University Press, Princeton, NJ, 1975. H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, Vol. 17, American Mathematical Society, 1991. A. Knightly, C.C. Li, Traces of Hecke Operators, in preparation. C.C. Li, Kuznietsov trace formula and asymptotic behavior of Hecke eigenvalues, UCLA thesis, 2001. J. Rogawski, Modular forms, the Ramanujan conjecture, and the Jacquet-Langlands correspondence, in: A. Lubotzky (Ed.), Discrete, Groups, Expanding Graphs and Invariant Measures, Birha¨user, Basel, 1994, pp. 135–176. J-P. Serre, Abelian l-adic representations and elliptic curves, W. A. Benjamin, Inc., New York, Amsterdam, 1968. J.-P. Serre, Re´partition asymptotique des valeurs propres de l’ope´rateur de Hecke Tp ; J. Amer. Math. Soc. 10 (1) (1997) 75–102. V.S. Varadarajan, An Introduction to Harmonic Analysis on Semisimple Lie Groups, Cambridge University Press, Cambridge, 1989. G.N. Watson, Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944.
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 193–209
http://www.elsevier.com/locate/jnt
Local–global problem for Drinfeld modules Gert-Jan van der Heiden Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, Netherlands Received 15 April 2002; revised 16 April 2003 Communicated by D. Goss
Abstract Let K be a function field with an A-algebra structure. The ring A arises in the definition of the Drinfeld module f over K: By EðKÞ we denote K together with the A-module structure induced on it by f: For any principal prime ideal ðaÞCA; we study the question whether an element xAEðKÞ which is an a-fold in EðKn Þ for every place n of K; is an a-fold in EðKÞ: In particular, we study the group ! Y Sða; KÞ :¼ ker EðKÞ=aEðKÞEðKn Þ=aEðKn Þ n
for Drinfeld modules of rank 2: We show that this finite group is trivial in many cases, but can become arbitrarily large. r 2003 Elsevier Inc. All rights reserved. Keywords: Drinfeld-modules; Elliptic curves; Local–global principle
1. The problem Let X be a projective, smooth, absolutely irreducible curve over Fq ; with charðFq Þ ¼ p; and let NAX be some fixed closed point on X : By Fq ðX Þ we will denote the function field of X and A will denote the ring of functions in Fq ðX Þ which are regular outside N: Let K be some separable, finite extension of Fq ðX Þ; which is also an A-algebra via the natural embedding g : A-K and let K s be the separable closure of K inside some algebraic closure K alg of K: Let Kftg be the skew P i polynomial ring consisting of elements i ki t ; ki AK: Multiplication in Kftg is E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00163-X
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given by the rule tk ¼ kq t for all kAK: Let f be a Drinfeld module over K of rank r; i.e., f is a ring homomorphism f : A-EndFq ðGa;K ÞDKftg such that for all aAA fa ¼
rdegðaÞ X
k i ti
with krdegðaÞ AK ; k0 ¼ gðaÞ:
i¼0
Note that we write fa instead of fðaÞ: For any field extension L of K; f induces an A-module structure on L via the embedding KCL; by ja ðlÞ ¼
X i
ki ti ðlÞ ¼
X
ki l q
i
for all lAL:
i
EðLÞ denotes L with this A-module structure. We write E½aðLÞ for kerðfa ÞðLÞ and moreover aQ for fa ðQÞ; with QAEðLÞ: Furthermore, we abbreviate E½a ¼ E½aðK s Þ: Let ðaÞCA be a principal prime ideal of A: The object of study in this paper is the kernel Sða; KÞ :¼ ker EðKÞ=aEðKÞ-
Y
! EðKn =aEðKn Þ ;
n
where the product is taken over all places n of K:
2. The group Sða; KÞ For any PAEðKÞ; let KP :¼ the splitting field of fa ðZÞ PAK½Z over K: Lemma 1. The finite field extensions KCK0 CKP are Galois. d Proof. Because dZ ðfa ðZÞ PÞ ¼ gðaÞa0; the polynomial fa ðZÞ P is separable and hence KP is a finite Galois extension of K: In particular, K0 is a Galois extension of K: &
Note that Fq ðX Þ is a function field, i.e., a finite separable extension of Fq ðtÞ; for some transcendental element tAFq ðX Þ: For function fields, as well as for number fields, we have Chebotarev’s density theorem, cf. [5]. The following lemma is a consequence of this theorem.
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Lemma 2. Let K be a function field and let L be a finite separable extension of K: If for all places n of K there exists a place o lying over n of L of degree 1; then K ¼ L: Proof. Let M be the normal closure of L=K; then both M=L and M=K are finite Galois extensions. Let H ¼ GalðM=LÞ and G ¼ GalðM=KÞ: By Chebotarev, every sAG is the Frobenius of some place m of M lying above some place n of K: This implies that s mod n generates the Galois group Galðkm =kn Þ; where km and kn denote the residue fields at m and n; respectively. Because both M=K and M=L are Galois, there is a tAG such that the conjugate m0 ¼ tðmÞ of m lies above a place o of L; which has degree 1 over n: In particular, we see that tst 1 generates Galðkm0 =kn Þ ¼ Galðkm0 =ko Þ; where the equality follows from the fact that degðo=nÞ ¼ 1: And thus we see that tst 1 AH; so sAt 1 Ht: We conclude that [ [ G¼ tHt 1 ¼ tHt 1 : tAG
tHAG=H
Note that 1AtHt 1 for all tAG: On the other hand, G equals the union of all distinct cosets tH; which is a disjoint union. By comparing the number of elements one sees that this is only possible if H ¼ G; hence K ¼ M G ¼ M H ¼ L: & Proposition 3. For every class ½P ¼ P þ aEðKÞASða; KÞ we have KP ¼ K0 : In particular Sða; K0 Þ ¼ 0: Proof. First note that for every QAaEðKÞ; we have KQ ¼ K0 ; hence the extension KP only depends on the class ½P ¼ P þ aEðKÞ: Let now PASða; KÞ and let n be a place of K: Then KCK0 CKP and correspondingly we have places n; n0 and nP ; with n0 a place of K0 lying above n and nP a place of KP lying above n0 : Because PASða; KÞ; there exists a solution of fa ðZÞ P ¼ 0 in Kn ; hence all solutions of this equation lie in ðK0 Þn0 : This means that K0 CKP CðK0 Þn0 : It follows that ðK0 Þn0 ¼ ðKP ÞnP and in particular degðnP =n0 Þ ¼ 1: From Lemmas 1 and 2, it follows that KP ¼ K0 : If ½PASða; K0 Þ; then ðK0 ÞP ¼ ðK0 Þ0 ¼ K0 ; hence PAaEðK0 Þ; thus ½P ¼ 0: & Once and for all, we fix the notation G ¼ GalðK0 =KÞ and as usual we write GL ¼ GalðLs =LÞ for a function field L: Proposition 4. We have Sða; KÞD
\
kerðReso Þ:
o
Here the intersection is taken over all places o of K0 and Reso is the restriction map Reso : H 1 ðG; E½aÞ-H 1 ðDo ; E½aÞ; where Do denotes the decomposition group at o:
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Proof. Consider the following diagram:
ðÞ
Clearly, the second and third row are exact as well as all columns. From this it follows that also the first row is exact. Hence we see that Sða; KÞDC: To determine the groups C and B; we use some Galois cohomology. Starting from the exact sequence fa
0-E½a-EðK s Þ ! EðK s Þ-0; we deduce that EðKÞ=aEðKÞ+H 1 ðGK ; E½aÞ: By additive Hilbert 90 the cokernel of this map is H 1 ðGKk ; EðK s ÞÞ ¼ 0; because EðK s Þ is here just ðK s Þþ ; hence EðKÞ=aEðKÞDH 1 ðGK ; E½aÞ: Similarly we deduce that EðK0 Þ=aEðK0 ÞDH 1 ðGK0 ; E½aÞ: In particular this means that FDH 1 ðG; E½aÞ: Applying the same arguments to Kn and K0;o ; for some place n of K and o of K0 lying above n; we deduce kerðEðKn Þ=aEðKn Þ-EðK0;o Þ=aEðK0;o ÞÞDH 1 ðDo ; E½aÞ; where Do denotes the decomposition group at o: This isomorphism implies that Y \ H 1 ðDo ; E½aÞ; BD n
ojn
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where the product runs over all places n of K: If we furthermore note that the map Reso Q only depends on the place n underlying o; it follows that C is the kernel of the map n Reso ; with Reso as in the proposition. & 3. The group H 1 ðG; E½aÞ In the following we will write F ¼ A=ðaÞ and V ¼ E½a: For all field extensions L*K0 ; we have V ¼ kerðfa ÞðLÞ: Note that F is a field extension of Fq ; because ðaÞ is prime. It is well-known that V DFr ; where r is the rank of f: The action of sAG on elements in K0 commutes with the action of ff for all f AA and hence we have a representation G+Glr ðFÞ; which is an embedding because K0 is given by adjoining the elements of V ; which are the zeroes of fa ðZÞ; to K: Proposition 5. For every Drinfeld module of rank 1; we have Sða; KÞ ¼ 0: Proof. Note that G+F and thus p[#G; but V H 1 ðG; V Þ ¼ 0: &
is a p-group, hence
Proposition 6. Let F be a finite field of characteristic p and let W be an F-vectorspace of dimension r: If FaF2 ; then H 1 ðGlr ðFÞ; W Þ ¼ 0: If gcdðr; #F Þ41; then H 1 ðSlr ðFÞ; W Þ ¼ 0: Proof. For the first part, note that if FaF2 ; then we may choose aAF ; such that aa1: Hence H ¼ /aIS is a non-trivial normal subgroup of Glr ðFÞ: Note that W H ¼ 0: Moreover H 1 ðH; W Þ ¼ 0; because this group is annihilated by both #H; which is prime to p; and p: By the exact sequence 0 ¼ H 1 ðGlr ðFÞ=H; W H Þ-H 1 ðGlr ðFÞ; W Þ-H 1 ðH; W Þ ¼ 0; the first statement follows. For the second part, the condition gcdðr; #F Þ41 implies that we may take aAF ; such that aa1 and ar ¼ 1: Then H ¼ /aIS is a normal subgroup of Slr ðFÞ: The same argument as above gives the result. &
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Remark 7. For rank r ¼ 2 the Galois group G is generically Gl2 ðFÞ; cf. [3]. It is conjectured that for arbitrary rank this is also true, i.e., the Galois group is generically Glr ðFÞ: Proposition 6 states that, given this conjecture, Sða; KÞ is generically 0: 4. The rank 2 case From now on we will assume that the rank of the Drinfeld module f is 2: Throughout the rest of this paper H will denote H :¼ G-Sl2 ðFÞ: Note that H is a normal subgroup of G and that p[½G : H; hence H 1 ðG=H; V H Þ ¼ 0 and we see by group cohomology that H 1 ðG; V Þ+H 1 ðH; V Þ: The classification of subgroups Sl2 ðFq Þ from [7] shows that H is one of the following. (1) p[#H: (2) D2n ; in this case p ¼ 2 and
n is odd. (3) 1 0 p ¼ 3 and H ¼ CSl2 ðF9 Þ with i2 ¼ 1: In this case, HDSl2 ðF5 Þ and 1 1 H=/71SDA5 : (4) Sl2 ðFpk Þ; where Fpk CFq :
(5) l 0 Sl2 ðFpk Þ; ; where l2 generates Fpk ; but leFpk : 0 l 1 (6) H is a Borel group, i.e., H has a normal abelian p-Sylow subgroup Q; such that H=Q is cyclic of order dividing #F :
In the following proposition, we deal with most of the subgroups in the classification. Theorem 8. If H is of type ð1Þ or ð2Þ or if p42 and 2j#H; then H 1 ðH; V Þ ¼ 0: Consequently, in all these cases Sða; KÞ ¼ 0: Proof. We consider the different types of H:
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Type (1): H 1 ðH; V Þ is annihilated by both p and #H; hence H 1 ðH; V Þ ¼ 0: Type (2): In this case p ¼ 2: We consider the following exact sequence, where x is one of the generators of order 2 of D2n Res
Cor
H 1 ðD2n ; V Þ ! H 1 ð/xS; V Þ ! H 1 ðD2n ; V Þ: Now by Serre [6] Cor3Res ¼ n: Because x is of order 2 and p ¼ 2; we know by the corollary to [6, Proposition VIII.4.6], that H 1 ð/xS; V ÞDH 1 ð/xS; V Þ: The latter group is isomorphic to the kernel kerð1 þ xÞ modulo the augmentation ideal. An easy computation shows that H 1 ð/xS; V Þ ¼ 0: Type p42 and 2#H: This implies that H contains the non-trivial normal subgroup /71S: Now the exact sequence 0 ¼ H 1 ðH=/71S; V /71S Þ-H 1 ðH; V Þ-H 1 ð/71S; V Þ ¼ 0; gives that H 1 ðH; V Þ ¼ 0:
&
Remark 9. The only cases of the classification which are not covered by this theorem are the following: p ¼ 2 and H is of type ð4Þ or ð5Þ; o`r H is of type ð6Þ (such that 2[#H). If p ¼ 2 and H is of type ð4Þ; then by Cline et al. [2, Table 4.5], we obtain that dimF H 1 ðG; V Þ ¼ 1: In Section 5 we discuss examples for p42 for which Sða; KÞ is non-trivial. For p ¼ 2 one can construct similar examples for H of type ð6Þ: For p ¼ 2 the types ð4Þ and ð5Þ might also give rise to non-trivial Sða; KÞ: We do not consider these two types in the sequel. H of type ð6Þ: In the rest of this section we will assume that H is of type ð6Þ and we compute H 1 ðH; V Þ: Let Q by the p-Sylow subgroup of H: Clearly H 1 ðH=Q; V Q Þ ¼ 0; because this group is annihilated by both p and #ðH=QÞ; which is prime to p: It follows that H 1 ðG; V Þ+H 1 ðH; V Þ+H 1 ðQ; V Þ:
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Let kAN such that pk ¼ #Q; then Q ¼ /s1 ; y; sk S and H ¼ /Q; rS; where si ¼
1 li 0
1
;
r¼
a
0
0
a 1
;
such that the li are linearly independent over Fp and aAF generates H=Q: Let tAQ and write Res/tS for the residue map Res/tS : H 1 ðH; V Þ-H 1 ð/tS; V Þ: Proposition 10. The F-vectorspace H 1 ðQ; V Þ has dimension ( dimFp Q if p42 1 dimF H ðQ; V Þ ¼ 1 þ dimFp Q if p ¼ 2: If H ¼ Q and sAQ is not the identity, then dimF kerðRes/sS Þ ¼ 1 þ dimFp Q: Proof. We write V ¼ Fe1 þ Fe2 : Note that V is an F½Q-module. The group ring F½Q is isomorphic to the commutative ring F½s1 ; y; sk with only relations spi ¼ 1: Write xi ¼ si 1; then F½Q is the commutative ring R ¼ F½x1 ; y; xk subject to the relations xpi ¼ 0 and these are the only relations. Note that F is isomorphic to R=ðx1 ; y; xk Þ: To compute H 1 ðQ; V Þ; we consider the following free resolution of the R-module F: 1
d1
d0
d 1
Rkþ2kðk 1Þ ! Rk ! R ! F-0; where the R-linear maps are given as follows: d 1 : 1/1 modðx1 ; y; xk Þ; write b1 ; y; bk for generators of Rk over R; then d0 : bi /xi ; 1
write c1 ; y; ck ; ci;j with 1piojpk for the generators of Rkþ2kðk 1Þ ; then bi ; d1 : ci /xp 1 i
d1 : ci;j /xi bj xj bi :
To see that the given sequence is exact, note that kerðd0 Þ is generated by the elements d1 ðci Þ; d1 ðci;j Þ for all i; j; because these exactly describe all relations in de the ring R:
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From this sequence we arrive at the cocomplex 1
d1
d0
V kþ2kðk 1Þ ’ V k ’ V ’0 with d0 ðvÞ ¼ ðx1 v; y; xk vÞ and p 1 d1 ðv1 ; y; vk Þ ¼ ðxp 1 1 v1 ; y; xk vk ; ðxi vj xj vi Þioj Þ:
To compute kerðd1 Þ and imðd0 Þ; note that the action of R on V is given by xi e1 ¼ 0 and xi e2 ¼ li e1 for all i: From this it follows immediately that imðd0 Þ is generated over F by the vector ðl1 e1 ; y; lk e1 Þ: Hence dimF imðd0 Þ ¼ 1: To compute kerðd1 Þ; note that xp 1 v ¼ 0 for all v if p42: So if p42; then an element of V k lies in kerðd1 Þ i iff xi vj ¼ xj vi : Write vi ¼ ai e1 þ bi e2 ; with ai ; bi AF; then xi ðaj e1 þ bj e2 Þ ¼ li bj e1 ¼ xj ðai e1 þ bi e2 Þ ¼ lj bi e1 : From this it follows that kerðd1 Þ is generated by ðe1 ; 0; y; 0Þ; y; ð0; y; 0; e1 Þ; ðl1 e2 ; y; lk e2 Þ; hence dimF kerðd1 Þ ¼ k þ 1: So we see that for p42; that dimF H 1 ðQ; V Þ ¼ k: If p ¼ 2; then elements in kerðd1 Þ must satisfy xp 1 vi ¼ xi vi ¼ 0; hence vi ¼ ai e1 ; i with ai AF: Hence kerðd1 Þ is contained in the span of ðe1 ; 0; y; 0Þ; y; ð0; y; 0; e1 Þ: For vectors in this span clearly also the other equations xi vj ¼ xj vi hold, hence this span equals kerðd1 Þ and thus dimF kerðd1 Þ ¼ k: So for p ¼ 2; we have dimF H 1 ðQ; V Þ ¼ k 1: Clearly, by this computation dimF H 1 ð/sS; V Þ ¼ 1 if p42 and 0 if p ¼ 2: From this the dimension formula for kerðRes/sS Þ follows. & Proposition 11. Suppose that HaQ; say H=QD/aS; with 2[ordðaÞ: Let d ¼ 1 if ordðaÞ ¼ 3 and p42 and d ¼ 0 otherwise. Let l ¼ dimFp ½a F: Then ( j 0 if ap aa2 for all j; 1 dimF H ðH; V Þ ¼ l þ d otherwise: Proof. First note that we may extend the restriction-inflation sequence as follows (cf. [6]): 0-H 1 ðH=Q; V Q Þ-H 1 ðH; V Þ-H 1 ðQ; V ÞH=Q -H 2 ðH=Q; V Q Þ;
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which induces an isomorphism H 1 ðH; V ÞDH 1 ðQ; V ÞH=Q ; because H 1 ðH=Q; V Q Þ ¼ H 2 ðH=Q; V Q Þ ¼ 0: We will now compute the H=Q-invariant cocycle classes in H 1 ðQ; V Þ: We will use the following notation: for a cocycle x : Q-V we write
xl 1 l xðsl Þ ¼ : ; where sl ¼ 0 1 yl We write x : F-F for the map x : l/xl and y : F-F for the second coordinate, then the cocycle relations and the relations between the elements in Q imply that x and y are determined by the following relations: xðm þ lÞ ¼ xðmÞ þ xðlÞ þ lmyð1Þ;
ð1Þ
yðlÞ ¼ lyð1Þ and yð1Þ ¼ 0 if p ¼ 2:
ð2Þ
So we see that in particular that y is F-linear. Let
a 0 r¼ ; 0 a 1 where a is as in the proposition. The action of H=Q on cocycles is given as follows: the cocycle ax maps s/r 1 xðrsr 1 Þ: An easy computation now shows that a cocycle class ½x represented by a cocycle x is invariant under H=Q when there is a coboundary Z; given by ðm1 ; m2 ÞAV ; such that for all sl AQ; r 1 xðsa2 l Þ ¼ xðsl Þ þ Zðsl Þ: m2 Let now Z* be the coboundary given by ð0; a 1 ÞAV : An easy computation shows that if we replace x by x Z* ; then for this x the following equation holds:
r 1 xðsa2 l Þ ¼ xðsl Þ: This x represents the class ½x uniquely and the relations read in coordinates: a 1 xða2 lÞ ¼ xðlÞ;
ð3Þ
ayða2 lÞ ¼ yðlÞ:
ð4Þ
So if we let W be the F-vectorspace consisting of tuples ðx; yÞ; where x : F-F is subject to relations (1) and (3) and y : F-F is subject to relations (2) and (4), then dimF H 1 ðH; V Þ ¼ dimF W : We will compute the latter dimension.
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Relations (2) and (4) imply that a3 yð1Þ ¼ yð1Þ: So either y ¼ 0 or a has order 3 and then y is determined by yð1Þ: If we let l1 ; y; ll be generators of F over Fp ½a; where l ¼ dimFp ½a F; then by (1) and (3), x is determined by xðli Þ; with i ¼ 1; y; l: Hence dimF W pl þ d: Suppose that yð1Þ ¼ 0; then x is Fp -linear. Let h be the minimal polynomial of a2 ; then for each lAF; we have 0 ¼ xðhða2 ÞlÞ ¼ hðaÞxðlÞ: Hence if hðaÞa0; i.e., if h is not the minimal polynomial of a; then xðlÞ ¼ 0: Note that h is the minimal j polynomial of a iff ap ¼ a2 for some jAN: Moreover if d ¼ 1; then the order of a is 3; i.e., a2 þ a þ 1 ¼ 0: One easily sees that a2 is the second root of 1 þ X þ X 2 besides j a; hence ap ¼ a2 : We conclude that if ap aa2 ; then xðlÞ ¼ yðlÞ ¼ 0; hence dimF H 1 ðH; V Þ ¼ 0: j Suppose now that ap ¼ a2 and let yð1Þ ¼ 0; then x is not only Fp -linear, but even Fp ½a-semi linear. This means that the F-subspace of W consisting of the tuples ðx; yÞ; with y ¼ 0; has dimension dimFp ½a F ¼ l: If d ¼ 0; then this subspace equals W and we see dimF H 1 ðH; V Þ ¼ l þ d: If d ¼ 1; then the dimension of W is either l or l þ 1: So let d ¼ 1; then p42 and ordðaÞ ¼ 3: Suppose that yð1Þa0 and let x : F-F be given by xðlÞ ¼ cl2 ; where c ¼ 12yð1Þ: Then one checks easily that x has property (1). And because ordðaÞ ¼ 3; it has property (3) as well. This shows that W contains an element ðx; yÞ with ya0; hence dimF H 1 ðH; V Þ ¼ l þ d: & In the following lemma, we show that kerðRes/sS Þ does not depend on the choice of 1asAQ: Lemma 12. For all s; tAQ; with both s and t not the identity, kerðRes/sS Þ ¼ kerðRes/tS Þ: Proof. For p ¼ 2; by the proof of Proposition 10, H 1 ð/sS; V Þ ¼ 0; hence kerðRes/sS Þ ¼ H 1 ðH; V Þ for all sAQ: Let now p42: Note that kerðRes/sS Þ ¼ H 1 ðH; V Þ-kerðH 1 ðQ; V Þ-H 1 ð/sS; V ÞÞ; because H 1 ðH; V Þ+H 1 ðQ; V Þ; so we may assume that H ¼ Q: Clearly, if s and t are linearly dependent over Fp ; then /sS ¼ /tS; so kerðRes/sS Þ ¼ kerðRes/tS Þ: If s and t are independent over Fp ; then we may extend them to a basis /s1 ; y; sk S; with s ¼ s1 ; t ¼ s2 and k ¼ dimFp Q:
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We write V ¼ Fe1 þ Fe2 such that the si ’s are upper triangular on the basis fe1 ; e2 g: Note that the kernel of Res/si S is the image of H 1 ðQ=/si S; V /si S ÞCHomðQ=/si S; F e1 Þ under the injective inflation map. The inflation map Inf : HomðQ=/si S; F e1 Þ-H 1 ðQ; V Þ # # j Þ ¼ aj AF; jai; then x is the cocycle is given by x/½x; such that if xð½s given by xðsj Þ ¼ ðaj ; 0ÞAV ;
jai
and
xðsi Þ ¼ ð0; 0Þ:
Now we will show that kerðRessk ÞCkerðRessl Þ: If ½xAkerðRessk Þ; it comes from a x# as mentioned above. Now let Z be a coboundary given by ðm1 ; m2 ÞAV ; hence Z : si /ðli m2 ; 0Þ
for all
i:
We choose m2 such that ll m2 þ al ¼ 0; then by construction there is a * * xAHomðQ=/s l S; F e1 Þ such that Ressl ðxÞ ¼ ½x þ Z: This shows that ½x ¼ ½x þ ZA kerðRessl Þ: & Recall that for any place o of K0 the map Reso is the restriction map Reso : H 1 ðG; V Þ-H 1 ðDo ; V Þ: Proposition 13. Let f : A-Kftg be a Drinfeld module of rank 2 and let H ¼ G-Sl2 ðFÞ be of type ð6Þ; with p-Sylow group Q: Let 1asAQ; then \ kerðReso Þ; Sða; KÞ ¼ kerðH 1 ðG; V Þ-H 1 ð/sS; V ÞÞ o:p2 j#Do
where the intersection is taken over places o of K0 : This intersection is finite. Proof. Suppose that o is any place of K0 : If p[#Do ; then kerðReso Þ ¼ H 1 ðG; V Þ; because H 1 ðDo ; V Þ ¼ 0: If pj#Do ; but p2 [#Do ; then kerðReso ÞCkerðH 1 ðG; V Þ-H 1 ð/sS; V ÞÞ; because H 1 ðDo ; V Þ+H 1 ð/sS; V Þ: By Chebotarev’s density theorem, it follows that there exists a place o of K0 with Do D/sS; from which the theorem follows.
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Note that the Sylow p-subgroup of Do has only elements of order p: This implies that if p2 j#Do ; then Do cannot be cyclic. Henceforth o is ramified if p2 j#Do : As there are only finitely many ramified primes, the finiteness of the intersection follows. & Remark 14. Clearly, kerðH 1 ðG; V Þ-H 1 ð/sS; V ÞÞCkerðRes/sS Þ; with Res/sS : H 1 ðH; V Þ-H 1 ð/sS; V Þ as before. Hence Proposition 13 combined with Propositions 11 and 10, gives a bound on dimF Sða; KÞ: Corollary 15. If f is a Drinfeld module of rank 2 over Fp ; p42 prime and ðaÞCA is a prime ideal of degree 1; then Sða; KÞ ¼ 0: Proof. In this case we have by Proposition 13 that Sða; KÞDkerðRes/sS Þ and because H 1 ðG; V Þ embeds into H 1 ðQ; V Þ; we have by Lemma 12, that dimF Sða; KÞp 1 þ dimFp Q: Note that now GCGl2 ðFp Þ; hence dimFp Q ¼ 1:
&
5. Examples of non-trivial Sða; KÞ In this section we show in Theorem 18 that there exist examples of Drinfeld modules over certain function fields K with non-trivial Sða; KÞ: Examples 16 and 17 prepare the proof of this theorem. Example 16. Let A ¼ Fq ½t; where charðFq Þ42 and let K ¼ Fq ðtÞ: Let f : A-Kftg be a Drinfeld module of rank 2 given by ft ¼ t þ tt þ t2 t2 : We let K0 ¼ Kðkerðft ÞÞ; then GCGl2 ðFq Þ: We consider the decomposition group Dt : Clearly, the Newton polygon of ft ðZÞ 1 has two slopes, namely 0 and qðq 1Þ : To factor ft ðZÞ in Fq ððtÞÞ½Z; we need at least a completely ramified extension of degree qðq 1Þ of Fq ððtÞÞ: Hence Dt -Sl2 ðFq Þ contains a subgroup of q elements. If we compare this with the classification of subgroups of Sl2 ðFq Þ in Section 4, we see that Dt contains Fq as a subgroup. We
1 Fq : conclude that G contains a subgroup isomorphic to 0 1 Recall that a system of Artin–Schreier equations over some Fp -field M zp1 z1 ¼ f1
with f1 AM;
^ zpn zn ¼ fn
with fn AM
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is called independent over M; if for all li AFp ; such that not all li are 0; the equation P zp z ¼ ni¼1 li fi has no solutions in M: Such a system gives rise to a tower of field extensions M ¼ M0 CM1 CyCMn where the extension Mi =Mi 1 is given by zpi zi ¼ fi and is of degree p: Example 17. Let q ¼ pk ; p42 and let f : Fq ½t-Kftg be a Drinfeld module of rank 2; such that GalðKðkerðft ÞÞ=KÞ contains a subgroup isomorphic to H¼
1
Fq
0
1
;
e.g. the examples of Example 16. We write ft ¼ t þ tc1 t þ tc2 t2 ;
c1 ; c2 AK;
hence K0 is the splitting field of the equation 1 þ c1 Z q 1 þ c2 Z q
2
1
¼ 0:
ð5Þ
Furthermore, there exist elements P; QAK0 ; such that kerðft Þ ¼ Fq P þ Fq Q: We let L ¼ K0H : Because GalðK0 =LÞ ¼ H; we may assume that PAL: If we substitute U ¼ Z q 1 in (5), we get 1 þ c1 U þ c2 U qþ1 ¼ 0:
ð6Þ
Let L1 be the splitting field of (6), then we have the field inclusion LCL1 CK0 ; where the latter field extension is given by the equation U ¼ Zq 1 : This implies that ½K0 : L1 jq 1; but ½K0 : L ¼ #H ¼ q; hence L1 ¼ K0 : This shows that K0 is the splitting field of (6) over L: Because PAL; we already know a solution of (6), namely u ¼ Pq 1 : Substituting V ¼ U u in (6) gives 1 þ c1 ðV þ uÞ þ c2 ðV þ uÞðV q þ uq Þ ¼ c1 V þ c2 Vuq þ c2 uV q þ c2 V qþ1 : Subsequently we divide out V and substitute W ¼ V 1 ; which shows that K0 =L is the splitting field of the equation Wq þ
c2 u c2 Wþ ¼ 0: c 1 þ c 2 uq c 1 þ c 2 uq
ð7Þ
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To simplify this equation a little more, we consider it over the extension LðbÞ of L; c2 u with bq 1 ¼ c1 þc q : Because ½LðbÞ : Ljq 1; the degree of M :¼ LðbÞ over L is 2u relatively prime to q; hence the splitting field M0 :¼ K0 ðbÞ of (7) over M also has Galois group GalðM0 =MÞDH: Substituting bX ¼ W in (7) gives Xq X ¼ f ;
where
f ¼
1 : bu
The following theorem shows that Sða; KÞ can be arbitrarily large. Theorem 18. For any kAN40 ; there exists a function field K; a Drinfeld module f : A-Kftg and a prime ideal ðaÞCA; such that dimF Sða; KÞ ¼ k: Proof. Let q ¼ pk for some integer k41 and p42 a prime. The computations of Examples 16 and 17 show that there is a Drinfeld module f over some function field M; such that M0 ¼ Mðkerðft ÞÞ is a Galois extension with Galois group H ¼
1 Fq and moreover this extension M0 =M is an Artin–Schreier extension 0 1 given by Xq X ¼ f ;
f AM:
ð8Þ
This extension is also given by the system of Artin–Schreier equations xp1 x1 ¼ b1 f ; ^ xpk
ð9Þ
xk ¼ bk f ;
P where the bi AFq are linearly independent over Fp : To see this, write z ¼ ki¼1 ai xi ; with ai AFq : An easy computation shows that z is a solution of X q X ¼ f if and only if 10 1 0 1 0 b1 ? bk a1 1 B p p CB B C ? b k C B a2 C B 0 C B b1 CB C ¼ B C B C: B ^ ^ C A@ ^ A @ ^ A @ k 1 k 1 0 ak ? bpk bp1 Because this matrix is invertible (cf. [4]), it follows that (9) is indeed equivalent to Xq X ¼ f : Now we consider the extension Mðz1 Þ=M; given by zp1 z1 ¼ b1 f g1 ; where we choose g1 as follows: for the finitely many places n of M; for which vn ð f Þo0; we let
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vn ðg1 Þ40 and for one place n0 for which vn0 ð f Þ40; we let vn0 ðg1 Þ ¼ 1: Such a g1 exists, cf. [1, Corollary VI.2.1]. The condition vn0 ðgÞ ¼ 1 makes sure that the system given by (9) and the equation for z1 are independent: Namely, by Hensel’s lemma all equations of (9) have solutions in Mn0 ; whereas the equation for z1 gives rise to a totally ramified extension of degree p: We conclude that (9) is still independent over Mðz1 Þ: Now we consider the field extension Mðz1 ; z2 Þ=Mðz1 Þ given by zp2 z2 ¼ b2 f þ g2 : We choose g2 in the same way as we chose g1 ; with M replaced by Mðz1 Þ: This implies that (9) is independent over Mðz1 ; z2 Þ: By repeating this we get that (9) is independent over the field Mðz1 ; y; zk 1 Þ: Let L=Mðz1 ; y; zk 1 Þ be the field extension given by (9), then its Galois group is H: Let n be a place of Mðz1 ; y; zk 1 Þ and let vn be its corresponding valuation. Let o be a place of L lying above n: We distinguish the following cases. (a) vn ð f Þ40; in this case we see that the equations of (9) are over the residue field given by xpi xi ¼ 0; hence they split completely over the residue field. Hensel’s lemma implies that Do is trivial. (b) vn ð f Þ ¼ 0; then also vn ðbi f Þ ¼ 0; hence all equations of (9) are over the residue field given by xpi xi ¼ ai ; with ai in the residue field. Hence all equations only give rise to a residue field extension. This shows that n is in L and thus Do is cyclic and can have at most p elements, because the elements of H have at most order p: (c) vn ð f Þo0: Note that the equations xpi xi ¼ bi f are equivalent to ypi yi ¼ gi by substituting yi ¼ z i x i ; for i ¼ 1; y; k 1: Because by construction vn ðgi Þ40; it follows that these equations give a trivial extension at n: So only the equation xpk xk ¼ bk f can give rise to a non-trivial extension, but this extension has at most degree p; hence Do can have at most p elements. We see that at any place o; the decomposition group Do has at most p elements. This means that for the non-trivial Do ; the kernel kerðReso Þ has dimension dimFq kerðReso Þ ¼ 1 þ dimFq H 1 ðH; V Þ ¼ k 1; by Proposition 10. Hence it follows by Proposition 12, that dimFq Sðt; Mðz1 ; y; zk 1 ÞÞ ¼ k 1: &
References [1] N. Bourbaki, Elements of Mathematics, Springer, Berlin, 1989. [2] E. Cline, B. Parshall, L. Scott, Cohomology of finite groups of Lie type I, Inst. Hautes E´tudes Sci. Publ. Math. 45 (1975) 169–191.
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[3] F. Gardeyn, t-Motives & Galois representations, Ph.D. Thesis, University of Ghent, 2001. [4] D. Goss, Basic Structures of Function Field Arithmetic, Springer, Berlin, 1996. [5] M. Jarden, The Cˇebotarev density theorem for function fields: an elementary approach, Math. Ann. 261 (1982) 467–475. [6] J.-P. Serre, Local Fields, Springer, New York, 1979. [7] M. Suzuki, Group Theory, Vol. I, Springer, Berlin, 1982.
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 210–254
http://www.elsevier.com/locate/jnt
On the global distance between two algebraic points on a curve Michel Laurenta, and Dimitrios Poulakisb a
Institut de Mathe´matiques de Luminy, CNRS, 163 Avenue de Luminy, Case 907, 13288 Marseille, Ce´dex 9, France b Department of Mathematics, Faculty of Sciences, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Received 14 May 2002; revised 10 July 2003 Communicated by J.-B. Bost
Abstract We prove diophantine inequalities involving various distances between two distinct algebraic points of an algebraic curve. These estimates may be viewed as extensions of classical Liouville’s inequality. Our approach is based on a transcendental construction using algebraic functions. Next we apply our results to Hilbert’s irreducibility Theorem and to some classes of diophantine equations in the circle of Runge’s method. r 2003 Elsevier Inc. All rights reserved. MSC: 11J85 Keywords: Diophantine inequalities; Algebraic functions; Hilbert’s irreducibility Theorem; Diophantine equations
1. Introduction and Liouville’s type inequalities on a curve We revisit some classical works on diophantine properties of algebraic functions in the framework of interpolation determinants. See Refs. [1,2,5,16,17, Chapters 5 and 6] as a non-exhaustive list of the topics we have in mind. The common feature of these works consists in applying the machinery of transcendence to the algebraic functions whose expansions in Taylor series are G-functions. Our approach enables
Corresponding author. E-mail address: [email protected] (M. Laurent).
0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.08.006
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us to establish, what we call ‘‘Liouville’s type inequalities’’ on a curve. These diophantine inequalities are symmetrical in the two points involved, since the origin and the ‘‘extrapolation point’’ play equivalent roles in the context of interpolation determinants. They are applied in Section 2 to Hilbert’s irreducibility Theorem and to some diophantine equations in the circle of Runge’s method, as in [6] or [17]. We specifically have taken care of the dependency upon the partial degrees of the polynomials involved. See [11] for an example of situation related to Hilbert’s irreducibility Theorem, in which this dependency is important. We have also investigated Runge’s method for the set of integral points rational over a totally real number field of arbitrary degree. Theorem 6 below furnishes a new result in this direction. Let us first introduce some notations which are standard. For any embedding of fields % s : Q+C s; where Cs denotes either the field C of complex numbers, or its p-adic analogue Cp ; equiped with their usual norms j:j (normalized by jpj ¼ p1 in the p-adic case), we % defined by jajs ¼ jsaj for any algebraic denote by j js the induced metric on Q number a: Let L be a number field and let x ¼ ðx1 ; y; xn Þ be any n-tuple in Ln ; we % denote by 1 hðxÞ ¼ ½L : Q %
X
logþ maxðjx1 js ; y; jxn js Þ;
s : L+Cs
where the summation index s ranges over all the distinct embeddings of the number field L into C or Cp ; the (absolute logarithmic) height of x: Recall that the height hðxÞ % the n-tuple x: % is independent of the choice of a field of rationality L for % Now the classical Liouville’s inequality can be stated as follows: for any distinct elements x1 and x2 of L; we have 1 ½L : Q
X s : L+Cs
logþ
1 phðx1 Þ þ hðx2 Þ þ log 2; jx1 x2 js
where s ranges as above along all the distinct embeddings of L into C or Cp : The lefthand side of this inequality can be viewed as a measure of global distance between x1 and x2 involving all the possible metrics. Our goal is to generalize this inequality to an arbitrary algebraic plane curve C; the classical Liouville’s inequality corresponding to the special case where C is a coordinate axis in the plane. It turns out that such extensions are known in arithmetical geometry and will be described in Section 3. We propose here an elementary and completely explicit formulation which enables us to deduce easily effective statements for the indicated applications.
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% and let Let K be a number field, viewed as a subfield of Q; F ðX ; Y Þ ¼ An ðX ÞY n þ ? þ A0 ðX Þ ¼
m X n X i¼0
ai; j X i Y j
j¼0
be an absolutely irreducible polynomial in K½X ; Y : Denote by hðF Þ ¼
1 ½K : Q
X
log
s : K+Cs
max
0pipm; 0pjpn
ðjai; j js Þ;
the (projective) height of the polynomial F : We shall suppose throughout this article that its partial degree n in the variable Y is X2 and that its degree m in X is X1: Let C be the plane curve with equation F ðx; yÞ ¼ 0; and for any field L containing K; denote by CðLÞ the set of L-rational points of C: To formulate our results, we need to introduce some definitions relying on the notion of algebraic functions and of Riemann surfaces. If Cs stands either for the fields C or Cp ; we denote by Bs ðx; rÞ the open disk in Cs with center xACs and radius r40: % be an algebraic point on C such that An ðxÞFY0 ðx; zÞa0: In Let P ¼ ðx; zÞACðQÞ other words, the projection ðx; yÞ/x restricted to C is supposed to be a local formal % isomorphism at the point P: Let s : Q+C s be an embedding as above. The branch yP;s is the meromorphic function yP;s : Bs ðsx; Rs ðPÞÞ-Cs ,fNg defined in a disk centered at sx with maximal radius Rs ðPÞ such that sF ðx; yP;s ðxÞÞ 0
and
yP;s ðsxÞ ¼ sz:
Notice that 0oRs ðPÞoN and that in the complex case Cs ¼ C; we have the lower bound Rs ðPÞXminjsx aj; where a ranges along the set of branch points in the x-plane of the Riemann surface uniformizing the algebraic function y defined by the equation sF ðx; yÞ ¼ 0: We say % belongs to the branch yP;s if that the point P0 ¼ ðx0 ; z0 ÞACðQÞ jx x0 js oRs ðPÞ
and
yP;s ðsx0 Þ ¼ sz0 :
Now let L be a number field containing K; let P1 ¼ ðx1 ; z1 Þ and P2 ¼ ðx2 ; z2 Þ be two points ACðLÞ: Suppose that An ðxi ÞFY0 ðxi ; zi Þa0 for i ¼ 1; 2: We define S ¼ SðL; P1 ; P2 Þ to be the set of embeddings s : L+Cs ¼ C or Cp such that either P2 belongs to the branch yP1 ;s or P1 belongs to the branch yP2 ;s : Roughly speaking, S is the set of embeddings s for which either sP2 is located on a circular patch of Cs ðCs Þ centered at sP1 ; or reversely sP1 lies on a similar disk centered at sP2 : Notice also that when jx1 x2 js ominðRs ðP1 Þ; Rs ðP2 ÞÞ; the assertions yP1 ;s ðsx2 Þ ¼ sz2 and
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yP2 ;s ðsx1 Þ ¼ sz1 are equivalent, since the functions yP1 ;s and yP2 ;s are then equal in the intersection Bs ðsx1 ; Rs ðP1 ÞÞ-Bs ðsx2 ; Rs ðP2 ÞÞ: We can now state our first extension of Liouville’s inequality relative to C:
Theorem 1. Let P1 ¼ ðx1 ; z1 Þ and P2 ¼ ðx2 ; z2 Þ be two points ACðLÞ: Suppose m ¼ degX F X1; x1 ax2
and
n ¼ degY F X2
An ðx1 ÞAn ðx2 ÞFY0 ðx1 ; z1 ÞFY0 ðx2 ; z2 Þa0:
Then X pffiffiffiffiffiffiffiffiffi 1 1 hðx ; x Þ logþ p 1 2 þ r þ h1 h2 þ log 4; ½L : Q sAS jx2 x1 js n where r¼
X 1 1 logþ ½L : Q sAS maxðRs ðP1 Þ; Rs ðP2 ÞÞ
and for i ¼ 1; 2; hi ¼ mð2n 1Þhðxi Þ þ ð2n 1ÞhðF Þ þ logð8ðm þ 1Þ2nþ2 ðn þ 1Þ2nþ3 Þ:
Remarks. (1) Since jx2 x1 js omaxðRs ðP1 Þ; Rs ðP2 ÞÞ for any sAS; the same statement remains valid with S replaced by any subset of S: (2) We have the upper bound ( ) X X 1 1 1 1 ; logþ logþ rpmin ½L : Q sAS Rs ðP1 Þ ½L : Q sAS Rs ðP2 Þ so that the term r is bounded for fixed P1 : An inequality of this type was first obtained by Sprindzuck [16]. See also Theorem 1.1 of [6] for a version with explicit constants. (3) We have hðx1 ; x2 Þphðx1 Þ þ hðx2 Þ: The upper bound furnished by Theorem 1 may be better than the classical Liouville’s inequality since nX2: (4) We have measured the distance of P1 and P2 by measuring the distance of their projections x1 and x2 on the x-axis. The process is obviously significant only when the map ðx; yÞ/x is locally smooth in a neighborhood of the two points. The proximity to the singular locus of this map is taken into account on the right-hand side of the inequality through the term r:
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Next we give a projective version of Theorem 1. We shall now measure the distance of the two points using the projective distance ds ðx1 ; x2 Þ ¼
jx2 x1 js ; maxð1; jx1 js Þmaxð1; jx2 js Þ
for any algebraic numbers x1 and x2 ; and any embedding s as above. Notice that this distance is equivalent (see Lemma 8 for a precise statement) with the quantity 1 1 % minð1; jx2 x1 js ; jx1 2 x1 js Þ: If we view the projective space P ðQÞ as the glueing 1 % identifying x with x ; our projective distance turns out to be of two copies of Q essentially the minimum of the distances in the two maps. Remark also that jx2 1 x1 js pjx1 2 x1 js exactly when jx1 x2 js p1; which explains the splitting of our definitions into the three cases below. Besides of the preceding notion of branch, we need its analogue for the map at infinity. For any non-zero xACs and any r40; denote B˜ s ðx; rÞ ¼
1 1 xACs : or : x x
Then B˜ s ðx; rÞ is a neighborhood of x which is either a disk or the complementary of a % be a point such that xAn ðxÞFY0 ðx; zÞa0: We define disk in Cs : Let P ¼ ðx; zÞACðQÞ the branch y˜ P;s to be the meromorphic function y˜ P;s : B˜ s ðsx; R˜ s ðPÞÞ-Cs ,fNg defined in a projective disk B˜ s ðsx; R˜ s ðPÞÞ with maximal radius R˜ s ðPÞ such that sF ðx; y˜ P;s ðxÞÞ 0
and
y˜ P;s ðsxÞ ¼ sz:
% belongs to the branch y˜ P;s when Analogously, we say that a point P0 ¼ ðx0 ; z0 ÞACðQÞ 1 1 oR˜ s ðPÞ and x0 x s
y˜ P;s ðsx0 Þ ¼ sz0 :
Let L be a number field containing K and let P1 and P2 be two points ACðLÞ as before. We shall suppose moreover that x1 x2 a0: For each embedding s : L+Cs denote 8 if jx1 x2 js o1; > < maxðRs ðP1 Þ; Rs ðP2 ÞÞ 0 ˜ ˜ if jx1 x2 js 41; Rs ðP1 ; P2 Þ ¼ maxðRs ðP1 Þ; Rs ðP2 ÞÞ > : ˜ ˜ maxðRs ðP1 Þ; Rs ðP2 Þ; Rs ðP1 Þ; Rs ðP2 ÞÞ if jx1 x2 js ¼ 1:
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We define now S0 to be the set of embeddings s : L+Cs such that 1 0 minðjx2 x1 js ; jx1 2 x1 js ÞoRs ðP1 ; P2 Þ
and either * * *
P1 belongs to the branch yP2 ;s or P2 belongs to the branch yP1 ;s if jx1 x2 js o1; P1 belongs to the branch y˜ P2 ;s or P2 belongs to the branch y˜ P1 ;s if jx1 x2 js 41; P1 belongs to the branch yP2 ;s or to the branch y˜ P2 ;s ; or P2 belongs to the branch y˜ P1 ;s or to the branch y˜ P1 ;s ; if jx1 x2 js ¼ 1:
In other words, S 0 is the set of embeddings s for which sP1 or sP2 lie on some circular patches of the preceding types associated to the coverings ðx; yÞ/x or ðx; yÞ/x1 ; and prescribed by the value of ds ðx1 ; x2 Þ: When jx1 x2 js ¼ 1; both coverings are convenient and we require that at least one of the points P1 or P2 belongs to the union of the two branches associated with the other point. Theorem 2. Under the assumptions of Theorem 1; suppose that x1 x2 a0: Then qffiffiffiffiffiffiffiffiffi X 1 1 hðx Þ þ hðx2 Þ þ r0 þ h01 h02 þ log 8 p 1 logþ ½L : Q sAS0 ds ðx1 ; x2 Þ n where X 1 1 logþ 0 r ¼ ½L : Q sAS0 Rs ðP1 ; P2 Þ 0
and
h0i
1 ¼ hi þ 2 1 hðxi Þ for i ¼ 1; 2: n
Suppose moreover that the polynomials F ðx1 ; Y Þ ¼ An ðx1 Þ
n Y
ðY z1; j Þ
and
F ðx2 ; Y Þ ¼ An ðx2 Þ
j¼1
n Y
ðY z2;k Þ
k¼1
have simple roots. Then we have the lower bound qffiffiffiffiffiffiffiffiffi X 1 1 hðx1 Þ þ hðx2 Þ nr00 ðn2 1Þ h01 h02 X logþ ½L : Q sAS0 ds ðx1 ; x2 Þ n ð2n2 þ n 1Þlog 2 with r00 ¼
½L0
1 : Q
X s : L0 +Cs
logþ
1 ; minj;k R0s ðP1; j ; P2;k Þ
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where L0 ¼ Lðz1;1 ; y; z2;n Þ denotes the field generated over L by all the roots z1; j ; z2;k and P1; j ¼ ðx1 ; z1; j Þ;
P2;k ¼ ðx2 ; z2;k Þ
for j; k ¼ 1; y; n:
Instead of the bidegree ðm; nÞ of the polynomial F ; we focus now on its total degree.
Theorem 3. Let n be an integer X2: Suppose that the polynomial F has the form X F ðX ; Y Þ ¼ ai; j X i Y j iþjpn
with a0;n a0: Then Theorems 1 and 2 hold with hi defined now by hi ¼ ð2n 2Þhðxi Þ þ ð2n 1ÞhðF Þ þ logð8ðn þ 1Þ4nþ6 Þ for i ¼ 1; 2: Remark. The above values for hi are smaller than those obtained by putting m ¼ n in Theorem 1. Notice however that the hypotheses of Theorem 3 are more restrictive, since we assume a0;n a0: The proofs of Theorems 1–3 follow the same lines. We shall give a detailed proof for Theorem 1 and will only indicate the modifications of the argumentation for Theorems 2 and 3. The scheme of proof is as follows. We first define some matrix M of evaluation of monomial functions on the curve C at the two points P1 and P2 with high multiplicities. It will be easily verified that the rows of the matrix M are linearly independent. Let D be a non-vanishing minor of maximal order extracted from M: The determinant D is an example of what we call an interpolation determinant. Its various absolute values jDjs can be bounded in different ways, the crucial estimate being provided by the analytical Lemma 7 below. Next, Theorems 1–3 follow easily from the product formula applied to D: In our approach, the multiplicities at P1 and P2 may be arbitrary positive integer parameters and we take advantage of this flexibility in the construction to get a reminder term of the expected size pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi O hðx1 Þhðx2 Þ :
2. Arithmetical applications It is known that diophantine inequalities, as those from Section 1, imply Hilbert irreducibility Theorem as well as Runge’s Theorem on Diophantine equations. This approach provides effective results and was developed in the papers [3,6,17]. In this
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section, we apply our estimations in these classical situations. We have not tried to be exhaustive, giving a simple example in the context of Hilbert irreducibility Theorem, and formalizing somehow Runge’s Theorem. In most cases, the method appears to be efficient only when the field of rationality L has degree ½L : Q on: Notice however that Theorem 6 below concerns algebraic points of any degree. It will be convenient to set HðF Þ ¼ ehðF Þ for any polynomial F and HðxÞ ¼ ehðxÞ for any algebraic number x: 2.1. Effective Hilbert irreducibility theorem In this section, we restrict to the case K ¼ L ¼ Q; which leads to a simple statement with a fairly precise upper bound. Compare for instance with the Addition to Corollary 1.3 of [6].
Theorem 4. Let F be an absolutely irreducible polynomial in Q½X ; Y with m ¼ degX F X1 and n ¼ degY F X2: Suppose F ð0; 0Þ ¼ 0 and
An ð0ÞFY0 ð0; 0Þa0:
Let k be a non-zero integer, p a prime number, and t a positive integer. Then the polynomial F ðx; Y Þ is irreducible in Q½Y for ( 2 3 1=k jkjXH 3mn ðn1Þ ; x¼ whenever 2 3 pt pt XH 3mn ðn1Þ ; where we have set H ¼ 8ððm þ 1Þðn þ 1ÞÞ2nþ3 HðF Þ2n1 : Suppose moreover that F satisfies the assumptions of Theorem 3. Then the same 2 3 statement holds for m ¼ n whenever jkj or pt are XH 3n ðn1Þ :
2.2. Diophantine equations Let us first complete the algebraic plane curve C in the two following ways. Homogeneizing the polynomial F ; first into a bihomogeneous polynomial with bidegree ðm; nÞ; and secondly with respect to the total degree in X and Y ; we obtain projective embeddings CDP1 P1
and
CDP2 :
We shall refer to the embedding in P2 only when F satisfies the assumptions of Theorem 3. Denote again by C the Zariski closure of the affine curve C in P1 P1 or
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in P2 : Notice that we can extend the projection ðx; yÞ/x into a morphism p : C-P1 ; since a0;n is supposed to be non-zero in Theorem 3. Set CN ¼ p1 ðNÞ: We shall say that p (or sometimes F ) is regular at N, if CN is made up with n distinct points which are different from ðN; NÞ in the case of an embedding in P1 P1 : This definition means in term of F that the leading polynomial FN ðTÞ ¼
n X
am; j T j ;
j¼0
or
FN ðTÞ ¼
n X
anj; j T j
j¼0
for the embedding in P2 ; has exactly n simple roots. For any number field L; we denote by OL its ring of integers. Theorem 5. Let d be an integer pn 1: Suppose that p is regular at N and that each point QACN is rational over some number field L containing K of degree ½L : Q pd: Let ðx; zÞAOK K be such that F ðx; zÞ ¼ 0: Then HðxÞpH 3mn
d =ðndÞ2
3 2
with H ¼ 8ððm þ 1Þðn þ 1ÞÞ2nþ3 HðF Þ2n1 :
Suppose moreover that F satisfies the assumptions of Theorem 3. Then HðxÞpH 3n
d =ðndÞ2
3 2
with H ¼ 8ðn þ 1Þ4nþ6 HðF Þ2n1 :
d Remarks. (1) Bounding nd pn 1; we obtain an effective version of Runge’s theorem, which is consistent with the results of [18] obtained for K ¼ Q: Notice however that we have assumed here that F should be regular at N: The results of [6] (see Corollary 1.3) are more general than our Theorem 5, but the bounds are larger. (2) If we know a priori that the point ðx; zÞ belongs to some specified Archimedean branches at infinity, the upper bound remains valid with d replaced by the maximal degree over Q of the corresponding points QACN :
In order to prove Theorems 4 and 5, we shall apply Theorems 1 and 3 to a subset of S made up with only one embedding, according to Remark 1 after Theorem 1. In some (unfortunately restrictive) situations, the set S turns out to be larger. Here is an example.
Theorem 6. Suppose that K is a totally real number field of degree d over Q and that p is regular at N: Let L be any totally real number field and let ðx; zÞAOL L satisfy F ðx; zÞ ¼ 0: Suppose that there exists a point Q0 ACN ; such that for each embedding % s : Q+C; sðQ0 Þ is the only real point of sðCN Þ: Then we have the uniform bound HðxÞpH 4dmn
3
=ðn1Þ2
with H ¼ 8ððm þ 1Þðn þ 1ÞÞ2nþ3 HðF Þ2n1 :
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Suppose moreover that F satisfies the assumptions of Theorem 3. Then 3
HðxÞpH 4dn
=ðn1Þ2
with H ¼ 8ðn þ 1Þ4nþ6 HðF Þ2n1 :
Remarks. (1) Our hypotheses imply that the point at infinity Q0 is necessarily rational over K: (2) Notice that when the sets sðCN Þ have no real point, a similar statement holds with a better upper bound. See [14] for the case of an embedding in P2 : We may apply Theorem 6 to equations of the shape f ðxÞ ¼ gðzÞ: See Chapter 5 of [1] and Theorem 3 of [18] for previous results of this kind in the special case of superelliptic equations zn ¼ f ðxÞ:
Corollary. Let K be a totally real number field of degree d; let n be an odd integer X3; and let f1 ; f2 ; g be three non-zero polynomials AK½X with deg g ¼ n: Suppose that the polynomial F ðX ; Y Þ ¼ f2 ðX ÞgðY Þ f1 ðX Þ is absolutely irreducible, that f1 ; f2 and g have leading coefficients which are nth powers in K; and that deg f1 deg f2 is divisible by n in Z: Let L be a totally real number field and let ðx; zÞAOL L be such that F ðx; zÞ ¼ 0: Then HðxÞpð8ðm þ 1Þ2nþ3 ðn þ 1Þ2nþ3 HðF Þ2n1 Þ4dmn
3
=ðn1Þ2
with m ¼ maxðdeg f1 ; deg f2 Þ:
3. Geometrical interpretation Bombieri gave in [3] an alternative proof of Sprindzuck’s inequalities [16], using the theory of heights on the curve C: We extend his arguments and give an interpretation of our Theorems 2 and 3 in terms of height functions associated to divisors on the product C C: We refer to Chapter 5 of [10] for the results quoted in this section, and more globally to the whole book for the algebraic geometry’s background. For simplicity, let us restrict to polynomials F as in Theorem 3. Embed C into P2 as in Section 2.2 and denote again by C its Zariski closure. Assume that the projective curve C is smooth and that its genus g ¼ ðn 1Þðn 2Þ=2 is X1: Let J be the jacobian of C: We fix a normalized embedding i : C+J; defined by iðPÞ ¼ ðPÞ a;
% PACðQÞ;
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for some divisor a on C of degree one, chosen in such a way that the theta divisor Y ¼ iðCÞ þ ? þ iðCÞ |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} ðg1Þ times
on J should be symmetrical. Let P ¼ p1 Y þ p2 Y s Y be the Poincare´ divisor on J J; where the maps s; p1 ; p2 : J J/J denote, respectively, the sum of the two components, the projections on the first and second factors. According to a theorem due to Mumford, we know that its restriction ði iÞ ðPÞ on C C is a divisor linearly equivalent to D a C C a; where D is the diagonal in C C: Let hˆY be the Ne´ron–Tate canonical height associated with the divisor Y: Remind that hˆY extends to a positive quadratic form % on JðQÞ#R; and denote by /:; :S the bilinear form associated to the quadratic form hˆY : In terms of height functions, we obtain the following equivalences modulo % bounded functions on ðC CÞðQÞ: hD ðP1 ; P2 Þ ha ðP1 Þ ha ðP2 Þ ¼ hP ðiðP1 Þ; iðP2 ÞÞ þ Oð1Þ ¼ hˆY ðiðP1 Þ þ iðP2 ÞÞ þ hˆY ðiðP1 ÞÞ þ hˆY ðiðP2 ÞÞ þ Oð1Þ ¼ 2/iðP1 Þ; iðP2 ÞS þ Oð1Þ: Remark that pffiffiffiffiffiffiffiffiffiffiffi 1 ha ðP1 Þ ¼ hðx1 Þ þ Oð hðx1 ÞÞ n
and
pffiffiffiffiffiffiffiffiffiffiffi 1 ha ðP2 Þ ¼ hðx2 Þ þ Oð hðx2 ÞÞ; n
while the term hD ðP1 ; P2 Þ measures a global distance on C between P1 and P2 ; whenever P1 aP2 : In Theorems 2 and 3, the quantity X 1 minf1; ds ðx1 ; x2 Þg 1 log ½L : Q sAS0 minf1; R0s ðP1 ; P2 Þg plays the role of hD ðP1 ; P2 Þ: Now Cauchy–Schwarz inequality reads 1 2jhP ðiðP1 Þ; iðP2 ÞÞj
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ j/iðP1 Þ; iðP2 ÞSjp hˆY ðiðP1 ÞÞhˆY ðiðP2 ÞÞ:
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We also know from Mumford that pffiffiffiffiffiffiffiffiffi g hˆY ðiðPÞÞ ¼ hðxÞ þ O hðxÞ ; n % Thus Theorems 2 and 3 can be viewed as explicit for any point P ¼ ðx; zÞACðQÞ: formulations in terms of naı¨ ve heights of the Cauchy–Schwarz inequality for the quadratic form hˆY : Notice however that in the situation of Theorem 3, we get values h1 and h2 of the shape hi ¼ ð2n 2Þhðxi Þ þ Oð1Þ which are slightly larger than the pffiffiffiffiffiffiffiffiffiffi expected quantities ð2g=nÞhðxi Þ þ O hðxi Þ : It may be enlightening to deduce alternatively the preceding facts from basic properties of arithmetic intersection. Let us fix some arithmetic surface X with generic fiber C K L and for each point PACðLÞ; denote by EP the horizontal divisor on X having P ¼ ðEP ÞL for generic point. Suppose for simplicity that the divisor a is rational over L: Then we can choose the Arakelov’s intersection product
ha ðPÞ ¼
ðEP Ea Þ ½L : Q
as a height function associated to the divisor a: The main observation is that the bilinear form 2/DL ; EL S connected with the Ne´ron–Tate height coincides with the intersection pairing ðD EÞ=½L : Q for any Arakelov’s divisors D and E on X which are orthogonal to the subgroup V generated by the irreducible components of the vertical fibers. See [8,9] for a proof. Select now bounded divisors F1 and F2 in V such that EP1 Ea þ F1 and EP2 Ea þ F2 are orthogonal to V : Then, by bilinearity of the intersection pairing, we can write ðEP1 Ea þ F1 EP2 Ea þ F2 Þ ½L : Q ðEP1 EP2 Þ ¼ ha ðP1 Þ ha ðP2 Þ þ Oð1Þ: ½L : Q
2/iðP1 Þ; iðP2 ÞS ¼
Thus we can choose
hD ðP1 ; P2 Þ ¼
ðEP1 EP2 Þ : ½L : Q
Now the adjunction formula ðEP EP Þ ¼ ðEP oX Þ; see [8] for more informations, pffiffiffiffiffiffiffiffiffiffiffiffi provides us with the estimate hˆY ðiðPÞÞ ¼ gha ðPÞ þ O ha ðPÞ :
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4. Auxiliary results We collect in this section the necessary lemmas. Most of the material is concerned with the properties of iterated derivatives on C: We shall also need to estimate the various radii involved in Theorems 1 and 2, which is the content of Section 4.2. 4.1. Derivations on a curve Let KðCÞ be the field of rational functions on the irreducible curve C: Thus KðCÞ ¼ Kðx; yÞ where x is transcendental over K and F ðx; yÞ ¼ 0: We shall be concerned with the two derivations @
and
@* : KðCÞ-KðCÞ
of the field KðCÞ which are the unique extensions to KðCÞ of the derivations @=@x and @=@x1 operating on the subfields KðxÞ and Kðx1 Þ; respectively. We also % % extend the derivations @ and @* to the field QðCÞ :¼ KðCÞ#K Q: % We are concerned with points P ¼ ðx; zÞACðQÞ such that An ðxÞFY0 ðx; zÞa0: The projections p : ðX ; Y Þ/X and p* : ðX ; Y Þ/X 1 ; whenever xa0; are then local % formal isomorphisms at P: To each rational function jAQðCÞ regular at P; we associate the two local functions fP ¼ j3p1
and
gP ¼ j3p* 1 :
In other words, fP and gP are the formal series in X x and X 1 x1 ; respectively, which represent the function j viewed as an algebraic function of the variable X or X 1 around x or x1 : Notice that formally fP ðzÞ ¼ gP ðz1 Þ: Then for each embedding % s : Q+C s we have the equalities sðjðP0 ÞÞ ¼ sfP ðsx0 Þ ¼ sgP ðsx01 Þ % belonging to the intersection of the branches yP ; s for any point P0 ¼ ðx0 ; z0 ÞACðQÞ 0 01 and y˜ P ; s; such that sx and sx lie in the disks of convergence of the series sfP and sgP : * For tX1 we Let us begin by some formal properties of the derivations @ and @: 1 shall denote by @ ½t ¼ ðt!Þ @ ðtÞ the operator @ iterated t times and divided by t!: % Lemma 1. Let jAQðCÞ be regular at the smooth point P for p and p: * Then for every tX0; we have ½t
@ ½t j3p1 ¼ fP
and
½t @*½t j3p* 1 ¼ gP ;
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where ½t fP
1 @ t ¼ fP t! @X
and
½t gP
t 1 @ ¼ gP t! @X 1
denote the divided derivatives of the one variable functions fP ðX Þ and gP ðX 1 Þ:
Proof. For t ¼ 0; these are the definitions of fP and gP : Notice that the first equality % is obvious for jAQðxÞ: For j ¼ y and t ¼ 1; this equality is also valid since @y ¼
FX0 ðx; yÞ and FY0 ðx; yÞ
ðy3p1 Þ0 ðX Þ ¼
FX0 ðX ; y3p1 ðX ÞÞ ; FY0 ðX ; y3p1 ðX ÞÞ
both relations coming by differentiating the identities F ðx; yÞ ¼ 0
and
F ðX ; y3p1 ðX ÞÞ 0:
The first equality is then satisfied for any j when t ¼ 1: Next we argue by induction for tX1: The proof for @* is similar. & Lemma 2. For any integer tX1; we have the formula t X t1 @ ¼ ð1Þ xtt @*½t : t 1 t¼1 ½t
t
* 1 Þ; observe that Proof. Since @ðx1 Þ ¼ x2 ¼ x2 @ðx * @ ¼ x2 @; which proves the lemma for t ¼ 1: From this relation and Lemma A3 from [13], we can write ½t
@ ¼
t X t¼1
at;t @*½t
with at;t
X t Y @ ½i1 ðx2 Þsi ¼ : P i s iX1 si ¼t % P isi ¼t
Since @ ½i1 ðx2 Þ ¼ @ ½i ðx1 Þ ¼ ð1Þi xi1 ; i
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we get
1
0
B X C B t C t1 C ¼ ð1Þt xtt C BP t1 @ si ¼t s A % P
t tt B
at;t ¼ ð1Þ x
isi ¼t
according to formula 3h from [4, p. 146]. & Let us now give some rational expression for @ ½t yv which is based on an explicit version of the formal implicit functions theorem, contained in the appendix of [13]. Write m X n X F ðX ; Y Þ ¼ ai; j X i Y j i¼0
j¼0
and denote by a the ðm þ 1Þðn þ 1Þ-tuple of its coefficients which we view for the % moment as indeterminates. Denote as usual by LðÞ the length of any polynomial with integral coefficients, that is the sum of the absolute value of its coefficients.
Lemma 3. For each integer tX0; vX1; there exists a polynomial Gv;t AZ½a; X ; Y such % that G ð a; x; yÞ v;t @ ½t yv ¼ % FY0 ðx; yÞ2t and
dega Gv;t p2t;
degX Gv;t pð2m 1Þt; degY Gv;t pð2n 2Þt þ v; tþv1 ðm þ 1Þ5t ðn þ 1Þ6t : LðGv;t Þp v1
%
Assume moreover that m ¼ n and that the total degree in X and Y of the polynomial F is pn: Then the total degree in X and Y of the polynomial Gv;t is pð2n 3Þt þ v:
Proof. The result is obvious for t ¼ 0: For brevity write Fi; j ¼ ði!j!Þ1 and for any tX1; set G1;t ¼
X t¼ðti; j Þi; j %
@ iþj F ðX ; Y Þ; @X i @Y j
ð2t
qðtÞF0;1 %
P
ti; j Þ
Y
ðFi; j Þti; j ;
ði;jÞ
where the summation index t ranges over the set of ðmn þ m þ n 1Þ-tuples of % integers t ¼ ðti; j Þ with 0pipm; 0pjpn and ði; jÞað0; 0Þ; ð0; 1Þ; %
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satisfying
X
iti; j ¼ t;
i; j
X
ð j 1Þti; j ¼ 1;
1p
X
i; j
225
ti; j p2t 1;
i; j
and where 1
qðtÞ ¼ P % i; j ti; j
P
i; j
ti; j
:
t % It is shown in Lemma A2 from [13] that @ ½t y ¼ G1;t =FY0 2t: Moreover the coefficients qðtÞ are integers, so that G1;t has integral coefficients. Now we easily check the % following estimations of degrees: ( ) X X dega G1;t p max 2t ti; j þ ti; j ¼ 2t; %
( degX G1;t p max m 2t t %
t %
i; j
X
ðn 1Þ 2t
t %
ti; j
þm
X
i; j
( degY G1;t p max
!
i; j
X i; j
! ti; j
þn
ti; j
X
i; j
i; j
X
X
i; j
ti; j
) iti; j
¼ ð2m 1Þt; )
jti; j
¼ ð2n 2Þt þ 1:
i; j
On the other hand, the polynomial m X n X m n am;n Fi; j ¼ X mi Y nj ; i j m¼0 n¼0 when viewed as a polynomial in Z½a; X ; Y ; has length % m X n X m n mþ1 nþ1 LðFi; j Þ ¼ ¼ pðm þ 1Þiþ1 ðn þ 1Þ jþ1 : i j i þ 1 j þ 1 m¼0 n¼0 We deduce that P Y X qðtÞLðF0;1 Þð2t ti; j Þ LðFi; j Þti; j LðG1;t Þp % t ði;jÞ % P P P P X 2t t þ ðiþ1Þti; j Þ i; j ðn þ 1Þð2ð2t ti; j Þþ ð jþ1Þti; j Þ p qðtÞðm þ 1Þð % t % ! X qðtÞ ðm þ 1Þ3t ðn þ 1Þ4t1 : p % t %
Since X t %
qðtÞ ¼ %
0
2X t1 k¼1
1 t1 X k C 2X 1B 1 B Cp ðmn þ m þ n 1Þk pððm þ 1Þðn þ 1ÞÞ2t1 ; @ A k Pt k t k¼1 % % ti; j ¼k
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we obtain the upper bound LðG1;t Þpðm þ 1Þ5t1 ðn þ 1Þ6t2 pðm þ 1Þ5t ðn þ 1Þ6t ; thus proving the lemma when v ¼ 1: For vX1; we use Leibnitz formula @ ½t yv ¼
X
@ ½t1 y ? @ ½tv y ¼
t1 þ?þtv ¼t
Gv;t FY0 2t
with X
Gv;t ¼
G1;t1 ? G1;tv :
t1 þ?þtv ¼t
The upper bounds for the partial degrees of Gv;t are then obvious and X tþv1 LðGv;t Þp LðG1;t1 Þ ? LðG1;tv Þp ðm þ 1Þ5t ðn þ 1Þ6t : v 1 t1 þ?þtv ¼t Suppose now that m ¼ n and degðX ;Y Þ F pn: Since degðX ;Y Þ Fi; j pn i j; we deduce from the explicit formula for G1;t that ( ! ) X X degðX ;Y Þ G1;t p max ðn 1Þ 2t ti; j þ ðn i jÞti; j ¼ ð2n 3Þt þ 1: t %
i; j
i; j
For any vX1; Leibnitz formula then implies the upper bound degðX ;Y Þ Gv;t pð2n 3Þt þ v:
&
4.2. Around Eisenstein’s Theorem % We are concerned here with the Taylor coefficients of a function jAQðCÞ at a % Eisenstein’s Theorem provides us with an upper bound for these point PACðQÞ: % Our results from coefficients relatively to any p-adic or Archimedean metric on Q: Section 4.1 imply a fairly precise version of this theorem for any smooth point P for p: See [7,15] for previous results of this kind. % Let s : Q+C s be an embedding as before and define 0 if Cs ¼ Cp ; e¼ 1 if Cs ¼ C: Denote by jF js ¼
max
0pipm;0pjpn
ðjai; j js Þ;
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227
where we recall that a ¼ ðai; j Þ stands for the tuple of the coefficients of the % polynomial F : % with An ðxÞFY0 ðx; zÞa0: Denote Lemma 4. Let P ¼ ðx; zÞ be a point ACðQÞ Hs ðPÞ ¼ 4e ðm þ 1Þeð2nþ2Þ ðn þ 1Þeð2nþ3Þ jF j2n1 maxð1; jxjs Þ2mnm1 : s For any integers tX0; vX1; we have the upper bound j@ ½t yv ðPÞjs pcvs;P
tþv1
e
!t
Hs ðPÞ jFY0 ðx; zÞj2s jAn ðxÞj2n3 s
v1
with cs;P ¼
ðm þ 1Þe ðn þ 1Þe jF js maxð1; jxjs Þm : jAn ðxÞjs
Moreover if we suppose xa0; we have *½t v
j@ y
ðPÞjs pcvs;P
tþv1 v1
e
jxjs Hs ðPÞ jFY0 ðx; zÞj2s jAn ðxÞj2n3 s
!t :
Proof. Let us first bound jzjs in term of jxjs : Since F ðx; zÞ ¼ 0; we can write the formula z¼
An1 ðxÞ þ ? þ A0 ðxÞznþ1 An ðxÞ
from which follows the upper bound maxð1; jzjs Þp
ððm þ 1Þðn þ 1ÞÞe jF js maxð1; jxjs Þm : jAn ðxÞjs
The lemma is then obvious for t ¼ 0: Suppose now v ¼ 1 and tX1 so that @ ½t yðPÞ ¼
G1;t ða; x; zÞ : % FY0 ðx; zÞ2t
We combine the estimations of Lemma 3 with (1). Remind the formula P X ð2t ti; j Þ Y qðtÞF0;1 ðFi; j Þti; j : G1;t ¼ % t¼ðt Þ ði;jÞ %
i; j
ð1Þ
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228
We start with the obvious upper bound jFi; j ðx; zÞjs p LðFi; j Þe jF js maxð1; jxjs Þmi maxð1; jzjs Þnj p ðm þ 1Þeðiþ1Þ ðn þ 1Þeð jþ1Þ jF js maxð1; jxjs Þmi maxð1; jzjs Þnj :
ð2Þ
For i ¼ 0 and jX1; we refine this inequality into jF0;j ðx; zÞjs p2e ðm þ 1Þe ðn þ 1Þeð jþ1Þ jF js maxð1; jxjs Þm maxð1; jzjs Þnj1 :
ð3Þ
Let us check this estimation in the Archimedean case, the p-adic case being similar n nj and simpler. We separate the leading term An ðxÞ j z from the others. Using (1) we find jF0;j ðx; zÞjs p ðm þ 1Þðn þ 1ÞjF js maxð1; jxjs Þm þ ðm þ 1ÞjF js maxð1; jxjs Þm
n
maxð1; jzjs Þnj1 j ! n1 X n maxð1; jzjs Þnj1 j n¼j
p 2ðm þ 1Þðn þ 1Þ jþ1 jF js maxð1; jxjs Þm maxð1; jzjs Þnj1 : Now from (2) and (3), we deduce an upper bound for the term indexed by t in the % sum defining G1;t : P ð2t ti; j Þ Y jFi; j ðx; zÞjtsi; j jF0;1 ðx; zÞjs
p2
e 2t
P
t iX1;j i; j
ðn þ 1Þ
ði;jÞ
ðm þ 1Þ
e 2 2t
e 2t
i; j
ti; j þ
m 2t
maxð1; jxjs Þ
maxð1; jzjs Þ
i; j
P
P
jF j2t s
P
ðn2Þ 2t
i; j
ti; j þ
P i; j
ð jþ1Þti; j
i; j
ti; j þ
i; j
P
P i; j
ti; j þ
jX2
P
P
ðiþ1Þti; j
ðmiÞti; j
ðnj1Þt0;j þ
P iX1
ðnjÞti; j
ð2m1Þt p4et ðm þ 1Þe3t ðn þ 1Þeð4t1Þ jF j2t maxð1; jzjs Þ s maxð1; jxjs Þ
Notice that X iX1;j
ti; j p
X i; j
iti; j ¼ t
ð2n4Þtþ1þ
P iX1;j
ti; j
:
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so that the last exponent is pð2n 3Þt þ 1: Using again (1), we find
ð2t jF0;1 ðx; zÞjs
P
ti; j Þ
Y
jFi; j ðx; zÞjtsi; j
ði;jÞ
p
et
4 ðm þ 1Þ
eð2ntþ1Þ
ðn þ 1Þeð2nþ1Þt jF jð2n1Þtþ1 maxð1; jxjs Þð2mnm1Þtþm s jAn ðxÞjð2n3Þtþ1 s
:
P Since t qðtÞpððm þ 1Þðn þ 1ÞÞ2t1 ; we obtain the expected upper bound in the case % % v ¼ 1: For vX1 , we use Leibnitz formula as in Lemma 3. The argumentation is similar for the second inequality, changing X into X 1 : Let ˜ ; Y Þ ¼ X m F ðX 1 ; Y Þ ¼ A˜ n ðX ÞY n þ ? þ A˜ 0 ðX Þ FðX ˜ 1 ; yÞ ¼ 0 and Fðx ˜ 1 ; zÞ ¼ 0; so that with A˜ n ðX Þ ¼ X m An ðX 1 Þ: Then Fðx Gv;t ð*a; x1 ; zÞ : @*½t yv ðPÞ ¼ % F˜0Y ðx1 ; zÞ2t Using the previous upper bound, we obtain !t ½t v tþv1 e H˜ s ðPÞ @* y ðPÞ p˜cv s;P s v1 jF˜0Y ðx1 ; zÞj2s jA˜ n ðx1 Þj2n3 s with ˜ 2n1 H˜ s ðPÞ ¼ 4e ðm þ 1Þeð2nþ2Þ ðn þ 1Þeð2nþ3Þ jFj maxð1; jx1 js Þ2mnm1 s ¼ jxjð2mnm1Þ Hs ðPÞ s and
c˜s;P ¼
m ˜ s maxf1; jxj1 ðm þ 1Þe ðn þ 1Þe jFj s g ¼ cs;P ; jA˜ n ðx1 Þjs
˜ s ¼ jF js and A˜ n ðx1 Þ ¼ xm An ðxÞ: Noting now that F˜0 ðx1 ; zÞ ¼ since jFj Y m 0 x FY ðx; zÞ; we obtain the desired upper bound. &
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Corollary. Let rs ðPÞ and r˜s ðPÞ be the convergence’s radii of the Taylor’s series of the analytical functions yP;s ðzÞ and y˜ P;s ðz1 Þ at the points x and x1 ; respectively. Then 1 1 Hs ðPÞ p p ; Rs ðPÞ rs ðPÞ jFY0 ðx; zÞj2s jAn ðxÞj2n3 s 1 1 jxjs Hs ðPÞ p : p 0 ˜ r ˜ ðPÞ Rs ðPÞ s jFY ðx; zÞj2s jAn ðxÞj2n3 s
Proof. On a neighborhood of x in Cs ; we find the Taylor’s expansions yP;s ðxÞ ¼ sz þ
þN X
s@ ½t yðPÞðx sxÞt ;
t¼1
y˜ P;s ðxÞ ¼ sz þ
þN X
s@*½t yðPÞðx1 sx1 Þt
t¼1
using Lemma 1 applied to j ¼ y: The convergence’s radii rs ðPÞ and r˜s ðPÞ of the series yP;s ðxÞ and y˜ P;s ðx1 Þ satisfy 1 Hs ðPÞ ¼ lim sup j@ ½t yðPÞj1=t ; p s rs ðPÞ t/þN jFY0 ðx; zÞj2s jAn ðxÞj2n3 s 1 jxjs Hs ðPÞ ¼ lim sup j@*½t yðPÞj1=t : p s 0 r˜s ðPÞ t/þN jFY ðx; zÞj2s jAn ðxÞj2n3 s The corollary follows immediately since Rs ðPÞXrs ðPÞ and R˜ s ðPÞX˜rs ðPÞ:
&
5. Proofs of Liouville’s type inequalities The proof of the inequalities differs only by various estimates concerning the same determinant D: We shall keep common notations in the proof of the three theorems. Let U; T1 ; T2 ; T be four positive integers satisfying nU ¼ T1 þ T2 ¼ T: Recall that P1 ¼ ðx1 ; z1 Þ and P2 ¼ ðx2 ; z2 Þ are the two points involved in our theorems. Denote for brevity c1 ¼ ðm 1Þðn 1Þ and consider the matrix M ¼ ð@ ½t1 xu yv ðP1 Þ j @ ½t2 xu yv ðP2 ÞÞ whose rows are labelled by the couples of integers ðu; vÞ with 0puoU; 0pvon; and whose first T1 columns are indexed by the integer t1 with 0pt1 oT1 ; while the
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following T2 þ c1 columns are indexed by t2 with 0pt2 oT2 þ c1 : Thus the matrix M contains T rows and T þ c1 columns. As usual when using interpolation determinants, we start with a zero lemma.
Lemma 5. The matrix M has maximal rank equal to T:
Proof. Suppose on the contrary that the rows of M are linearly dependent. Then there exist algebraic numbers gu;v ; 0puoU; 0pvon; not all zero, such that the polynomial GðX ; Y Þ ¼
U 1 X n1 X u¼0
gu;v X u Y v
v¼0
satisfies @ ½t1 GðP1 Þ ¼ 0 for 0pt1 oT1
and
@ ½t2 GðP2 Þ ¼ 0 for 0pt2 oT2 þ c1 :
Let RðX Þ be the resultant with respect to Y of the polynomials F and G: Since there exist polynomials A and B such that R ¼ AF þ BG; observe that the polynomial R vanishes with multiplicity XT1 at the point P1 ; and multiplicity XT2 þ c1 at the point P2 : Since deg RpðU 1Þn þ ðn 1Þm ¼ T þ c1 1; we deduce that R is the zero polynomial. Reminding now that F is absolutely irreducible, it follows that F divides G; which contradicts the upper bound degY Gpn 1on ¼ degY F :
&
Fix now a non-zero minor D extracted from M with maximal format T T: Thus D ¼ detð@ ½t1 xu yv ðP1 Þ j @ ½t2 xu yv ðP2 ÞÞ0puoU;0pvon t1 AE1 ;t2 AE2
for some subsets E1 and E2 made up, respectively, with integers oT1 and oT2 þ c1 corresponding to the selected columns from M: Then their cardinalities T10 :¼ Card E1
and
T20 :¼ Card E2
satisfy
T1 c1 pT10 pT1 ;
T2 pT20 pT2 þ c1 ;
T10 þ T20 ¼ T:
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5.1. Proof of Theorem 1 We shall bound jDjs for each embedding s : L+Cs ¼ C; Cp in two different ways according as sAS or not. For i ¼ 1; 2; denote for simplicity hi; s ¼ logð2e maxð1; jxi js ÞHs ðPi ÞÞ ¼ logð8e ðm þ 1Þeð2nþ2Þ ðn þ 1Þeð2nþ3Þ jF j2n1 maxð1; jxi js Þ2mnm Þ; s fi; s ¼ logðjFY0 ðxi ; zi Þj2s jAn ðxi Þj2n3 Þ: s Our first upper bound is valid for any such embedding s:
Lemma 6. For large T; we have logjDjs p
T1 T2 ðlog maxð1; jx1 js ; jx2 js Þ þ e log 2Þ þ n T2 þ ðh2;s f2;s Þ þ OðT log TÞ: 2
T1
2
ðh1;s f1;s Þ
Moreover, the remainder term OðT log TÞ vanishes for almost all s:
Proof. Observe first that if P ¼ ðx; zÞ stands either for P1 or P2 jFY0 ðx; zÞj2s jAn ðxÞj2n3 pmaxf1; jxjs gHs ðPÞ s
ð5Þ
since, combining (1) and (3), we get jFY0 ðx; zÞjs p 2e ðm þ 1Þe ðn þ 1Þ2e jF js maxf1; jxjs gm maxf1; jzjs gn2 p
2e ðm þ 1Þeðn1Þ ðn þ 1Þen jF jn1 maxf1; jxjs gmnm s jAn ðxÞjn2 s
Write now
½t u v
@ x y ¼
minðt;uÞ X t¼0
u t
xut @ ½tt yv
:
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from Leibnitz formula. Using Lemma 4 and (5), we obtain !e minðt;uÞ X u ½t u v maxð1; jxjs Þut max fmaxð1; jxjs Þt j@ ½t yv ðPÞjs g j@ x y ðPÞjs p 0ptpt t t¼0 p cvs;P 2eu maxð1; jxjs Þut ( !t ) t þ v 1 e maxð1; jxjs ÞHs ðPÞ max 0ptpt v1 jFY0 ðx; zÞj2s jAn ðxÞj2n3 s !t e Hs ðPÞ u tþv1 v eu p cs;P 2 maxð1; jxjs Þ v1 jFY0 ðx; zÞj2s jAn ðxÞj2n3 s for tX0 and vX1: For v ¼ 0; we obviously bound j@ ½t xu ðPÞjs p2eu maxð1; jxjs Þu : Let us enumerate consecutively the two sets of columns E1 ¼ ft1;1 ; y; t1;T10 g
and
E2 ¼ ft2;T10 þ1 ; y; t2;T g
involved in the minor D and expand 0
T1 T X Y Y D¼ 7 @ ½t1; j xuj yvj ðP1 Þ @ ½t2; j xuj yvj ðP2 Þ; j¼T10 þ1
j¼1
where the sum is taken over all the bijections j/ðuj ; vj Þ of the set f1; y; Tg onto f0; y; U 1g f0; y; n 1g: From the above upper bounds, we obtain !e U Y tþn1 Y tþn1 e ðn1ÞT en jDjs p ðT!Þ maxðcs;P1 ; cs;P2 Þ 2 2 n1 n1 t1 AE1 t2 AE2 !P !P t t t1 AE1 1 t2 AE2 2 Hs ðP1 Þ Hs ðP2 Þ jFY0 ðx1 ; z1 Þj2s jAn ðx1 Þj2n3 jFY0 ðx2 ; z2 Þj2s jAn ðx2 Þj2n3 s s ( ) PT PT 0 uj 1 u 0 sup maxð1; jx1 js Þ j¼1 j maxð1; jx2 js Þ j¼T1 þ1 ; where the supremum is taken over all the bijections j/ðuj ; vj Þ: Suppose for instance PT10 that jx1 js Xjx2 js : The maximal value in the supremum is then reached when j¼1 uj is P maximal and Tj¼T 0 þ1 uj is minimal. This occurs exactly when the sequence u1 ; y; uT10 1 contains the value U 1 repeated n times,y, U ½T10 =n repeated n times and U ½T10 =n 1 repeated T10 n½T10 =n times. Then
0
T1 X j¼1
uj ¼ n
U 2
U ½T10 =n 2
þ OðTÞp
T1 T2 þ n
T1 2
þ OðTÞ
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T X
uj ¼ n
½T20 =n 2
j¼T10 þ1
Notice also that X T1 t1 p ; 2 t1 AE1
X
t2 p
t2 AE2
T2 2
T2 þ OðTÞp þ OðTÞ: 2
þ OðTÞ;
n
T1 U T2 T1 T2 þ p þ : n 2 2 2
Putting these estimates in the upper bound for jDjs ; we obtain the lemma. The proof is similar when jx1 js pjx2 js : & Now we give an other upper bound for jDjs which involves the distance of the points P1 and P2 : It will be used for sAS: Here is the point where interpolation determinants are used. Lemma 7. Let Z40 and let s : L+Cs be an embedding such that jx2 x1 js o2e minð1; Rs ðP1 Þ ZÞ and
yP1 ;s ðsx2 Þ ¼ sz2 :
Then log jDjs pT1 T2 log jx2 x1 js þ
T
2
1 þ e log 2 þ OZ ðT log TÞ: log Rs ðP1 Þ Z þ
Proof. Denote for brevity R ¼ minð1; Rs ðP1 Þ ZÞ: By multilinearity on the rows of the determinant D; we can write D ¼ det ð@ ½t1 ðx x1 Þu yv ðP1 Þ j @ ½t2 ðx x1 Þu yv ðP2 ÞÞ0puoU;0pvon : t1 AE1 ;t2 AE2
In a first step we eliminate the poles, thanks to some process reminiscent of the Borel–Dwork Criterion. Set QðxÞ ¼ An ðxÞn1 and observe that the functions QðxÞðx x1 Þu yv ¼ ðx x1 Þu ðAn ðxÞyÞv An ðxÞn1v
with 0puoU; 0pvon
are regular on the algebraic curve C: Using Leibnitz formula we find the formula 0
D¼
T1 XY j¼1
@ ½t1; j t1; j Q1 ðP1 Þ
T Y
@ ½t2; j t2; j Q1 ðP2 Þ
j¼T10 þ1
det ð@ ½t1 QðxÞðx x1 Þu yv ðP1 Þ j @ ½t2 QðxÞðx x1 Þu yv ðP2 ÞÞ
t1 ¼t1;1 ;y;t1;T 0 ; 1 t2 ¼t2;T 0 þ1 ;y;t2;T 1
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where the sum is taken over all the sequences of integers t1;1 ; y; t1;T10 ; t2;T10 þ1 ; y; t2;T with 0pt1;i pt1;i and 0pt2; j pt2; j : Remark that we can restrict the summation index to the sequences for which the t1;i and the t2; j are pairwise distinct, since otherwise the determinant on the right hand side vanishes. Then we have 0
T1 X j¼1
t1; j t1; j pc1 T10
T X
and
t2; j t2; j pc1 T20 :
j¼T10 þ1
Note that there exists c2 40; depending only upon F ; x1 ; x2 ; s such that j@ ½t Q1 ðPi Þjs pctþ1 2 for any tX0 and i ¼ 1; 2: We deduce an upper bound of the shape 0 Y T Y T1 ½t t 1 ½t2; j t2; j 1 1; j 1; j @ Q ðP1 Þ @ Q ðP2 Þ pcT3 j¼1 j¼T10 þ1 s
for all the factors occurring in the above expression of D: We are thus reduced to bound determinants of the shape D0 ¼ det ð@ ½t1 QðxÞðx x1 Þu yv ðP1 Þ j @ ½t2 QðxÞðx x1 Þu yv ðP2 ÞÞ: Since the functions yP1 ;s and yP2 ;s are equal in a neighborhood of sx2 ; on account of Lemma 1 we can write ½t1 ½t2 ðsx1 Þ j fu;v ðsx2 ÞÞ sD0 ¼ det ð fu;v
as an interpolation determinant made up with the power series X fu;v;k ðz sx1 Þk fu;v ðzÞ :¼ sQðzÞðz sx1 Þu yP1 ;s ðzÞv ¼ kX0
converging in the disk Bs ðsx1 ; Rs ðP1 ÞÞ: Then Theorem 1 of [12] provides us with the expansion X ki ki t1 ki ki t2 0 sD ¼ det ð fu;v;kj Þdet 0 : t ðsx2 sx1 Þ t1 2 0pk1 o?okT Set for simplicity R ¼ minð1; Rs ðP1 Þ ZÞ: Cauchy’s formula applied to the functions fu;v on the disk Bs ðx1 ; RÞ implies the upper bounds j fu;v;k jpc4 Rk ;
kX0;
where c4 is an upper bound on the disk Bs ðsx1 ; RÞ of all the functions sQðzÞyP1 ;s ðzÞv for 0pvon: Observe that non-zero contributions in the above sum come from sequences 0pk1 o?okT which contain necessarily all the values t1;1 ; y; t1;T10 :
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Bounding now tk2 p2ek ; we find s
jD0 js p ðT!T20 !Þe cT4 R
PT 0
X
0pk1 o?okT
1 t j¼1 1; j
PT
t j¼T 0 þ1 2; j 1
PT
2
e
t j¼T 0 þ1 2; j 1
e PT kj PT10 t1; j PT 0 t2; j j¼1 j¼1 j¼T þ1 2 jx2 x1 js 1 : R
By assumption r :¼ 2e jx2 x1 js =R is o1; so that PT
X
k j¼1 j
r
T ¼r 2
X
r
PT
k j¼1 j
0pk1 p?pkT
0pk1 o?okT
T T 1 pr 2 : 1r
Estimating
0
T1 X
t1; j ¼
j¼1
T1 2
þ OðT1 Þ and
T X
t2; j ¼
j¼T10 þ1
we find the required upper bound since T1 T2 ¼
T 2
T 1
2
T2
þ OðT2 Þ;
2
T 2
2
: &
We are now able to prove Theorem 1. For any embedding s : L+Cs denote Rs ¼ maxfRs ðP1 Þ; Rs ðP2 Þg; and for any ZX0; set SZ ¼ fsAS: jx2 x1 js o2e minð1; Rs ZÞg: Notice that when Z40 and sASZ ; we can apply Lemma 7 either to the point P1 or to P2 ; so that the estimation for jDjs provided by this lemma is valid for any sASZ : For the moment fix Z40; and write the product formula: X
logjDjs ¼ 0:
s : L+Cs
We bound each term logjDjs ; using Lemma 6 when seSZ ; or Lemma 7 when sASZ : Note that T2 ¼ T1 T2 þ T21 þ T22 and gather the respective coefficients of T1 T2 ; T21 ; T22 :
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We obtain the basic inequality 8 <X 1 þ þ e log 2 T1 T2 logjx2 x1 js þ log :sAS Rs Z Z 9 X log maxð1; jx1 j ; jx2 j Þ e log 2= s s þ þ ; n n seSZ 8 9 X < X = T1 1 þ þ e log 2 þ logþ ðh1;s f1;s Þ ; Rs Z 2 :sASZ seSZ 8 9 X < X = T2 1 þ e log 2 þ þ logþ ðh2;s f2;s Þ þ OZ ðT log TÞX0: ; Rs Z 2 :sAS seS Z
Z
Divide now this inequality by T1 T2 and choose T1 and T2 tending to þN with pffiffiffiffiffiffiffiffiffiffiffiffi limðT2 =T1 Þ ¼ h1 =h2 : This process enables us to eliminate the term OZ ðT log TÞ: Next select values of Z tending to 0, in order to replace Z by 0 in the inequality. Finally, we find X X 1 log maxð1; jx1 js ; jx2 js Þ e log 2 þ logjx2 x1 js þ logþ þ e log 2 þ Rs n n sAS0 seS0 sffiffiffiffiffi( ) X 1 h2 X þ 1 þ log þ e log 2 þ ðh1;s f1;s Þ 2 h1 sAS Rs seS0 0 sffiffiffiffiffi( ) X 1 h1 X 1 þ logþ þ e log 2 þ ðh2;s f2;s Þ X0: 2 h2 sAS Rs seS 0
0
Using the corollary of Lemma 4, we bound
logþ
1 1 Hs ðPi Þ plogþ plogþ phi;s fi;s e log 2 0 Rs Rs ðPi Þ jFY ðxi ; zi Þj2s jAn ðxi Þj2n2 s
for i ¼ 1; 2 (observe that hi;s Xfi;s þ e log 2 is equivalent to (5)). Replace now in the previous inequality logþ ð1=Rs Þ by its upper bound h1;s f1;s e log 2
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on the second line and by h2;s f2;s e log 2 on the third line. We obtain X
logjx2 x1 j1 s p
sAS0
1 X ðlog maxð1; jx1 js ; jx2 js Þ þ e log 2Þ n seS 0 X þ 1 þ log þ e log 2 Rs sAS0 sffiffiffiffiffi sffiffiffiffiffi ! ! 1 h2 X 1 h1 X þ ðh1;s f1;s Þ þ ðh2;s f2;s Þ : 2 h1 s 2 h2 s
P Now for i ¼ 1; 2; we have s fi;s ¼ 0 by the product formula and hi ¼ ½L : P Q 1 s hi;s by definition of the height. It follows that X X 1 1 hðx ; x Þ 1 1 log p 1 2 þ logþ ½L : Q sAS jx2 x1 js n ½L : Q sAS Rs 0
! þ
pffiffiffiffiffiffiffiffiffi h1 h2 þ log 2:
0
On the other hand, the definition of S0 shows that X X 1 1 1 þ þ 1 log p log þ e log 2 ½L : Q sAS\S jx2 x1 js ½L : Q sAS\S Rs 0 0 ! X 1 þ 1 p log þ log 2: ½L : Q sAS\S Rs 0
Adding the two last inequalities gives Theorem 1. 5.2. Proof of Theorem 2 Let us first compare the projective distance with the usual distance in the two maps covering P1 : % we have the % Lemma 8. For any embedding s : Q+C s and any non-zero x1 ; x2 AQ; inequalities 1 1 1 1 2e min 1; jx1 x2 js ; pds ðx1 ; x2 Þpmin 2e ; jx1 x2 js ; : x1 x2 s x1 x2 s
Proof. Let us begin with the right-hand side. The triangle inequality implies ds ðx1 ; x2 Þp2e :
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Since 1 jx1 x2 js 1 ¼ min jx1 x2 js ; ds ðx1 ; x2 Þp x1 x2 s maxf1; jx1 x2 js g we obtain the upper bound. The proof of the lower bound splits into several cases. When jx1 js p1 and jx2 js p1; we have 1 1 e ds ðx1 ; x2 Þ ¼ jx1 x2 js X2 min 1; jx1 x2 js ; : x1 x2 s When jx1 js X1 and jx2 js X1; we have 1 1 1 1 e ds ðx1 ; x2 Þ ¼ X2 min 1; jx1 x2 js ; : x1 x2 s x1 x2 s When jx1 js p1pjx2 js ; we have ds ðx1 ; x2 Þ ¼
jx1 x2 js : jx2 js
Suppose jx2 js p2e : Then 1 jx1 x2 js jx1 x2 js 1 e X X2 min 1; jx1 x2 js ; : x1 x2 s jx2 js 2e Suppose jx2 js 42e : Since jx1 js p1; we can write 1 jx1 x2 js jx2 js jx1 js 1 X X21 X21 min 1; jx1 x2 js ; ; x1 x2 s jx2 js jx2 js in the Archimedean case and 1 jx1 x2 js 1 ¼ 1Xmin 1; jx1 x2 js ; ; x1 x2 s jx2 js in the non-Archimedean case.
&
In addition to Lemma 7, we shall also use an analytical argument in the map at N on P1 : Lemma 9. Suppose x1 x2 a0: Let Z40 and let s : L+Cs be an embedding such that 1 1 o2e minð1; R˜ s ðP1 Þ ZÞ and y˜ P ;s ðsx2 Þ ¼ sz2 : 1 x x1 s 2
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Then 1 1 1 1 logjDjs p T1 T2 log þlogþ þ logjx1 x2 js þ e log 2 x2 x1 s R˜ s ðP1 Þ Z n T1 1 1 þ 2 1 logjx1 js þ e log 2 logþ n 2 R˜ s ðP1 Þ Z T2 1 1 þ þ 2 1 logjx2 js þ e log 2 log n 2 R˜ s ðP1 Þ Z
þ OZ ðT log TÞ:
Proof. We first transform D into a linear combination of analogous determinants involving @* instead of @: Using multilinearity on the columns, we deduce from Lemma 2 the formula D ¼7
T10 X Y t1; j s %
j¼1
s1; j
T Y t2; j
t s x1 1; j 1; j
s2; j
j¼T10 þ1
t2; j s2; j
x1
det ð@*½s1 xu yv ðP1 Þ j @*½s2 xu yv ðP2 ÞÞ
s1 ¼s1;1 ;y;s1;T 0 1 s2 ¼s2;T 0 þ1 ;y;s2;T 1
where the sum is taken over the sequences of integers s1;1 ; y; s1;T10 ; s2;T10 þ1 ; y; s2;T with 0psi; j pti; j and si; j X1 whenever ti; j X1: We may also suppose that the integers s1;1 ; y; s1;T10 and s2;T10 þ1 ; y; s2;T are pairwise distinct. Next we use the same process as in Lemma 7 in order to eliminate the poles. Set Q ¼ xUþ1 ðxm An ðxÞÞn1 AK½x1 so that the functions Qxu yv ¼ ðx1 ÞU1u ðxm An ðxÞÞn1v ðxm An ðxÞyÞv are integral over the ring K½x1 : Using Leibnitz formula and multilinearity on the columns, we obtain the expression X Cs;t det ð@*½t1 Qxu yv ðP1 Þ j @*½t2 Qxu yv ðP2 ÞÞ t1 ¼t1;1 ;y;t1;T 0 D¼7 s;t %%
%%
1
t2 ¼t2;T 0 þ1 ;y;t2;T 1
with coefficients Cs;t %%
T10 T Y Y t1; j t2; j t1; j s1; j *½s1; j t1; j 1 t s ¼ Q ðP1 Þ @ x1 x2 2; j 2; j @*½s2; j t2; j Q1 ðP2 Þ; s1; j s2; j j¼1 j¼T 0 þ1 1
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where the summation’s indices t ¼ ðti; j Þi; j satisfy ti; j psi; j pti; j : Again we can % restrict to indices t for which the t1; j and the t2;k are pairwise distinct. Then % 0
T1 X
s1; j t1; j pc1 T10
T X
and
s2; j t2; j pc1 T20 :
j¼T10 þ1
j¼1
For such s and t; let us prove an upper bound of the shape % % T1 T2 logjx1 x2 js 2 2 logjCs;t js p T1 T2 2 logjx1 js 2 logjx2 js n n n %% 2 2 þ OðT log TÞ: Using again Leibnitz formula @*½st Q1 ðX Þ ¼
st X
ð1Þk
U 2þk k
k¼0
and writing
X U1þk
@ @X 1
½stk
1 A˜ n ðX 1 Þ
t st t ts x ; we obtain easily ¼ x2t ts s x 0
0
logjCs;t js p logjx1 js @2@ %%
0
X
1
t1 A þ UT10 A
t1 AT10
0
þ logjx2 js @2@
1
X
1
1
t2 A þ UT20 A þ OðT log TÞ
t2 AT20
which implies the expected upper bound. As in Lemma 7, we are thus reduced to bound determinants of the shape D0 ¼ det ð@*½t1 Qxu yv ðP1 Þ j @*½t2 Qxu yv ðP2 ÞÞ: Using now multilinearity on the rows, we replace in D0 the functions Qxu yv ¼ xðU1uÞ ðxm An ðxÞÞn1 yv by u m ðx1 x1 An ðxÞÞn1v ðxm An ðxÞyÞv 1 Þ ðx
which are clearly integral over K½x1 : Denote R ¼ minð1; R˜ s ðP1 Þ ZÞ and set u m sAn ðzÞÞn1v ðzm sAn ðzÞy˜ P1 ;s ðzÞÞv : f˜u;v ðzÞ ¼ ðz1 sx1 1 Þ ðz
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The functions f˜u;v are analytical and uniformly bounded in the projective disk B˜ s ðsx1 ; RÞ; so that the power series X k gu;v ðzÞ :¼ f˜u;v ðz1 Þ ¼ gu;v;k ðz sx1 1 Þ kX0
converge and are uniformly bounded in the disk Bs ðsx1 1 ; RÞ: Using now the second formula from Lemma 1, we can again write 1 ½t2 sD0 ¼ 7det ðg½tu;v1 ðsx1 1 Þ j gu;v ðsx2 ÞÞ
as an interpolation determinant. Then, the same argumentation as in Lemma 7 leads to the upper bound 1 T 1 1 logjD0 js p T1 T2 log þ þ e log 2 logþ x1 x2 s 2 R˜ s ðP1 Þ Z þ OZ ðT log TÞ: Collecting our upper bounds for jD0 js and jCs;t js ; we obtain Lemma 9. %%
&
The proof of the first part of Theorem 2 follows the same steps as for Theorem 1. Let us detail only the points which differ. Remind the notation 8 if jx1 x2 js o1; > < maxfRs ðP1 Þ; Rs ðP2 Þg 0 ˜ ˜ if jx1 x2 js 41; Rs ðP1 ; P2 Þ ¼ maxfRs ðP1 Þ; Rs ðP2 Þg > : maxfRs ðP1 Þ; Rs ðP2 Þ; R˜ s ðP1 Þ; R˜ s ðP2 Þg if jx1 x2 js ¼ 1 and define for any ZX0 the subset 1 1 0 0 e 0 SZ ¼ sAS : min jx2 x1 js ; o2 minð1; Rs ðP1 ; P2 Þ ZÞ : x2 x1 s Observe that 8 < jx2 x1 js 1 1 min jx2 x1 js ; ¼ 1 1 : x2 x1 s x2 x1 s
if jx1 x2 js p1; if jx1 x2 js X1:
Let Z40 and let sASZ0 : In order to estimate jDjs ; we use either Lemma 7 when jx1 x2 js o1; or Lemma 9 when jx1 x2 js 41: For the third case jx1 x2 js ¼ 1; it may happen that both lemmas are valid; in this situation among the 4 possibilities, we use any point (P1 or P2 ) and any lemma (7 or 9) for which the corresponding radius reach the maximal value R0s ðP1 ; P2 Þ: pffiffiffiffiffiffiffiffiffiffiffiffi Select now T1 and T2 tending to N with limðT2 =T1 Þ ¼ h01 =h02 and next choose Z tending to 0: Taking again the same estimations as in Theorem 1, we find
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the basic inequality X 1 þ log minðjx2 x1 js ; jx1 2 x1 js Þ þ log sAS00
1 1 þ logþ jx1 x2 js þ e log 2 R0s ðP1 ; P2 Þ n
X log maxð1; jx1 j ; jx2 j Þ e log 2 s s þ n n 0 seS0 8 9 sffiffiffiffiffi X = 1 h02 < X 1 þ þ l log logjx j þ e log 2 þ ðh f Þ s 1;s 1;s 1 s ; 2 h01 :sAS0 R0s ðP1 ; P2 Þ seS00 0 9 sffiffiffiffiffi8 X = 1 h01 < X 1 þ l þ log logjx j þ e log 2 þ ðh f Þ X0; s 2;s 2;s 2 s ; 2 h02 :sAS0 R0s ðP1 ; P2 Þ seS 0 þ
0
0
where ls ¼ 0 if we apply Lemma 7 and ls ¼ 2ð1 1=nÞ if we use Lemma 9 relatively to the embedding s: Thanks to the usual Liouville’s inequality, we bound
X sAS00
1 ls logjxi js p2 1 ½L : Q hðxi Þ n
for i ¼ 1; 2: Next, we bound obviously X sAS00
logþ jx1 x2 js þ
X
log maxð1; jx1 js ; jx2 js Þp½L : Q ðhðx1 Þ þ hðx2 ÞÞ:
seS00
Using again the corollary of Lemma 4, notice that logþ
1 phi;s fi;s e log 2 R0s ðP1 ; P2 Þ
for i ¼ 1; 2; and replace in the above basic inequality logþ ð1=R0s ðP1 ; P2 ÞÞ by its upper bound h1;s f1;s e log 2 on the third line and by h2;s f2;s e log 2 on the fourth line. Noting that X 1 1 logþ 1 ½L : Q sAS0 \S0 minðjx2 x1 js ; jx1 2 x1 js Þ 0 X 1 1 þ þ e log 2 ; p log ½L : Q sAS0 \S0 R0s ðP1 ; P2 Þ 0
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we obtain the upper bound X 1 1 logþ 1 ½L : Q sAS0 minðjx2 x1 js ; jx1 2 x1 js Þ ffiffiffiffiffiffiffiffiffi q hðx Þ þ hðx2 Þ þ r0 þ h01 h02 þ log 4: p 1 n The first part of Theorem 2 follows immediately using Lemma 8. Let us now prove the lower bound. It turns out that the lower bound is a formal consequence of the upper bound, as was remarked by Debes in Theorem 2 of [5]. P First, observe that the global distance ½L : Q 1 sAS0 logþ ds ðx1 ; x2 Þ1 remains unchanged by extension of the field L: Thus may assume without loss of generality that L0 ¼ L: Let S 00 be the set of embeddings s : L+Cs such that 1 00 0 minðjx2 x1 js ; jx1 2 x1 js ÞoRs :¼ min Rs ðP1; j ; P2;k Þ: j;k
0 For any j; k ¼ 1; y; n; denote Sj;k the set S0 associated to the pair of points 00 00 0 ðP1; j ; P2;k Þ: Set now Sj;k ¼ S -Sj;k and
dj;k
0 1 1 @X 1 A hðx1 Þ hðx2 Þ: ¼ logþ 1 1 ½L : Q sAS00 n n minðjx2 x1 js ; jx2 x1 js Þ j;k
00 Let sAS 00 : Observe that the number of pairs ð j; kÞ for which sASj;k is exactly n: To 00 see this, fix sAS and suppose for simplicity that jx1 x2 js o1: Write j-k if P2;k belongs to the branch yP1; j ;s and j’k if P1; j belongs to the branch yP2;k ;s : Then s 00 belongs to Sj;k whenever j-k or j’k: Since jx2 x1 js omaxðRs ðP1; j Þ; Rs ðP2;k ÞÞ for each pair ð j; kÞ; there exists either j 0 or k0 such that j-k0 or j 0 ’k: Moreover, the right and left arrows are injective since all the roots z1; j and z2;k are supposed to be simple. It follows easily from these properties that there are exactly n pairs ð j; kÞ for which j and k are linked either by a right arrow or a left arrow. In the remaining cases jx1 x2 js o1 or jx1 x2 js ¼ 1; we argue in a similar way, using branches of the type y˜ P;s ; or unions of the type yP;s ,y˜ P;s : Thus,
n n X X j¼1
k¼1
! ! X 1 1 dj;k ¼ n log hðx1 Þ hðx2 Þ 1 ½L : Q sAS00 minð1; jx2 x1 js ; jx1 2 x1 j s Þ ! ! X 1 1 e log 2 hðx1 Þ hðx2 Þ Xn log ½L : Q sAS00 ds ðx1 ; x2 Þ
ARTICLE IN PRESS M. Laurent, D. Poulakis / Journal of Number Theory 104 (2004) 210–254
X X n 1 ¼ log e log 2 ½L : Q ds ðx1 ; x2 Þ sAS00 seS 00
245
!
! X n 1 X log n log 2 1 ½L : Q seS00 minð1; jx2 x1 js ; jx1 2 x1 j s Þ ! X n 1 logþ 00 n log 2 X ½L : Q seS00 Rs by Lemma 8 and the product formula. On the other hand, for any pair ð j; kÞ with j; k ¼ 1; y; n; we have the upper 0 bound 1 qffiffiffiffiffiffiffiffiffi 1 @X 1 dj;k p logþ 00 A þ h01 h02 þ log 4 ½L : Q sAS00 Rs j;k
by the proof of the first part of Theorem 2 (use also Remark 1 following Theorem 1). Putting together the upper and lower bounds, we deduce for each fixed pair a; b the lower bound qffiffiffiffiffiffiffiffiffi h01 h02 þ log 4 n log 2; da;b X nr00 ðn2 1Þ 00 for n values of ð j; kÞ: Now ðP1 ; P2 Þ is one noting again that each sAS 00 appears in Sj;k of the pairs ðP1;a ; P2;b Þ: Using again Lemma 8, we obtain the required lower bound 00 DS 0 ðP1 ; P2 Þ: since Sa;b
5.3. Proof of Theorem 3 Assume now that m ¼ n and that the polynomial F has total degree pn: Let X F ðX ; Y Þ ¼ ai; j Xi Y j iþjpn
with a0;n a0: The underlying idea is to embed the curve C into P2 ; rather than P1 P1 ; which produces better estimates in our situation. The scheme of proof remains formally identical with the one of Theorems 1 and 2. We shall only display the points which differ. In relation with Sections 4, 5.1 and 5.2, define henceforth Hs ðPÞ ¼ 4e ðn þ 1Þeð4nþ6Þ jF j2n1 maxð1; jxjs Þ2n3 ; s hi;s ¼ logð2e maxð1; jxi js ÞHs ðPi ÞÞ ¼ logð8e ðn þ 1Þeð4nþ6Þ jF j2n1 maxð1; jxi js Þ2n2 Þ; s fi;s ¼ logðjFY0 ðxi ; zi Þj2s ja0;n j2n3 Þ; s for i ¼ 1; 2:
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Let us first bound jzjs in term of jxjs : Now the refined inequality ðn þ 1Þ2e jF js maxf1; jxjs g maxf1; jzjs gp ja0;n js
ð6Þ
holds. Notice that the factor maxð1; jxjs Þ was raised to a power m ¼ n in (1). Let us write Pn1 Pnj j j¼0 i¼0 ai; j xi z n z ¼ : a0;n We obtain maxð1; jzjs Þn p
ðn þ 1Þ2e jF js max0pjpn1 fmaxð1; jxjs Þnj maxð1; jzjs Þ j g : ja0;n js
Remark now that ja0;n js pjF js pðn þ 1Þ2e jF js so that we may suppose without restriction that maxð1; jzjs ÞXmaxð1; jxjs Þ: Then the above maximum is reached for j ¼ n 1; which implies (6). In relation with Lemma 4, we prove now upper bounds of the shape !t tþv1 e Hs ðPÞ ½t v v ; ð7Þ j@ y ðPÞjs pcs;P v1 jFY0 ðx; zÞj2s ja0;n j2n3 s *½t v
j@ y
ðPÞjs pcvs;P
tþv
e
tþv1 v1
v
(
e
max jxjs ;
jxjs Hs ðPÞ
)t
jFY0 ðx; zÞj2s ja0;n j2n3 s
ð8Þ
for tX0 and vX1: According to Lemma 3, the polynomial Gv;t has a total degree in X and Y bounded by ð2n 3Þt þ v: Thus we deduce from Lemma 3 and (6) that j@ ½t yv ðPÞjs p
ð2n3Þtþv LðGv;t Þe jF j2t s maxð1; jxjs ; jzjs Þ
jFY0 ðx; zÞj2t s
ðn þ 1Þ2e jF js maxð1; jxjs Þ p ja0;n js
!v
tþv1 v1
e
Hs ðPÞ
!t
jFY0 ðx; zÞj2s ja0;n j2n3 s
which proves (7). For (8), we change again X into X 1 : Set ˇ ; YÞ ¼ FðX
X iþjpn
anij; j Xi Y j ¼
n X
X nj Aj ðX 1 ÞY
j
j¼0
ˇ 1 ; YX 1 Þ ¼ X n F ðX ; Y Þ: Then Fðx ˇ 1 ; yx1 Þ ¼ 0 and Fðx ˇ 1 ; zx1 Þ ¼ 0: so that FðX
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Thus we can write the formula ˇ x1 ; zx1 Þ Gv;t ða; : @*½t ðy=xÞv ðPÞ ¼ Fˇ0Y ðx1 ; zx1 Þ2t Applying (7) to the polynomial Fˇ we obtain an upper bound of the shape v j@*½t ðy=xÞv ðPÞjs pðjxj1 s cs;P Þ
tþv1
e
jxjs Hs ðPÞ jFY0 ðx; zÞj2s ja0;n j2n3 s
v1
!t ;
observing that Fˇ0Y ðx1 ; zx1 Þ ¼ xnþ1 FY0 ðx; zÞ;
aˇ 0;n ¼ a0;n
and
Hs ððx1 ; zx1 ÞÞ ¼ jxjsð2n3Þ Hs ðPÞ:
Write now Leibnitz formula @*½t yv ¼
t X
ð1Þ
t¼0
t
t þ v 1 vþt *½tt ðy=xÞv : x @ t
We deduce the upper bound j@*½t yv ðPÞjs pcvs;P
!e t X vþt1 tþv1 e t¼0
v1
t
( max jxjs ;
jxjs Hs ðPÞ
)t
jFY0 ðx; zÞj2s ja0;n j2n3 s
vþt P ¼ v : which gives (8) since tt¼0 vþt1 t Let us now verify the inequality hi;s Xfi;s e log 2:
ð9Þ
It is equivalent to the previous upper bound jFY0 ðx; zÞj2s ja0;n j2n3 pmaxð1; jxjs ÞHs ðPÞ s
ð5Þ
with our new value of Hs ðPÞ: We restrict to the Archimedean case, the p-adic case leading to similar computations. As for (3), we isolate the leading term a0;n Y n in F : We deduce now from (6) that jFY0 ðx; zÞjs p nðja0;n js jzjs Þjzjn2 þ ðn þ 1ÞjF js s
n1 X j¼1
p
2ðn þ 1Þ
2n1
jF jn1 maxð1; jxjÞn1 s : ja0;n jn2
Noting that ja0;n js pjF js ; we obtain (5).
maxð1; jxjs Þnj jjzjsj1
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Finally, the corollary of Lemma 4 becomes
1 1 maxð1; jxjs ÞHs ðPÞ ; max ; p rs ðPÞ r˜s ðPÞ jFY0 ðx; zÞj2s ja0;n j2n3 s
ð10Þ
which follows immediately from (7) and (8). Now we can read again the proofs of Theorems 1 and 2. Lemmas 5 and 7, and their proofs, remain valid mutatis mutandis with our new values of the parameters hi;s and fi;s ; on account of estimations (7)–(10). As an example, the inequality logþ
1 R0s ðP1 ; P2 Þ
pminfh1;s f1;s e log 2; h2;s f2;s e log 2g;
which was used in the proof of Theorem 2, follows immediately from (5), (9) and (10). In conclusion, Theorems 1 and 2 hold with hi ¼
X
hi;s ¼ ð2n 2Þhðxi Þ þ ð2n 1ÞhðF Þ þ logð8ðn þ 1Þ4nþ6 Þ
s
for i ¼ 1; 2:
6. Deduction of the arithmetical applications Their proofs follow the same principle. We fix some point P1 and look at the points P2 which are close to P1 : Under suitable conditions, Theorems 1–3 provide a lower bound for the degree ½L : Q of the field L ¼ KðP1 ; P2 Þ: For Hilbert’s theorem, we choose P1 ¼ ð0; 0ÞACðQÞ and P2 ¼ ðx; zÞ; where z is some root of F ðx; Y Þ ¼ 0: We expect that L ¼ QðzÞ would have the maximal degree n; meaning that the polynomial Pðx; Y Þ is irreducible in Q½Y : In the case of Runge’s Theorem, the origin P1 is an algebraic point ACN ; with degree don; and we expect that P2 ¼ ðx; zÞ cannot belong to CðKÞ: Thus the two situations may be viewed as reverse from each other, and lead to parallel computations. 6.1. Proof of Theorem 4 Recall that K ¼ Q in this section and denote by s the natural embedding s : QDCs ¼
C
if x ¼ 1=k;
Cp
if x ¼ pt
of Q into Cs : Let us consider the point P1 ¼ ð0; 0ÞACðQÞ and the corresponding meromorphic function yP1 ;s : Notice first that rs ðP1 Þ4H 1 ; since by corollary
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of Lemma 4 we know that 1 4e ðm þ 1Þeð2nþ2Þ ðn þ 1Þeð2nþ3Þ jF j2n1 s p oH; 2 2n3 rs ðP1 Þ ja0;1 js ja0;n js the last upper bound coming from the product formula Y Y Y Y ja0;1 js ¼ ja0;1 j1 jF j1 and ja0;n js ¼ ja0;n j1 jF j1 t X t t X t : tas
tas
tas
tas
It follows from the assumptions that 3
jxjs pH 3mn
ðn1Þ2
oH 1 ors ðP1 Þ:
Define now ( z ¼ yP1 ;s ðxÞA
% Q-R % Q-Q p
if x ¼ 1=k; if x ¼ pt
and set L ¼ QðzÞ;
P2 ¼ ðx; zÞACðLÞ:
Let us verify that An ðxÞFY0 ðx; zÞa0: Denote by R the resultant of F ðX ; Y Þ and FY0 ðX ; Y Þ with respect to Y : Since R is equal to the product of An ðX Þ with the discriminant of F ; we have to prove that RðxÞa0: Assuming the contrary, we obtain pffiffiffi2n1 HðF Þ2n1 oH 2 HðxÞp2HðRÞp2 ðm þ 1Þðn þ 1Þ n using Lemma 4 of [15]. Noting that HðxÞ is equal either to jkj or to pt ; we find a contradiction with our hypotheses. The number field L is contained in Cs by its definition. Denote again by s the natural inclusion s : LDCs ; which obviously extends the preceding embedding s : QDCs : By construction, the point P2 belongs to the branch yP1 ;s : Then we may apply Theorem 1 with S ¼ fsg; according to Remark 1 after Theorem 1. We obtain the inequality pffiffiffiffiffiffiffiffiffi 1 1 hðx; 0Þ 1 1 logþ þ logþ þ h1 h2 þ log 4: p ½L : Q jxjs n ½L : Q Rs ðP1 Þ Noting that logþ jxj1 s ¼ hðxÞ in both cases, we find pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞ p þ log H þ log Hðlog H þ mð2n 1ÞhðxÞÞ þ log 4: ½L : Q n Suppose on the contrary that the polynomial Pðx; Y Þ is reducible in Q½Y ; so that ½L : Q pn 1: Writing now hðxÞ ¼ lmn3 ðn 1Þ2 log H with lX3; we deduce the
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inequality log 4 pffiffiffiffiffi 1 þ log 1 H o ; l 2lp 2 mn ðn 1Þ 2
which is incompatible with the lower bound lX3: Suppose now that the polynomial F is as in Theorem 3. Applying this last result leads by the same argument to the sharper inequality pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞ p þ log H þ log Hðlog H þ ð2n 2ÞhðxÞÞ þ log 4: ½L : Q n Then we obtain in this case the same exponent divided by m:
&
6.2. Proof of Theorem 5 First we perform a change of coordinates ðx; yÞ/ðx1 ; yÞ;
or
ðx; yÞ/ðx1 ; yx1 Þ
whether we stay on P1 P1 or on P2 : Let QACN : In this new frame, the point Q has coordinates ð0; ZÞ for some root Z of FN : In order to unify the proof in both cases, denote by G the polynomial ˜ ; Y Þ ¼ X m F ðX 1 ; Y Þ; FðX
or
ˇ ; Y Þ ¼ X n F ðX 1 ; YX 1 Þ FðX
in the case of an embedding in P2 : The curve C has local equation Gðx; yÞ ¼ 0 in the new coordinates. With some abuse of notations set Q ¼ ð0; ZÞ; regarded now as a % point on the curve Gðx; yÞ ¼ 0; and for any embedding s : Q+C; denote by rs ðQÞ be the convergence’s radius of the Taylor’s series of the function yQ;s ðzÞ at the origin. Let us verify that rs ðQÞ4H d : Denote by Bn ðX Þ the coefficient of Y n in the polynomial GðX ; Y Þ; which is equal either to A˜ n ðX Þ; or to the constant a0;n : Notice that Gð0; Y Þ ¼ FN ðY Þ: Since F is regular at infinity, it follows that both GY0 ð0; ZÞ ¼ 0 ðZÞ and Bn ð0Þ are non-zero. For each embedding t : L+Ct of the field L ¼ KðZÞ; FN recall from (5) the upper bound jGY0 ð0; ZÞj2t jBn ð0Þj2n3 pð4ðm þ 1Þ2nþ2 ðn þ 1Þ2nþ3 ÞeðtÞ jGj2n1 : t t It follows from the corollary of Lemma 4, joined with the product formula, that 1 4ðm þ 1Þ2nþ2 ðn þ 1Þ2nþ3 jGj2n1 s p rs ðQÞ jGY0 ð0; ZÞj2s jBn ð0Þj2n3 s Y p ð4ðm þ 1Þ2nþ2 ðn þ 1Þ2nþ3 ÞeðtÞ jGj2n1 oH d : t t : L+Ct
Suppose now that jxjs 4H maxfd;2dg where d :¼ ½K : Q : Let us first verify that Bn ðx1 ÞGY0 ðx1 ; oÞa0 for any algebraic number o with Gðx1 ; oÞ ¼ 0: Assuming the
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contrary and arguing as in Theorem 4, we obtain d pffiffiffi2n1 jxjs pHðxÞd p 2 ðm þ 1Þðn þ 1Þ n HðF Þ2n1 oH 2d which contradicts our assumption. Now jx1 js ors ðQÞ for any QACN by the preceding remarks. Define oQ;s ¼ yQ;s ðsx1 Þ
PQ;s ¼ ðsx1 ; oQ;s Þ:
and
Thus when Q ranges along CN ; the complex numbers oQ;s are the n distinct roots of the polynomial sGðsx1 ; Y Þ: On the other hand we have Gðx1 ; zÞ ¼ 0;
or
Gðx1 ; zx1 Þ ¼ 0
in the case of an embedding in P2 ; since F ðx; zÞ ¼ 0: Denote P ¼ ðx1 ; zÞ; or P ¼ ðx1 ; zx1 Þ in the second case. It follows that there exists a unique Qs ¼ ð0; Zs ÞACN such that sP ¼ PQs ;s whenever jxjs 4H maxfd;2dg : Then the point P belongs to the branch yQs ;s and L ¼ KðZs Þ is a field of rationality for both P and Qs : Applying Theorems 1 and 3 to the points P and Qs ; we obtain the respective inequalities pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi logþ jxjs hðxÞ þ log H þ log Hðlog H þ mð2n 1ÞhðxÞÞ þ log 4 p n d and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi logþ jxjs hðxÞ þ log H þ log Hðlog H þ ð2n 2ÞhðxÞÞ þ log 4: p n d Notice that both upper bounds remain obviously true when jxjs pH maxfd;2dg : Choose % now d embeddings s : Q+C which restrict on K to distinct embeddings and sum the corresponding inequalities. Dividing by d; we find pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞ p þ log H þ log Hðlog H þ mð2n 1ÞhðxÞÞ þ log 4 d n and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞ p þ log H þ log Hðlog H þ ð2n 2ÞhðxÞÞ þ log 4; d n from which follow the respective upper bounds hðxÞp3mn3 d2 =ðn dÞ2 log H as in the proof of Theorem 4.
and
hðxÞp3n3 d2 =ðn dÞ2 log H
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6.3. Proof of Theorem 6 We use the same setting as in Section 6.2, and we recall the changes of coordinates ðx; yÞ/ðx1 ; yÞ or ðx; yÞ/ðx1 ; x1 yÞ: Observe that each point QACN is rational % and over some number field containing K of degree over Q bounded by nd: Let xAQ nd 1 % let s : Q+C be such that jxjs 4H : It follows from Section 6.2 that sðp ðxÞÞ is made up with the n distinct points whose (new) coordinates are PQ;s ¼ ðsx1 ; oQ;s Þ with oQ;s ¼ yQ;s ðsx1 Þ for QACN : Suppose now that jxjs 42H nd : Since minQACN rs ðQÞ4H nd ; we deduce from the triangle inequality that min
PAp1 ðxÞ
rs ðPÞX min rs ðQÞ 12H nd 412H nd : QACN
Then, we can interchange the roles of x and N; so that sðCN Þ is the set of the n points QP;s ¼ ð0; yP;s ð0ÞÞ
for PAp1 ðxÞ:
The two applications P/Q and Q/P thus defined are bijections which preserve real points when sxAR: By construction, corresponding points P and Q belong to the same branch relative to the embedding s: Now let L be a totally real number field containing K (this is not restrictive) and let P ¼ ðx; zÞAOL L with F ðx; zÞ ¼ 0: We assume that for any embedding % s : Q+C; the set sðCN Þ contains a unique real point sðQ0 Þ: It follows from the preceding remarks that P belongs to the branch yQ0 ;s for all the embeddings % s : Q+C such that jxjs 42H nd ; since then sðPÞ turns out to be the unique real point located on sðp1 ðxÞÞ: The points P and Q are both rational over L: Now Theorem 1 gives the inequality 0 1 ½L : Q
X s : L+C jxjs X2H nd
logjxjs p
1
X C hðxÞ 1 B B þ logðH nd ÞC @ A n ½L : Q s : L+C jxjs X2H nd
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ log Hðlog H þ mð2n 1ÞhðxÞÞ þ log 4:
We deduce that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðxÞ hðxÞp þ nd log H þ log Hðlog H þ mð2n 1ÞhðxÞÞ þ log 8: n
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3
mn Write hðxÞ ¼ lðn1Þ 2 log H and suppose lX1: It follows that
l
log 8 pffiffiffiffiffi nd þ log H 2lp mn2 pd; n1
which implies lp4d: When F satisfies the assumptions of Theorem 3, we can replace in the above inequapffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lities the term log Hðlog H þ mð2n 1ÞhðxÞÞ by log Hðlog H þ ð2n 2ÞhðxÞÞ; which leads to exponents divided by m: This concludes the proof of Theorem 6. For the proof of the corollary, suppose for instance that deg f1 Xdeg f2 ; the case deg f1 odeg f2 being similar. Write f1 ðX Þ ¼ anm1 X m1 þ ? þ a0 ;
f2 ðX Þ ¼ bnm2 X m2 þ ? þ b0
and gðX Þ ¼ cnn X n þ ? þ c0 with am1 bm2 cn a0: The equation F ðx; zÞ ¼ 0 is then equivalent to Gðx; zxðm1 m2 Þ=n Þ ¼ 0
with GðX ; Y Þ ¼ f2 ðX ÞgðX ðm1 m2 Þ=n Y Þ f1 ðX Þ:
Notice that G is the product of a power of X by an absolutely irreducible polynomial, since F is absolutely irreducible. Dividing eventually by some power of X ; we may assume without loss of generality that G is absolutely irreducible. For any % complex embedding s : Q+C; the polynomial sGN ðY Þ ¼ sðbm2 cn Þn Y n sanm1 ¼
n Y
sðbm2 cn ÞY e2p
pffiffiffiffiffi 1n=n
sam1
n¼0
has an unique real root sðam1 =bm2 cn Þ: Thus Theorem 6 can be applied, which gives the required upper bound since HðGÞ ¼ HðF Þ:
References [1] Y. Andre´, G-Functions and Geometry, Vieweg, Braunschwieg, 1989. [2] E. Bombieri, On G-functions, Recent Progress in Analytic Number Theory, Vol. 2, Academic Press, New York, 1981, pp. 1–67. [3] E. Bombieri, On Weil’s The´ore`me de de´composition, Amer. J. Math. 105 (1983) 295–308. [4] L. Comtet, Analyse combinatoire, tome premier, Collection Sup. Le mathe´maticien, Presses universitaires de France, 1970. [5] P. Debes, G-fonctions et the´ore`me d’irre´ductibilite´ de Hilbert, Acta Arith. 47 (1986) 371–401. [6] P. Debes, Hilbert subsets and s-integral points, Manuscripta Math. 89 (1996) 107–137. [7] B. Dwork, A. Van der Poorten, The Eisenstein constant, Duke Math. J. 65 (1992) 23–43 (corrigendum 76 (1994) 669–672). [8] G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. 119 (1984) 387–424.
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[9] P. Hriljac, Heights and Arakelov’s intersection theory, Amer. J. Math. 107 (1) (1985) 23–38. [10] S. Lang, Fundamentals of Diophantine Geometry, Springer, Berlin, 1983. [11] M. Laurent, Some remarks on the approximation of complex numbers by algebraic numbers, Bull. Greek. Math. Soc. 42 (1999) 49–557. [12] M. Laurent, Interpolation determinants of exponential polynomials, Publ. Math. Debrecen 56 (3–4) (2000) 457–473. [13] M. Laurent, D. Roy, Sur l’approximation alge´brique en degre´ de transcendance un, Ann. Inst. Fourier 49 (1999) 27–55. [14] D. Poulakis, Polynomial bounds for the solution of a class of Diophantine equations, J. Number Theory 66 (1997) 271–281. [15] W. Schmidt, Eisenstein’s theorem on power series expansion of algebraic functions, Acta Arith. 56 (1990) 161–179. [16] V.G. Sprindzuck, Arithmetic specializations of polynomials, J. Reine. Angew. Math. 340 (1983) 26–52. [17] V.G. Sprindzuck, Classical Diophantine Equations, Springer, Berlin, 1993. [18] P.G. Walsh, A quantitative version of Runge’s theorem on diophantine equations, Acta Arith. 62 (1992) 157–172.
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 255–262
http://www.elsevier.com/locate/jnt
When subset-sums do not cover all the residues modulo p Jean-Marc Deshouillersa,,1 and Gregory A. Freimanb,2 a
Statistique Mathe´matique et Applications, Universite´ Victor Segalen Bordeaux 2, F-33076 Bordeaux, Cedex France b School of Mathematical Sciences, Tel-Aviv University, 69978 Tel-Aviv, Israel Received 15 June 2002; revised 24 July 2003 Communicated by R.C. Vaughan
Abstract pffiffiffi Let c4 2: We prove that a subset A of Z=pZ; where p is a prime number, with cardinality pffiffiffi larger than c p such that its subset sums do not cover Z=pZ has an automorphic image which is rather concentrated; more precisely, there exists s prime to p such that X as o1 þ Oðp1=4 ln pÞ: p aAA r 2003 Elsevier Inc. All rights reserved. MSC: 11P70 Keywords: Inverse problems of additive number theory
Corresponding author. E-mail addresses: [email protected] (J.-M. Deshouillers), [email protected] (G.A. Freiman). 1 Supported by Universite´ Victor Segalen Bordeaux 3 (EA 2961), Universite´ Bordeaux 1 and CNRS (UMR 5465). 2 Supported by Tel Aviv University and ADEMAS Association. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.08.009
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1. Introduction 1:1: Let p be a prime number and A be a set of distinct non-zero residue classes modulo p: We denote by A the set of the subset sums of A; that is to say ( ) X b; BCA : A ¼ bAB
+ and Heilbronn [3] and In 1968, Olson [7] solved a conjecture of Erdos pffiffiffiffiffiffiffiffiffiffiffiffiffiffi showed that if jAj4 4p 3; then A ¼ Z=pZ: One readily sees that this pffiffiffiffiffiffiffiffiffiffiffi is close to the best possible result, since for p43 and ko2ð p 1Þ; the set B ¼ fb1 ¼ 1; b2 ¼ 1; y; bk ¼ ð1Þk1 Ikþ1 is such that B aZ=pZ 2 mg (cf. [3]). The minimal value s0 for which A ¼ Z=pZ as soon as jAj4s0 was determined by Dias da Silva and Hamidoune [2] in 1994. In this paper, we are concerned with subsets A of Z=pZ such that A aZ=pZ: A first remark is that, for any t prime to p; the set t A ¼ fta; aAAg satisfies also the relation ðt AÞ aZ=pZ: A second remark is that we have P a˜ a 1 a A aZ=pZ as soon as aAA jjpjjo1 p; where jjpjj is defined as minzAZ jp zj; where a˜ is any representative in Z of the class a: We prove here the following result. pffiffiffi Theorem 1. Let c4 2: Let p be a prime number and A be a subset of Z=pZ with pffiffiffi cardinality larger than c p; such that its subset sums do not cover Z=pZ: There exists s prime to p such that X as o1 þ Oðp1=4 ln pÞ: ð1Þ p aAA 1:2: Notation For a real number u; we use the traditional notation ep ðuÞ ¼ expð2piu p Þ
and
jjujj ¼ minzAZ ju zj;
expression ep ðbÞ ˜ ˜ common value of all the ep ðbÞ’s (resp. jjb=pjj), representing the class b; we further let jbj over all the representative b˜ of b; or equivalently when
bAZ=pZ;
the
(resp. jjb=pjj) denotes the where b˜ is any integer ˜ denote the minimum of jbj jbj ¼ pjjb=pjj The letter p denotes a prime number which is sufficiently large to satisfy all the implicit or explicit inequalities. When a multiset (family of elements where repetition is allowed) is described by its elements, we list them between two double curly brackets, e.g. A ¼ ff0; 1; 1gg ¼ ffa1 ; a2 ; a3 gg: If A is a multiset, the elements of which P lie in an abelian group, we denote by A the set (sic) of its subset sums, i.e. A ¼ f bAB b; BCAg; where the sum over the empty set is taken to be 0:
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2. Arithmetic progressions in subset sums In the early 1990s, Freiman [4] and Sa´rko¨zy [8] independently proved that for a 1
subset D included in ½1; N with more than ðN ln NÞ2 elements, the set D contains an arithmetic progression with more than jDj2 ðln NÞ1 elements. For our purpose, we need some further connection between the arithmetic properties of the elements of D and the difference of the arithmetic progression, in the spirit of the work [1] by Chaimovitch. Theorem 2. Let I4L4100 and B42C ln L be positive integers such that C 2 4500Lðln LÞ2 þ 2000I ln L: Let B be a set of B integers included in ½L; L : Then, there exist d40 and a subset C of B with cardinality C such that (i) all the elements of C are divisible by d; (ii) C contains an arithmetic progression with I terms and difference d; (iii) at most C ln L elements of B are not divisible by d: The proof of Theorem 2 is based on the two following lemmas. Lemma 1 (Galil and Margalit [5, Theorem 4.1]). Let D be a subset of Z included in an interval of length l and let c be an integer larger than l satisfying ðCard DÞ2 450lðlog2 lÞ2 þ 200cðlog2 lÞ;
ð2Þ
then the set D contains an arithmetic progression of length 2c: Lemma 2. Let G be a finite abelian group with qX2 elements and let A be a multiset consisting of m non-zero elements of G with mXq 1: Then, either there exists a subfamily B of A with q 1 elements such that B ¼ G; or there exists a subgroup H of G; different from f0g and G such that less than CardðG=HÞ 1 elements of A are not in H: Proof. The lemma is obviously valid when q ¼ 2: Let q42 be an integer such that the lemma is valid for any group with at most q 1 elements, an induction hypothesis that will be used only at the very end of the argument. We proceed by induction and construct step by step sub-multisets Bk of A in the following way: we let B0 ¼ |; we then choose an arbitrary element from A; call it a1 and let B1 ¼ ffa1 gg: We have jB1 j ¼ jffa1 gg j ¼ jf0; a1 gj ¼ 241 ¼ jB0 j:
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When, for some kpq 2; the multiset Bk ¼ ffa1 ; y; ak gg has been constructed, we construct Bkþ1 by attaching to Bk an element a (which will then be called akþ1 ) from A\Bk ; if there exists such an a satisfying jffa1 ; y; ak ; agg j4jBk j: We begin by assuming that this programme can be performed up to the step q 2 inclusively, i.e. up to the construction of Bq1 : In this case we have jBq1 j4jBq2 j4y4jB1 j ¼ 2; this implies jBq1 jXq; whence Bq1 ¼ G; in this case the conclusion of our lemma is reached. We now assume that, at some level kpq 2; we have jffa1 ; y; ak ; agg j ¼ jBk j for any a in A\Bk : For any such a; we have Bk þ Ha ¼ Bk ;
ð3Þ
where Ha denotes the subgroup of G generated by a: Since aa0; we have jHa X2; which implies that the largest subgroup H of G such that Bk ¼ Bk þ H has cardinality at least 2. If jHj ¼ q; then Bk ¼ G and our lemma is proved. Otherwise, we have 2pjHjoq: Since, for any a in A\Bk ; the subgroup Ha is a subgroup of H; the element a is in H: We thus have ðA\Bk ÞCH: If k is strictly less than CardðG=HÞ 1; our lemma is proved. We may thus assume that k is at least equal to q0 1; where q0 denotes the order of G0 ¼ G=H; let us denote by a0i the canonical image of ai in G=H and by B0k that of Bk : Since Bk is not equal to G; it cannot have a representative in every coset modulo H and thus the set ðB0k Þ is not equal to G0 : Since 2pq0 oq; we may apply Lemma 2 to the multiset B0k : there exists a subgroup F0 of G0 ; different from {0} and G0 such that less than CardðG0 =F0 Þ 1 elements of B0k are not in F0 : Let F be the reciprocal image of F0 in G; the last statement implies that less than CardðG0 =F0 Þ 1 elements of A are not in F; but F is different from f0g and G and we have CardðG0 =F0 Þ ¼ CardðG=FÞ: Thus, our lemma is proved for q; and thus, it is proved by induction. & We formulate a result that will be used in the proof of Theorem 2 and has its own independent interest. Proposition 1. Let B4C and I4L4100 be positive integers such that C 2 4500Lðln LÞ2 þ 2000I ln L: Let B be a set of B integers included in ½L; L : Then, either there exists a subset C of B with cardinality C such that C contains an interval I with I elements, or there exists rX2 which divides at least B C elements of B: Proof. Let us consider a subset D of B with IC=2m elements. By Lemma 1, the set D contains an arithmetic progression with at least 4I elements. This arithmetic progression is included in an interval of length at most LC=2 : its difference d is thus at most LC=8I; which is at most C=8: Let P ¼ fn0 þ jd; 0ojpNg be such an arithmetic progression, and recall that N is at least 4I:
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First case: For each divisor rX2 of d; we can find at least r 1 elements in B\D which are not divisible by r: Then, by Lemma 2, we can find a subset E in B\D with d 1 elements, such that E covers all the residue classes modulo d: It is clear that C ¼ D,E is a subset of B with at most C elements. Let us prove that C contains an interval with at least I elements. By the properties of E; we can find d pairwise distinct residue classes modulo d in E -½dL; dL : Let x be an integer in ½n0 þ IN4 md; n0 þ I3N 4 md ; there exists y in E -½dL; dL such that x y n0 (mod d). Moreover, since NX4I; we have n0 ox ypn0 þ Nd; whence x yAD and so xAD þ E CðD,EÞ : We N have proved that C contains an interval with at least I3N 4 m I 4 m þ 1 terms, and so it contains at least I terms. Second case: There exists a divisor rX2 of d such that less than r 1 elements in B\D are not divisible by r: Then the number of elements of B which are not divisible by r is at most jDj þ rpjDj þ dpC: We actually showed the existence of a subset C of B with at most C elements, satisfying the other required properties. It is now straightforward to obtain Proposition 1. & Proof of Theorem 2. If there exists a subset C of B with cardinality C such that C contains an interval I with I elements, Theorem 2 is true with d ¼ 1: Otherwise, by Proposition 1, there exists r1 X2 which divides at least B C elements of B: We let B1 ¼ fb=r1 AB; r1 jbg; which is included in ½L=r1 ; L=r1 C½L; L ; and we repeat the argument. If there exists a subset C1 of B1 with cardinality C such that C1 contains an interval I1 with I elements, Theorem 2 is true with d ¼ r1 and C ¼ fr1 c1 ; c1 AC1 g: Otherwise, since we have jB1 j4Cð2 ln L 1Þ; we can apply Proposition 1, which implies that there exists r2 X2 which divides at least jB1 j C elements of B1 : We then construct B2 ; etc. If we assume that we can proceed inductively up to the construction of Bk ; for some k; it means that we can construct a set with at least Cð2 ln L kÞ integers included in ½L=2k ; L=2k ; this is clearly impossible if k4ln L=ln 2: Thus, there exists ko1:5 ln L such that Bk admits a subset Ck with cardinality c such that Ck contains an interval Ik with I elements. Then Theorem 2 is true with d ¼ r1 :yrk and C ¼ fdck ; ck ACk g: &
3. Proof of Theorem 1 3:1: The set A is dense around 0: We cannot apply directly Theorem 2 to our set A (nor to its canonical image in p=2; p=2½) because we miss the required density by a logarithmic factor. For that reason, we first need to show that some large part of A is located in a short interval.
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Proposition 2. Let p be a prime and A be a subset of Z=pZ such that A aZ=pZ: There exists a non-zero tAZ=pZ such that ! X at2 41 ð4Þ exp p p p aAA and, for any u40
at 1 Card aAA; Xu pu2 ln p: p
ð5Þ
Proof. Let x be an element in Z=pZ which is not in A : For any subset fa1 ; y; ar g of pairwise distinct elements in A; we have X ep ðtða1 þ ? þ ar xÞÞ ¼ 0; t
which implies X Y t
ð1 þ ep ðtaÞÞep ðtxÞ ¼ 0;
aAA
whence we deduce, by taking aside t=0, X Y ð1 þ ep ðtaÞÞep ðtxÞ ¼ 0: 2jAj þ ta0 aAA
This implies that there exists ta0 such that Y aAA
1 1 2jAj 4 2jAj : j1 þ ep ðtaÞjX p1 p
ð6Þ
By Taylor’s formula, we have, for jyjp12
ðpyÞ2 ðpyÞ4 p p3 þ p1 cos pyp1 py2 p1 py2 p expðpy2 Þ; 2 96 2 24 whence j1 þ expð2piyÞjp2 expðpjjyjj2 Þ:
ð7Þ
Relation (4) comes from (6) and (7). If we assume that (5) does not hold, we would have ! X at2 p expðp log pÞ ¼ pp op1 ; exp p p aAA in contradiction to (4).
&
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3:2: Some set ðs AÞ contains a long interval: This step is an intermediate one and the choice of the value of the parameters is very flexible. The following result can be extended to any A with at least p0:8 ðln pÞa for any a43:6: Proposition 3. Let p be a prime number which is sufficiently large. Let A be a subset of pffiffiffi Z=pZ such that jAjX p and A aZ=pZ and let I ¼ Ip0:9 m: There exists s prime to p such that jfaAA; jjas=pjjXI 1=4 gjp2I 1=2 ln2 p and s A contains a subset C with at most I 1=2 ln2 p elements such that C contains an interval with at least I elements. Proof. We let u ¼ I 1=4 : By Proposition 2, there exits a non-zero t such that jfaAA; jjat=pjjXI 1=4 gjpI 1=2 ln p: Let us consider B1 ; the image of fat; aAA; jjat=pjjoI 1=4 g in ½pI 1=4 ; pI 1=4 through the natural map from Z=pZ to ½p=2; p=2 : We readily see that we may apply Theorem 2 with I ¼ I; L ¼ IpI 1=4 m; B ¼ B1 ; C ¼ 30II 1=2 ln pm: Thus, there exists d40 and C1 CB1 with C elements, such that d divides all the elements of C1 ; C contains an arithmetic progression with I terms and difference d and fbAB1 ; djbg has at least B1 I 1=2 ln2 p elements. Since d is less than p; we can consider the inverse d% of d modulo p: Let B denote the canonical image of fb=d; bAB1 g in Z=pZ: It is a subset % which has the following properties: of s A (with s ¼ dt), (i) 8b1 AB : jb1 jppI 1=4 ; (ii) jðs AÞ\Bjp2I 1=2 ln2 p; (iii) B contains a subset C (namely the canonical image of fc=d; cAC1 g) with at most I 1=2 ln2 p elements, such that C contains an interval with at least I elements. Proposition 3 is thus proved.
&
3:3: End of the proof: Since the hypothesis and conclusion of Theorem 1 are invariant under the multiplication by a non zero element, we may assume without loss of generality that Proposition 3 holds with s ¼ 1: Let B ¼ faAA; jajop0:8 g and let us write E ¼ B\C: Since the difference between two elements in E is at most 2p0:8 oIp0:9 m ¼ I; the set E þ C is an interval, and since it Pis included in A which does not cover Z=pZ; its length is less than p: We thus have eAE jejop; which implies that for any l40 we have pffiffiffi pffiffiffi jfeAE; jejXl pgjpl1 p:
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pffiffiffi pffiffiffi Let us choose l ¼ 2ðc 2Þ1 ; and define D ¼ faAA; jajol pg and D1 its natural pffiffiffi pffiffiffi image in ½l p; l p : We have pffiffiffi pffiffiffi pffiffiffi pffiffiffi jD1 jXjEj l1 pXjAj 3p0:45 ln2 p l1 pXc0 p with c0 ¼ ðc þ 2 2Þ=3: pffiffiffi We apply Theorem 2 with L ¼ Il pm; I ¼ 2L þ 1; B ¼ D1 and C1 ¼ 30IL1=2 ln Lmð¼ Oðp1=4 ln pÞÞ: there exists d40 and C1 CD1 with cardinality pffiffiffi pffiffiffi C1 such that d divides all the elements in C1 and at least ðc þ 3 2Þ=4 p elements in D1 \C1 ; and such that C1 contains an arithmetic progression with difference d and 2L þ 1 terms. We define d% as the inverse of d modulo p; and we write d% A as the disjoint union of the following three subsets : C; the image of fc=d; cAC1 g in Z=pZ; B; the image of fb=d; bAD1 \C1 and djbg in Z=pZ and F ¼ d% A\ðB,CÞ: We have BC½L; L ; pffiffiffi jBj43 þ p; jCj ¼ Oðp1=4 ln pÞ and C contains an interval of length 2L þ 1: P pffiffiffi The set B þ C is an interval with at least bAB jbj elements. Since jBj43 þ p; we have X X jbjX2 k4p=2: pffiffi bAB kp1þ p=2 length at least Since B þ C is an interval withP P p=2; the set B þ C þ F is also an interval and its length is at least bAB jbj þ f AF jf j; but this interval is included in P P ðd% AÞ and we thus have bAB jbj þ f AF jf jpp: Finally, since C contains
Oðp1=4 ln pÞ elements which have all a size which is Oðp1=2 Þ; we have X % jdajpp þ Oðp3=4 ln pÞ; aAA
which is equivalent to (1), and proves Theorem 1.
&
Acknowledgments The authors are thankful to the referee for his/her very careful reading of a previous version of this paper. References [1] M. Chaimovich, New algorithm for dense subset sum problem, Aste´risque 258 (1999) 363–373. [2] J.A. Dias da Silva, Y.O. Hamidoune, Cyclic spaces for Grassman derivatives and additive theory, Bull. London Math. Soc. 26 (1994) 140–146. + H. Heilbronn, On the addition of residue classes mod p; Acta Arithmetica 9 (1964) 149–159. [3] P. Erdos, [4] G.A. Freiman, New analytical results in subset sum problem, Discrete Math. 114 (1993) 205–217. [5] Z. Galil, O. Margalit, An almost linear-time algorithm for the dense subset sum problem, SIAM J. Comput. 20 (6) (1991) 1157–1189. [6] J. Olson, An addition theorem modulo p; J. Combin. Theory 5 (1968) 45–52. [7] A. Sa´rko¨zy, Finite addition theorems II, J. Number Theory 48 (2) (1994) 197–218.
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http://www.elsevier.com/locate/jnt
An alternate proof of Cohn’s four squares theorem Jesse Ira Deutsch Mathematics Department, University of Botswana, Private Bag 0022, Gaborone, Botswana Received 2 July 2002; revised 23 May 2003 Communicated by D. Goss
Abstract While various techniques have been used to demonstrate the classical four squares theorem for the rational integers, the method of modular forms of two variables has been the standard way of dealing with sums of squares problems for integers in quadratic fields. The case of pffiffiffi representations by sums of four squares in Qð 5Þ was resolved by Go¨tzky, while those of pffiffiffi pffiffiffi Qð 2Þ and Qð 3Þ were resolved by Cohn. These efforts utilized modular forms. In previous work, the author was able to demonstrate Go¨tzky’s theorem by means of the geometry of pffiffiffi numbers. Here Cohn’s theorem on representation by the sum of four squares for Qð 2Þ is proven by a combination of geometry of numbers and quaternionic techniques. r 2003 Elsevier Inc. All rights reserved. MSC: primary 11D09; 11D57; 11P05; secondary 11Y99 Keywords: Sums of squares; Quaternions; Geometry of numbers
1. Introduction In papers in the early 1960’s, H. Cohn found analogues of sums of squares theorems for certain quadratic number fields. These papers extended the work of pffiffiffi Go¨tzky in the 1920s demonstrating that every totally positive integer in Qð 5Þ is the sum of four squares of integers from that field. In particular, Cohn showed that pffiffiffi every totally positive integer with even coefficient on the radical term in Qð 2Þ and E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.08.005
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pffiffiffi Qð 3Þ is the sum of four integer squares from their respective fields. The above work of Go¨tzky and Cohn was based on the theory of modular forms of two variables, and gave the number of representations in addition to an existence proof (see [2– 4,6]). Here Cohn’s Theorem on the existence of sum of four squares representations pffiffiffi is proven in the case of Qð 2Þ without recourse to the theory of modular forms. Other techniques have been used to prove the classical four squares theorem that every positive rational integer is the sum of four rational integer squares. Lagrange used infinite descent, Grace worked with the geometry of numbers, and Hurwitz utilized a special ring of quaternions (see [8,9]). In a previous paper, the author gave pffiffiffi an alternate proof of Go¨tzky’s result for Qð 5Þ by means of geometry of numbers. Using a convex figure in R8 called a spherical diamond, it was possible by means of geometry of numbers to demonstrate that for each algebraic integer prime r in pffiffiffi Qð 5Þ there exist algebraic integers a; b; g; d and k such that a2 þ b2 þ g2 þ d2 ¼ kr;
ð1:1Þ
pffiffiffi where jkk jp8: Here means conjugation with respect to Qð 5Þ: From this point, pffiffiffi some number theory in the ring of integers of Qð 5Þ eliminated all cases except that of k a unit. The four square theorem then follows (see [5]). Due to larger bounds and differences in number theory of the corresponding ring, the above approach is pffiffiffi insufficient by itself for the case of sums of four squares in Qð 2Þ: Hurwitz’s approach to the classical four squares theorem used the norm Euclidean property of the ring Z 1; i; j; 12ð1 þ i þ j þ kÞ : However, one key step in the proof required the use of the result in elementary number theory that for every rational prime p there exists rational integers a and b such that a2 þ b2 þ 1 ¼ kp with kop: While it is not clear if there is an analogue to this result in quadratic fields, this difficulty can be avoided by using the conclusions available from the geometry of numbers. In contrast to Hurwitz’s quaternions, we use the ring of cubian quaternions. pffiffiffi pffiffiffi Letting Oð 2Þ represent the ring of algebraic integers in Qð 2Þ; the cubians are the pffiffiffi pffiffiffi pffiffiffi Oð 2Þ-module with generators f1; 2ð1 þ iÞ=2; 2ð1 þ jÞ=2; 12ð1 þ i þ j þ kÞg: This set is a ring and a principal ideal domain (see [1,11]). Recall than an associate of an element of a ring is the product of that element with a unit of the ring on the left or right. While each Hurwitz quaternion has an associate with integer coefficients in the standard quaternion basis f1; i; j; kg; it is not the case that every cubian has an pffiffiffi associate with Oð 2Þ coefficients in that basis. However, some weaker results are pffiffiffi proven that combined with knowledge of factorization into primes of Oð 2Þ suffice pffiffiffi to obtain Cohn’s result for Qð 2Þ: For information on the classical case see Hardy and Wright [8, Chapter XX] and Herstein [9]. To fix notation, let i; j; and k be the standard quaternions whose squares are 1 and whose products are anti-commutative, i.e. ij ¼ ji; etc. We use H to represent the ring of all real quaternions, that is, all real linear combinations of the elements 1, i; j; k: Bold type is used to represent a quaternion variable, so that a typical element
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of H is of the form q ¼ q1 þ q2 i þ q3 j þ q4 k with q1 ; y; q4 real numbers. Quaternion conjugation is denoted by placing a bar above the variable. Thus q% is the same as q except for sign changes in the coefficients of i; j; and k: N represents the quaternion norm, so that NðqÞ ¼ q q% ¼ q21 þ ? þ q24 : For R a subring of the real numbers, and s1 ; y; sk real quaternions, we define R½s1 ; y; sk as the R module generated by s1 ; ysk : When R is a subring of a quadratic field, we use a star to show conjugation with respect to the field. Thus the pffiffiffi pffiffiffi pffiffiffi cubians are K ¼ Oð 2Þ½1; 2ð1 þ iÞ=2; 2ð1 þ jÞ=2; 12ð1 þ i þ j þ kÞ : We define w1 ¼ 1;
w2 ¼
pffiffiffi 2ð1 þ iÞ=2;
w3 ¼
pffiffiffi 2ð1 þ jÞ=2;
w4 ¼ 12ð1 þ i þ j þ kÞ:
ð1:2Þ
2. Results from the geometry of numbers A key result in the alternate proof of Go¨tzky’s theorem was that for every prime r pffiffiffi pffiffiffi pffiffiffi in the algebraic integers Oð 5Þ of Qð 5Þ there exist k; a; b; g and d in Oð 5Þ for which kr ¼ a2 þ b2 þ g2 þ d2 with k a unit. This can be generalized using the same geometry of numbers technique employed in Deutsch [5], though the restriction on k must be loosened. Lemma 1. Let p be a rational prime which splits in the ring of algebraic integers O of a real quadratic field of discriminant d. Let r be a prime of O dividing p. Then there exist k; a; b; g and d in O such that kr ¼ a2 þ b2 þ g2 þ d2 and jkk jp1:70 d: Proof. We closely follow the proof in Deutsch [5]. Since p splits, jrr j ¼ p: Choose rational integers a and b such that a2 þ b2 þ 1 0ðmod pÞ: Hence a2 þ b2 þ 1 0ðmod rÞ: Choose e so that 1 and e are Z module generators of O; i.e. O ¼ Z½1; e : To employ the geometry of numbers approach, we need to choose a lattice and a centrally symmetric convex subset in R8 : For the lattice we choose the set ða; a ; b; b ; aa þ bb þ mr; aa þ bb þ m r ; ba ab þ nr; ba ab þ n r Þ;
ð2:1Þ
where a; b; m and n run through all of O: It can be seen that this lattice is generated by the following basis elements: 9 8 ð1; 1; 0; 0; a; a; b; bÞ; ðe; e ; 0; 0; ae; ae ; be; be Þ; > > > > > = < ð0; 0; 1; 1; b; b; a; aÞ; ð0; 0; e; e ; be; be ; ae; ae Þ; > : > > ð0; 0; 0; 0; r; r ; 0; 0Þ; ð0; 0; 0; 0; er; e r ; 0; 0Þ; > > > > ; : ð0; 0; 0; 0; 0; 0; r; r Þ; ð0; 0; 0; 0; 0; 0; er; e r Þ
ð2:2Þ
The size of the lattice is the absolute value of the determinant of a basis, which is jðe e Þ4 r2 ðr Þ2 j or d 2 p2 : We apply Minkowski’s convex body theorem to the
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spherical diamond CðrÞ in R8 defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x22 þ x23 þ x24 þ x25 þ x26 þ x27 þ x28 pr:
ð2:3Þ
Using Deutsch [5, Lemma 8] we choose r such that p4 8 r X28 d 2 p2 280
3 r8 X
28 d 2 280 2 p p4
ð2:4Þ
so r ¼ 2:2822d 1=4 p1=4 suffices. Minkowski’s geometry of numbers gives us algebraic integers, not all zero, such that a2 þ b2 þ g2 þ d2 ¼ kr
ð2:5Þ
ffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi kr þ k r p2:2822d 1=4 p1=4
ð2:6Þ
and
so 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi krk r p kr þ k r p2:2822d 1=4 p1=4
jkrk r j1=4 pð2:2822=2Þd 1=4 p1=4 jkk jp1:6955d: That proves the Lemma.
ð2:7Þ
&
pffiffiffi Specializing to the case of Qð 2Þ we can get a tighter bound. pffiffiffi pffiffiffi Lemma 2. Let r be a prime in Oð 2Þ: Then there exist k; a; b; g and d in Oð 2Þ such that kr ¼ a2 þ b2 þ g2 þ d2 and jkk jp9: Proof. Let r lie above the rational prime p: There are three cases. If p is inert then r is an associate of p: The classical four squares theorem for pffiffiffi rational integers shows that we can choose k as a unit of Oð 2Þ: pffiffiffi If p ramifies then p ¼ 2 and r must be an associate of 2: Noting the trivial pffiffiffipffiffiffi pffiffiffi equation 2 2 ¼ 12 þ 12 we may choose k as an associate of 2: Thus jkk j ¼ 2 which suffices for the Lemma. pffiffiffi Suppose that p splits. Note the discriminant of Oð 2Þ is 8: By Lemma 1 there pffiffiffi exists kAOð 2Þ such that kr is the sum of four squares and jkk jp1:7d ¼ 13:56: Hence jkk jp13: pffiffiffi Let q be a rational prime that is inert in Oð 2Þ and which divides kk : Being prime implies q divides k or k : But if q divides one, it must also divide the other. Hence q2
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267
pffiffiffi divides kk : Since 3, 5, 11 and 13 are inert in Oð 2Þ we observe that jkk j cannot equal any of f3; 5; 6; 10; 11; 12; 13g: Hence jkk jp9: &
3. The cubians It has already been noted that the cubians K form a ring and are a principal ideal domain for each of the left ideals and the right ideals. There are 48 units of norm one in K of which only a subset will be needed for our purposes. For details see Vigne´ras [11]. Lemma 3. The following are units in K of norm 1. 9 8 pffiffiffi pffiffiffi pffiffiffi > > w ; w ; w ; w ; 2 w
w ; 2 w
w ; 2 w
w ; 1 2 3 4 2 1 4 3 3 1 > > > > pffiffiffi pffiffiffi pffiffiffi pffiffiffi > > > > > >
w þ 2 w þ 2 w
w ; w þ w
2 w ; 2 w
w ; > > 4 3 2 1 3 2 1 4 2 > > p ffiffi ffi p ffiffi ffi p ffiffi ffi > > = < w w ; w 2w ; w 2w ; w 2w ; 4 1 2 1 4 3 3 1 pffiffiffi pffiffiffi pffiffiffi : 7 > > 2w4 þ 2w3 þ w2 2w1 ; w4 2w3 þ w1 ; > > > > pffiffiffi pffiffiffi pffiffiffi pffiffiffi > > > > >
w4 þ 2w3 þ 2w2 2w1 ; 2w4 w3 w2 þ 2w1 ; > > > > > > > ; : pffiffiffi 2w4 w3 w2
ð3:1Þ
pffiffiffi Proof. First note that K is closed under conjugation since w% 1 ¼ w1 ; w % 2 ¼ 2w1
pffiffiffi w2 ; w % 3 ¼ 2w1 w3 ; w % 4 ¼ w1 w4 : The quaternions listed above all have norm 1 as is easily demonstrated by computer algebra. The inverse of any quaternion of norm 1 is its conjugate. For the quaternions above this means the inverse is an element of K: Hence the quaternions in (3.1) are units of K: These units were derived by multiplying w1 through w4 together numerous times and in various combinations. & pffiffiffi Lemma 4. K-R ¼ Oð 2Þ: pffiffiffi Proof. As 1AK it is clear that Oð 2ÞDK-R: In the other direction, write a typical pffiffiffi aAK as a ¼ a1 w1 þ a2 w2 þ a3 w3 þ a4 w4 where a1 ; y; a4 AOð 2Þ: Suppose that aAR: Then, as the k coefficient of a is zero we must have a4 ¼ 0: Considering the j coefficient we find a3 ¼ 0: Similarly for the i coefficient, implying a2 ¼ 0: Thus a ¼ pffiffiffi a1 w1 AR: Therefore, K-RDOð 2Þ and the Lemma follows. & pffiffiffi Lemma 5. For all aAK the quaternionic norm NðaÞAOð 2Þ: Proof. NðaÞ ¼ a%a is an element of the ring K as this ring is closed under conjugation. pffiffiffi By the previous Lemma, NðaÞAOð 2Þ: &
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Hurwitz’s proof of the rational four squares theorem depends upon the fact that every quaternion in his special ring has an associate in Z½1; i; j; k : In contrast, numerical computation tends to imply that even if we permit multiplication by units on both sides of an element of K we do not always get at least one such two sided pffiffiffi associate into Oð 2Þ½1; i; j; k : However, the following weaker results suffice for our purposes. Lemma 6. For all qAK there exists a quaternion unit uAK of norm 1 such that pffiffiffi pffiffiffi pffiffiffi has Oð 2Þ–integer coefficients, i.e. 2 quAOð 2Þ½1; i; j; k :
pffiffiffi 2qu
pffiffiffi Proof. Let q ¼ a1 w1 þ a2 w2 þ a3 w3 þ a4 w4 ; where a1 ; y; a4 AOð 2Þ: Write q ¼ 2ðg1 w1 þ g2 w2 þ g3 w3 þ g4 w4 Þ þ ðd1 w1 þ d2 w2 þ d3 w3 þ d4 w4 Þ;
ð3:2Þ
pffiffiffi pffiffiffi where g1 ; y; g4 AOð 2Þ and d1 ; y; d4 AOð 2Þ=ð2Þ: Note that a full set of residues of pffiffiffi pffiffiffi pffiffiffi Oð 2Þ modulo 2 can be chosen as f0; 1; 2; 1 þ 2g: Computation shows that for all possible values of d1 ; y; d4 there exists a quaternion unit u in (3.1) such that pffiffiffi pffiffiffi 2ðd1 w1 þ ? þ d4 w4 Þu is an element of Oð 2Þ½1; i; j; k : See Table 1 for examples of the relevant computations. Also ! 4 4 X pffiffiffi pffiffiffi X 2 2 gt w t u ¼ 2 2 ð3:3Þ g# t wt t¼1
t¼1
pffiffiffi with g# 1 ; y; g# 4 AOð 2Þ as K is closed under quaternion multiplication. However, two pffiffiffi times any element of K must be in Oð 2Þ½1; i; j; k as the denominators of the
Table 1 pffiffiffi pffiffiffi Some associates of elements of K whose dilations by 2 are in Oð 2Þ½1; i; j; k pffiffiffi Quaternion q Unit u 2qu ^ pffiffiffi 2w pffiffiffi 1 2w1 þ w4 pffiffiffi pffiffiffi 2w þ 2w4 pffiffiffi pffiffiffi 1 2w þ ð 2 þ 1Þw4 pffiffiffi 1 2w þ w3 pffiffiffi 1 2w þ w3 þ w4 pffiffiffi pffiffiffi 1 2w1 þ w3 þ 2w4 pffiffiffi pffiffiffi 2w þ w þ ð 2 þ 1Þw4 pffiffiffi 1 p3ffiffiffi 2w þ 2w pffiffiffi 1 pffiffiffi 3 2w þ 2w þ w pffiffiffi 1 pffiffiffi 3 p4ffiffiffi 2w1 þ 2w3 þ 2w4 ^
^ w1 w2 w1 w2 w1 w3 w1 w3 w1 w2 w1 ^
^ 2 pffiffiffi pffiffiffi 2 þ ð 2 þ 1Þi þ j 3þiþjþk pffiffiffi pffiffiffi pffiffiffi 2 þ ð2 2 þ 1Þi þ ð 2 þ 1Þj 3þj pffiffiffi pffiffiffi 2 þ ð2 2 þ 1Þj þ k 4 þ i þ 2j þ k pffiffiffi pffiffiffi pffiffiffi 2 þ ð3 2 þ 1Þj þ ð 2 þ 1Þk pffiffiffi pffiffiffi ð 2 þ 2Þ þ 2j pffiffiffi pffiffiffi ð 2 þ 1Þ þ ð 2 þ 2Þi þ 2j k pffiffiffi pffiffiffi ð 2 þ 3Þ þ i þ ð 2 þ 1Þj þ k ^
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generators are 1,
269
pffiffiffi 2 or 2. Hence
pffiffiffi pffiffiffi 2qu ¼ 2 2
4 X t¼1
! gt w t
pffiffiffi uþ 2
4 X
! dt wt
u
ð3:4Þ
t¼1
pffiffiffi is an element of Oð 2Þ½1; i; j; k : & We now need a more delicate result on associates. If we consider certain primes in pffiffiffi pffiffiffi pffiffiffi Oð 2Þ such as 3 þ 2 and 5 þ 2 we know that these cannot be written as sums of pffiffiffi pffiffiffi pffiffiffi squares. Yet the product ð3 þ 2Þð5 þ 2Þ ¼ 17 þ 8 2 can be written as the sum of four squares. This situation is related to the following lemma. pffiffiffi pffiffiffi Lemma 7. Let qAK: Thus NðqÞAOð 2Þ: Suppose NðqÞ ¼ a þ b 2 where a is an odd rational integer and b is an even rational integer. Then there exists a quaternion unit pffiffiffi uAK of norm 1 such that quAOð 2Þ ½1; i; j; k : Proof. Consider the norm of a typical element of K: pffiffiffi pffiffiffi pffiffiffi pffiffiffi Nðða1 þ b1 2Þw1 þ ða2 þ b2 2Þw2 þ ða3 þ b3 2Þw3 þ ða4 þ b4 2Þw4 Þ;
ð3:5Þ
pffiffiffi where a1 ; y; a4 ; b1 ; y; b4 AZ: By Lemma 5 this expression must simplify to A þ B 2 where A and B are rational integers. Computer algebra shows that A and B are polynomials in the a’s and b’s with rational integer coefficients. Also A a21 þ a22 þ a23 þ a24 þ a1 a4 þ a2 a3 ðmod 2Þ; B a1 a2 þ a1 a3 þ a2 a4 þ a3 a4 þ a4 b1 þ a3 b2 þ a2 b3 þ a1 b4 ðmod 2Þ:
ð3:6Þ
Note that if we take rational integers c1 ; y; c4 and d1 ; y; d4 such that ct at ðmod 2Þ and dt bt ðmod 2Þ for t ¼ 1; 2; 3; 4 and compute the quaternion norm of ðc1 þ pffiffiffi pffiffiffi d1 2Þw1 þ ? þ ðc4 þ d4 2Þw4 then the values of A and B modulo 2 do not change. pffiffiffi As in the previous Lemma let q ¼ a1 w1 þ a2 w2 þ a3 w3 þ a4 w4 where at AOð 2Þ for pffiffiffi t ¼ 1; y; 4: Again decompose q as in (3.2), where g1 ; y; g4 AOð 2Þ and pffiffiffi pffiffiffi d1 ; y; d4 AOð 2Þ=ð2Þ: Let d ¼ d1 w1 þ ? þ d4 w4 : Then at dt ðmod 2Þ in Oð 2Þ for t ¼ 1; y; 4: Thus the rational coefficients of at and dt are congruent modulo 2 in Z and the same holds for the corresponding irrational coefficients. Consequently, the rational coefficient of NðqÞ is congruent modulo 2 to the rational coefficient of NðdÞ; and the irrational coefficient of NðqÞ is also congruent modulo 2 to the irrational coefficient of NðdÞ: Computation shows that for every possible value of d such that the rational pffiffiffi coefficient of NðdÞ is odd, and the coefficient of 2 in NðdÞ is even, there exists a pffiffiffi quaternion unit u in (3.1) such that du is an element of Oð 2Þ½1; i; j; k : See Table 2
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Table 2 pffiffiffi Some restricted elements of K with associates in O ð 2Þ ½1; i; j; k Quaternion q
Norm
Unit u
qu
^ w1 þ w4 pffiffiffi w1 þ 2w3 pffiffiffi w1 þ 2w3 þ w4 pffiffiffi w1 þ 2w2 pffiffiffi w1 þ 2w2 þ w4 pffiffiffi pffiffiffi w1 þ 2w2 þ 2w3 pffiffiffi pffiffiffi w þ 2w2 þ 2w3 þ w4 p1ffiffiffi 2w þ w3 pffiffiffi pffiffiffi 1 2w þ w3 þ 2w4 pffiffiffi pffiffiffi 1 2w þ ð 2 þ 1Þw3 pffiffiffi pffiffiffi pffiffiffi 1 2w1 þ ð 2 þ 1Þw3 þ 2w4 ^
^ 3 5 9 5 9 11 17 5 11
^ w4 w1 w4 w1 w4 w1 w4 w3 w3 w3 w3 ^
^ iþjþk 2þj 2i þ 2j þ k 2þi 2i þ j þ 2k 3þiþj 3i þ 2j þ 2k 1 þ 2j 1 þ 3j þ k pffiffiffi 1 þ ð 2 þ 2Þj pffiffiffi 1 þ ð 2 þ 3Þj þ k ^
pffiffiffi 7þ4 2 pffiffiffi 13 þ 6 2 ^
for examples of the corresponding computation. Thus
qu ¼ 2
4 X t¼1
! gt wt
uþd u¼2
4 X
! g# t wt
þd u
ð3:7Þ
t¼1
pffiffiffi with g# t AOð 2Þ for all t as K is closed under quaternion multiplication. Since twice pffiffiffi pffiffiffi any element of K is in Oð 2Þ½1; i; j; k we find qu is also in Oð 2Þ½1; i; j; k : & pffiffiffi Suppose r is a totally positive prime of the ring Oð 2Þ lying over the rational odd pffiffiffi prime p: If r ¼ p does not split in Oð 2Þ then it is the sum of four squares by the classical theorem of Lagrange. However, the case where p does split, i.e. p 71ðmod 8Þ; must be taken care of by other means. We proceed to do this with the next few Lemmas. pffiffiffi pffiffiffi Lemma 8. Suppose r is a prime of the ring Oð 2Þ: Then there exists a unit l of Oð 2Þ and a quaternion q of K such that NðqÞ ¼ lr: Proof. Suppose r lies over the rational prime p: We consider three cases depending pffiffiffi on how p factors in Oð 2Þ: pffiffiffi If p is inert, then r and p differ only by a factor of a unit of Oð 2Þ: By the classical four squares theorem we can find rational integers a; b; c and d such that p ¼ pffiffiffi pffiffiffi a2 þ b2 þ c2 þ d 2 : Note that 1AK; 2w2 w1 ¼ iAK; 2w3 w1 ¼ jAK and
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271
2w4 w1 i j ¼ kAK: Let q ¼ aw1 þ bi þ cj þ dk: Then qAK; NðqÞ ¼ p and we pffiffiffi have already observed that p is a unit of Oð 2Þ times r: pffiffiffi If p ramifies then p ¼ 2: We note that 2 þ 2 ¼ Nðw1 þ w2 Þ: Since r itself must pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi equal 2 times a unit of Oð 2Þ; and 2 þ 2 ¼ 2 ð1 þ 2Þ it is clear that r times a pffiffiffi unit of Oð 2Þ is the norm of w1 þ w2 AK: Suppose p splits. Then p 71ðmod 8Þ and 7p ¼ r r where r is the conjugate pffiffiffi of r with respect to Qð 2Þ: Suppose further that jr r j49: Applying Lemma 2, pffiffiffi there exists a; b; g and d in Oð 2Þ such that a2 þ b2 þ g2 þ d2 ¼ kr with jkk jp9 and pffiffiffi kAOð 2Þ: Let q ¼ aw1 þ bi þ gj þ dk: Then qAK and NðqÞ ¼ a2 þ b2 þ g2 þ d2 ¼ kr: Also NðriÞ ¼ r2 and riAK: Since K is a principal right ideal domain, the right ideal generated by ri and q is also generated by a single element which we denote r: As right K ideals we have ðrÞ ¼ ðri; qÞ: Thus there exists s; tAK such that r ¼ s ri þ t q: Also there exists v1 ; v2 AK such that v1 r ¼ ri and v2 r ¼ q: Thus NðrÞ divides NðriÞ and NðqÞ in the pffiffiffi pffiffiffi ring Oð 2Þ: This implies that NðrÞ divides r2 and kr: Since Oð 2Þ is a unique factorization domain and a principal ideal domain, NðrÞ divides the greatest common divisor of r2 and kr: But jkk jp9ojrr j and r is prime, so k is relatively prime to r: Thus NðrÞ divides r which implies that NðrÞ is a unit or an associate of r: Consider the equation r ¼ s ri þ t q and take quaternion conjugates to find r% ¼ r%i s% þ q% %t: Multiplying together we find NðrÞ ¼ r%r ¼ r#s þ NðqÞNðtÞ
ð3:8Þ
for some s#AK: Note that NðrÞ; NðqÞ; and NðtÞ are real numbers. This implies that s# pffiffiffi is real, and by Lemma 4 must be an element of Oð 2Þ: Hence r divides NðrÞ in pffiffiffi Oð 2Þ: Together with the previous result we conclude that NðrÞ must be an associate of r; i.e. equal to r times a unit. pffiffiffi Suppose p splits and jr r jp9: The only case is p ¼ 7: We note that 3 þ 2 ¼ pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi Nðw2 þ 2w3 Þ and 3 2 ¼ Nðw2 2w3 Þ: Since 7 ¼ ð3 þ 2Þð3 2Þ any pffiffiffi pffiffiffi pffiffiffi Oð 2Þ prime r of norm 7 is only a factor of a unit off from 3 þ 2 or 3 2: This completes the proof. & pffiffiffi Lemma 9. Let r be a totally positive prime of the ring Oð 2Þ lying over the rational prime p. Suppose p is congruent to 1 modulo 8. Then r can be written as the sum of four pffiffiffi squares of algebraic integers from Oð 2Þ: pffiffiffi Proof. Write r ¼ a þ b 2 where a; bAZ: Since r is totally positive rr ¼ 7p ) rr ¼ þp:
ð3:9Þ
So a2 2b2 ¼ p 1ðmod 8Þ: All odd numbers squared are congruent to one modulo 8 while the squares of even numbers are congruent to zero or four. Since a2 2b2 is congruent to one modulo 8 we must have a odd. That implies 2b2 0ðmod 8Þ which
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pffiffiffi implies b2 0ðmod 4Þ so b must be even. Thus we find that r a þ b 2 1ðmod 2Þ pffiffiffi in Oð 2Þ: pffiffiffi By the previous Lemma there exists a quaternion qAK and a unit lAOð 2Þ such that NðqÞ ¼ lr: Considering the basis of K we may write q ¼ a1 þ bi þ gj þ dk pffiffiffi where a; b; g; dA12Oð 2Þ: Thus lr ¼ NðqÞ ¼ a2 þ b2 þ g2 þ d2 40:
ð3:10Þ
pffiffiffi Taking conjugates with respect to Oð 2Þ we find l r is also a sum of four squares and strictly greater than zero. These last two relations imply that lr is totally positive. Since r is totally positive we conclude that the unit l is totally positive. As a pffiffiffi unit, we may write l as 7ð1 þ 2Þn ; nAZ: From total positivity, the sign on l must pffiffiffi be positive, which then forces the exponent to be even, n ¼ 2m: So l ¼ ð3 þ 2 2Þm : pffiffiffi It is now easy to see that NðqÞ simplifies to 1 modulo 2 in Oð 2Þ: By Lemma 7 pffiffiffi above, there exists a quaternion unit uAK of norm 1 such that quAOð 2Þ½1; i; j; k : # þ g# j þ dk: # We find Write qu ¼ a# 1 þ bi ð1 þ
pffiffiffi 2m # 2 þ ð#gÞ2 þ ðdÞ # 2: 2Þ r ¼ NðqÞ ¼ NðquÞ ¼ ð#aÞ2 þ ðbÞ
Dividing both sides by the unit square factor ð1 þ pffiffiffi as the sum of four squares from Oð 2Þ: &
ð3:11Þ
pffiffiffi 2m 2Þ gives a representation of r
pffiffiffi Lemma 10. Let r; n be totally positive primes of the ring Oð 2Þ lying over rational primes p and q respectively. Suppose that each of p and q are congruent to 1 modulo pffiffiffi 8. Then the product rn can be written as the sum of four squares from Oð 2Þ: pffiffiffi Proof. Write r ¼ a þ b 2 where a; bAZ: Since r is totally positive we find that jrr j ¼ p implies rr ¼ þp: Thus a2 2b2 ¼ p 1ðmod 8Þ: For the above to hold, a must be odd. Then a2 1ðmod 8Þ yields 2b2 2ðmod 8Þ: That implies b2 pffiffiffi 1ðmod 4Þ which tells us that b is also odd. We conclude that r 1 þ 2ðmod 2Þ in pffiffiffi pffiffiffi Oð 2Þ: Similarly n 1 þ 2ðmod 2Þ: pffiffiffi By Lemma 8 there exist units l1 ; l2 AOð 2Þ and quaternions q1 ; q2 AK such that l1 r ¼ Nðq1 Þ and l2 n ¼ Nðq2 Þ: Consider l1 l2 rn ¼ Nðq1 q2 Þ: Since l1 l2 rn is the norm of an element of K; it can be written as the sum of four squares of elements from pffiffiffi pffiffiffi Qð 2Þ and is thus greater than zero. Taking conjugates with respect to Qð 2Þ we pffiffiffi find that ðl1 l2 rnÞ is also the sum of four squares of elements from Qð 2Þ and is also greater than zero. Hence l1 l2 rn is totally positive. Since r and n are totally pffiffiffi positive, we find that l1 l2 is a totally positive unit of Oð 2Þ: This implies that pffiffiffi l1 l2 ¼ ð1 þ 2Þ2m for some mAZ: A short computation shows that Nðq1 q2 Þ reduces pffiffiffi to 1 modulo 2 in Oð 2Þ: Hence by Lemma 7 there exists a quaternion unit uAK of pffiffiffi norm 1 such that q1 q2 uAOð 2Þ½1; i; j; k : Write q1 q2 u as a þ bi þ gj þ dk with a; b; g
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pffiffiffi and d in Oð 2Þ: Then l1 l2 rn ¼ Nðq1 q2 Þ ¼ Nðq1 q2 uÞ pffiffiffi ) ð1 þ 2Þ2m rn ¼ a2 þ b2 þ g2 þ d2 : Dividing both sides by the square unit ð1 þ pffiffiffi squares from the ring Oð 2Þ: &
ð3:12Þ
pffiffiffi 2m 2Þ we find that rn is the sum of four
4. Cohn’s theorem on sums of four squares pffiffiffi Theorem 11. Every totally positive algebraic integer in Oð 2Þ with even coefficient on pffiffiffi the radical term is the sum of four squares from Oð 2Þ: pffiffiffi pffiffiffi Proof. Let aAOð 2Þ be as in the statement of the theorem. Then a ¼ a þ 2b 2: pffiffiffi Taking this modulo 2 we find a is congruent to 0 or 1 modulo 2 in Oð 2Þ: By unique factorization we may write a as a unit times a product of primes, and with no loss of generality we may choose the primes to be totally positive. Thus a ¼ m ð2 þ
pffiffiffi h 2Þ p1 p2 ?pk g1 g2 ?gm b1 b2 ?bn ;
ð4:1Þ
pffiffiffi pffiffiffi pffiffiffi pffiffiffi where m is a unit, 2 þ 2 ¼ 2ð1 þ 2Þ is a totally positive associate of 2; the p’s are nonsplitting primes of Z; the g’s are primes lying over rational primes congruent to 1 modulo 8, and the b’s are primes lying over rational primes congruent to 1 modulo 8. We already know that the p’s and the g’s can each be written as the sum of pffiffiffi four squares from Oð 2Þ: Since a and the primes in (4.1) are totally positive, m must be totally positive also. pffiffiffi Hence m ¼ ð1 þ 2Þ2t for some tAZ: Taking (4.1) modulo 2 pffiffiffi pffiffiffi a ð3 þ 2 2Þt ð2 þ 2Þh p1 ?pk g1 ?gm b1 ?bn ðmod 2Þ; pffiffiffi pffiffiffih 2 ð1 þ 2Þn ; f0 or 1g
ðmod 2Þ:
ð4:2Þ
The congruences for the g’s and the b’s were established in the proofs of the lemmas on representation by four squares in the previous section. There are three cases to consider, depending on the value of h: If h is zero, then n must be even. Thus the b’s can be paired up. The product of any pair of b’s is the sum of four squares by Lemma 10. Since m is a square, by the identity expressing the product of two sums of four squares as again a sum of four pffiffiffi squares, we find that a can be written as a sum of four squares from Oð 2Þ:
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If h is one and n is even then (4.2) reduces to the claim that 0 or 1 must be pffiffiffi pffiffiffi pffiffiffi pffiffiffi congruent to 2 1 modulo 2. If n is odd, (4.2) reduces to 2ð1 þ 2Þ 2ðmod2Þ being congruent to 0 or 1. In either case this is impossible. If hX2 then pffiffiffi pffiffiffi a ¼ m 2ð3 þ 2 2Þ ð2 þ 2Þh 2 p1 ?pk g1 ?gm b1 ?bn :
ð4:3Þ
We previously showed that for each totally positive prime b lying over pffiffiffi a rational prime congruent to 1 modulo 8 that there exists a unit lAOð 2Þ and a quaternion qAK such that lb ¼ NðqÞ: By the usual reasoning, l is pffiffiffi totally positive and thus must equal ð1 þ 2Þ2d for some dAZ: This gives pffiffiffi d pffiffiffi pffiffiffi b ¼ Nðð1 þ 2Þ qÞ: Since 1 þ 2 is a unit in Oð 2Þ we find that b is the norm of some element of K: By the norm multiplication property there exists rAK for which b1 b2 ?bn ¼ NðrÞ: pffiffiffi pffiffiffi pffiffiffi Also ð2 þ 2Þh 2 ¼ Nððw1 þ w2 Þh 2 Þ; and mð3 þ 2 2Þ ¼ ð1 þ 2Þ2tþ2 : Thus pffiffiffi pffiffiffi pffiffiffi m ð3 þ 2 2Þ ð2 þ 2Þh 2 b1 ?bn ¼ Nðð1 þ 2Þtþ1 ðw1 þ w2 Þh 2 rÞ:
ð4:4Þ
Let s be the expression inside the norm symbol of (4.4). It is clear that sAK: By pffiffiffi Lemma 6 there exists a quaternion unit uAK of norm 1 such that 2su is an element pffiffiffi pffiffiffi of Oð 2Þ½1; i; j; k : Write 2su as y1 1 þ y2 i þ y3 j þ y4 k where y1 ; y; y4 are elements pffiffiffi of Oð 2Þ: Then pffiffiffi pffiffiffi m 2ð3 þ 2 2Þ ð2 þ 2Þh 2 b1 ?bn 1 pffiffiffi ¼ Nð 2 s uÞ ¼ y21 þ y22 þ y23 þ y24
ð4:5Þ
pffiffiffi is the sum of four squares of Oð 2Þ: Since p1 ?pk g1 ?gm is the sum of four squares pffiffiffi of Oð 2Þ; the product, a; is also the sum of four such squares. This proves the theorem of Cohn. & 5. The Icosians A similar approach can be used to obtain an alternate proof of Go¨tzky’s four pffiffiffi squares theorem for Qð 5Þ: Only an overview will be given as the demonstration is close in style to that of Cohn’s four squares theorem. The corresponding ring is pffiffiffi pffiffiffi called the ring of icosians, denoted I: We let Oð 5Þ denote the integers in Qð 5Þ and pffiffiffi pffiffiffi t ¼ ð1 þ 5Þ=2 the fundamental unit. Then I can be thought of as the Oð 5Þ module
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with generators e1 ¼ 12ð1 þ t 1 i þ tjÞ;
e2 ¼ 12ðt 1 i þ j þ tkÞ;
e3 ¼ 12ðti þ t 1 j þ kÞ;
e4 ¼ 12ði þ tj þ t 1 kÞ
ð5:1Þ
as in Vigne´ras [11]. While there are 120 units of norm one in I we need only a small portion of these for our results. Lemma 12. The following are units in I of norm 1: 1; e1 ; e2 ; e3 ; e4 ; te3 e2 ; te4 e3 ; te2 e4 ;
e1 þ t e2 þ t e3 þ ð1 þ tÞe4 ; e1 þ t e2 þ e3 þ e4 ;
e1 te2 þ e3 þ e4 ; e1 te2 þ e3 þ te4 ; te1 e2 þ ð1 þ tÞe4 ;
e1 te2 þ te4 ; te1 e2 þ te4 :
ð5:2Þ
pffiffiffi Lemma 13. I-R ¼ Oð 5Þ: pffiffiffi Lemma 14. For all aAI the quaternionic norm NðaÞAOð 5Þ: Lemma 15. For all qAI there exist quaternion units u1 ; u2 AI of norm 1 such that u1 qu2 pffiffiffi pffiffiffi has Oð 5Þ–integer coefficients, i.e. u1 qu2 AOð 5Þ½1; i; j; k : pffiffiffi Proof. Let q ¼ a1 e1 þ a2 e2 þ a3 e3 þ a4 e4 ; where a1 ; y; a4 AOð 5Þ: Write q ¼ 2ðg1 e1 þ g2 e2 þ g3 e3 þ g4 e4 Þ þ ðd1 e1 þ d2 e2 þ d3 e3 þ d4 e4 Þ;
ð5:3Þ
pffiffiffi pffiffiffi where g1 ; y; g4 AOð 5Þ and d1 ; y; d4 AOð 5Þ=ð2Þ: Note that a full set of residues of pffiffiffi Oð 5Þ modulo 2 can be chosen as f0; 1; t; t g: Computation shows that for all possible values of d1 ; y; d4 there exists quaternion units u1 ; u2 in (5.2) such that pffiffiffi u1 ðd1 e1 þ ? þ d4 e4 Þu2 is an element of Oð 5Þ½1; i; j; k : The proof continues in an analogous fashion to Lemma 6. & pffiffiffi Lemma 16. Suppose r is a prime of the ring Oð 5Þ: Then there exists a unit l of pffiffiffi Oð 5Þ and a quaternion q of I such that NðqÞ ¼ lr: Proof. The proof is completely analogous to that of Lemma 8. One small difference is in the case where r lies over the rational prime p which splits. In this case, by pffiffiffi geometry of numbers there exists a; b; g and d in Oð 5Þ such that a2 þ b2 þ g2 þ
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pffiffiffi d2 ¼ kr with jkk jp8 and kAOð 5Þ: The proof then continues in the same fashion as in Lemma 8. & pffiffiffi Lemma 17. Let r be a totally positive prime of the ring Oð 5Þ lying over the rational prime p: Then r can be written as the sum of four squares of algebraic integers from pffiffiffi Oð 5Þ: pffiffiffi Proof. By Lemma 16 there exists a quaternion qAI and a unit lAOð 5Þ such that pffiffiffi NðqÞ ¼ lr: We may write q as a1 þ bi þ gj þ dk; where a; b; g and d are in Qð 5Þ: Thus lr ¼ NðqÞ ¼ a2 þ b2 þ g2 þ d2 40:
ð5:4Þ
pffiffiffi Taking conjugates with respect to Qð 5Þ we find l r is also a sum of squares and greater than zero. Thus lr is totally positive from which it follows that l itself is totally positive. Note that l ¼ 7tn for some rational integer n: Together this implies that l ¼ tn and n is even. We can therefore write NðqÞ ¼ t2m r for some mAZ: By Lemma 15 there exist quaternion units of norm one, u1 ; u2 ; such that pffiffiffi u1 qu2 AOð 5Þ½1; i; j; k : Write pffiffiffi # g# ; d; # AOð 5Þ: u1 qu2 ¼ a# 1 þ b# i þ g# j þ d# k; a# ; b; ð5:5Þ Then # 2 þ ð#gÞ2 þ ðdÞ # 2: t2m r ¼ NðqÞ ¼ Nðu1 qu2 Þ ¼ ð#aÞ2 þ ðbÞ
ð5:6Þ
Dividing by t2m gives the desired representation of r: & pffiffiffi Theorem 18. Every totally positive algebraic integer in Oð 5Þ is the sum of four pffiffiffi squares from Oð 5Þ: Proof. This follows as in Deutsch [5]. &
6. Further directions It appears plausible that the technique of combining geometry of numbers bounds with special rings of quaternions can lead to more results of similar type. Due to a result of Siegel [10] sums of squares are not universal for any totally real number pffiffiffi fields besides the rational integers and the integers in Qð 5Þ: Restricting to cases analogous to Cohn’s four squares theorem does not improve matters. Assuming the discriminant of the quadratic field is divisible by four and totally positive integers under consideration have even irrational coefficients, numerical computations imply
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pffiffiffi pffiffiffi that only in Qð 2Þ and Qð 3Þ are the sum of four squares universal for this limited class of integers. It would be interesting to formulate a condition in general number fields which insures that a totally positive integer satisfying such condition is the sum of four integer squares. Such general results would most likely require much deeper techniques than those used in this paper.
7. The computation The computations utilized the PUNIMAX variant of MAXIMA on a Pentium 133 chip personal computer with 32 megabytes of RAM. The operating system was LINUX 2.0.35. The running time to perform the computations relevant to Lemmas 6 and 7 totaled 8 min and 24 s: Two SNOBOL4 programs were written to transform the MAXIMA output into TEX readable form for Tables 1 and 2. The computation for Lemma 15 required 85:58 min: A table for the products of pairs of basis elements w1 ; y; w4 was created. Using this, the entries appearing in Tables 1 and 2 were checked by manual calculation.
Acknowledgments The author express his appreciation to the Mathematics Department of the University of Botswana for the use of their facilities. The author thanks B. Haible, the maintainer of PUNIMAX, for permitting its free use for academic purposes [7], Prof. Vigne´ras for clarifying an essential point about the cubians, and Michael Rosen for raising the issue of representations in general number fields. He also thanks Don Hadwin and Jiankui Li for helping him find the paper of Siegel [10], and Harvey Cohn for many encouraging E-mail communications.
References [1] M. Baake, R. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999) 1258–1276. [2] H. Cohn, Decomposition into four integral squares in the fields 21=2 and 31=2 ; Amer. J. Math. 82 (1960) 301–322. [3] H. Cohn, Calculation of class numbers by decomposition into three integral squares in the fields of 21=2 and 31=2 ; Amer. J. Math. 83 (1961) 33–56. [4] H. Cohn, Cusp forms arising from hilbert’s modular functions for the field of 31=2 ; Amer. J. Math. 84 (1962) 283–305. [5] J.I. Deutsch, Geometry of numbers proof of Go¨tzky’s four squares theorem, J. Number Theory 96 (2) (2002) 417–431. [6] F. Go¨tzky, U¨ber eine zahlentheoretische Anwendung von Modulfunktionen zweier Vera¨nderlicher, Math. Ann. 100 (1928) 411–437. [7] B. Haible, Private communication, 1997.
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[8] G. Hardy, E. Wright, An Introduction to the Theory of Numbers, 4th Edition, Oxford University Press, London, 1971. [9] Herstein, Topics in Algebra, J Wiley, New York, 1975. [10] C.L. Siegel, Sums of mth Powers of Algebraic Integers, Ann. Math. 46 (2) (1945) 313–339. [11] M.-F. Vigne´ras, Arithme´tique des Alge`bres de Quaternions, Lecture Notes in Mathematics, Vol. 800, Springer, New York, 1980.
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Rank determines semi-stable conductor H. Kisilevsky1 Department of Mathematics and Statistics and CICMA, Concordia University, 1455 de Maisonneuve Blvd. West, Montre´al, Quebec, Canada H3G 1M8 Received 5 February 2003; revised 6 May 2003; accepted 1 July 2003 Communicated by D. Goss
Abstract Suppose that E1 and E2 are elliptic curves over the rational field, Q; such that ords¼1 LðE1 =K; sÞ ords¼1 LðE2 =K; sÞ ðmod 2Þ for all quadratic fields K=Q: We prove that their conductors NðE1 Þ; and NðE2 Þ are equal up to squares. If rankZ ðE1 ðKÞÞ rankZ ðE2 ðKÞÞ ðmod 2Þ for all quadratic fields K=Q; then the same conclusion holds, provided the 2-parts of their Tate–Shafarevich groups are finite. r 2003 Elsevier Inc. All rights reserved.
Y. Zarhin posed the following question: Suppose that X1 and X2 are abelian varieties defined and isogenous over a number field k: Then for any finite extension field K=k; rankZ ðX1 ðKÞÞ ¼ rankZ ðX2 ðKÞÞ: (Here rankZ ðX ðKÞÞ is the number of free generators of the Mordell–Weil group X ðKÞ:) Zarhin’s question is whether the converse holds: Let X1 and X2 be abelian varieties defined over k: If rankZ ðX1 ðKÞÞ ¼ rankZ ðX1 ðKÞÞ for every finite extension K=k; then are X1 and X2 isogenous over k? In this article we consider this question for elliptic curves E1 and E2 defined over the rational field Q: For an elliptic curve E; and for finite extensions K=Q; LðE=K; sÞ will denote the L-function of E viewed as a curve over K: By the recent proof [B-C-D-T,T-W,W] of the modularity of elliptic curves over Q; we know that LðE=Q; sÞ has an analytic continuation for all sAC; and satisfies the
1
E-mail address: [email protected]. This work was supported in part by grants from NSERC and FCAR.
0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00157-4
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functional equation LðE; sÞ ¼
pffiffiffiffiffiffiffi s NE =2p GðsÞLðE; sÞ ¼ wE LðE; 2 sÞ;
where NE is the conductor of E=Q; and wE ¼ 71: The Birch and Swinnerton-Dyer conjecture asserts that ords¼1 LðE=K; sÞ ¼ rankZ ðEðKÞÞ; and that the Tate–Shafarevich group V ðKÞ is finite. The ‘‘parity conjecture’’ for E asserts that ords¼1 LðE=K; sÞ rankZ ðEðKÞÞ ðmod 2Þ: We note that ords¼1 LðE=K; sÞ is sometimes referred to as the ‘‘analytic rank’’ of E over K: In view of Faltings’ Theorem, we can rephrase Zarhin’s question in terms of the analytic rank of E over K viz., Given elliptic curves E1 and E2 defined over Q such that ords¼1 LðE1 =K; sÞ ¼ ords¼1 LðE2 =K; sÞ for all finite extensions K=Q; is it true that LðE1 =Q; sÞ ¼ LðE2 =Q; sÞ? It is a pleasure to thank Ken Kramer for a number of valuable discussions regarding this paper and Karl Rubin for several useful suggestions. Theorem 1. Let E1 and E2 be elliptic curves defined over Q; with conductors NðE1 Þ and NðE2 Þ: Suppose that for every number field K with ½K : Qp2 ords¼1 LðE1 =K; sÞ ords¼1 LðE2 =K; sÞ ðmod 2Þ: Then NðE1 Þ and NðE2 Þ are equal up to square factors. Proof. Since the curves Ei are modular, their L-functions, LðEi =Q; sÞ; have analytic continuations to the entire plane and satisfy functional equations. Consider the twists of the curves Ei by fundamental discriminants D of quadratic fields which are pffiffiffiffi ðDÞ coprime to NðE1 ÞNðE2 Þ: For such a discriminant D; let K ¼ Q D ; let Ei denote ðDÞ
the twisted curves, and let LðEi =Q; sÞ be the corresponding L-functions. Then ðDÞ
LðEi =Q; sÞ is the twist of LðEi =Q; sÞ by the quadratic Dirichlet character wD corresponding to K; and is modular by Hecke’s theorem. Since ords¼1 LðE1 =Q; sÞ ords¼1 LðE2 =Q; sÞ ðmod 2Þ
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it follows that LðE1 =Q; sÞ and LðE2 =Q; sÞ have the same signs in their functional equations. Also, by assumption we have ords¼1 LðE1 =K; sÞ ords¼1 LðE2 =K; sÞ ðmod 2Þ: Since ðDÞ
LðEi =K; sÞ ¼ LðEi =Q; sÞLðEi =Q; sÞ; it follows that ðDÞ
ðDÞ
ords¼1 LðE1 =Q; sÞ ords¼1 LðE2 =Q; sÞ ðmod 2Þ for all such fundamental discriminants D: Applying the functional equations [Sh, ðDÞ Theorem 3.66] to the L-functions LðEi =Q; sÞ we see that the sign of the functional ðDÞ
equation for LðEi =Q; sÞ changes from that of LðEi =Q; sÞ by the factor wD ðNðEi ÞÞ and therefore wD ðNðE1 ÞÞ ¼ wD ðNðE2 ÞÞ: Hence wD ðNðE1 ÞNðE2 ÞÞ ¼ 1 for all such D; and this implies that NðE1 Þ and NðE2 Þ are equal up to square factors. & Theorem 2. Let E1 and E2 be elliptic curves defined over Q; with conductors NðE1 Þ and NðE2 Þ: Suppose that for every number field K with ½K : Qp2 rankZ ðE1 ðKÞÞ rankZ ðE2 ðKÞÞ ðmod 2Þ: If the 2-primary parts of Tate–Shafarevich groups, V 2 ðEi ðKÞÞ; are finite for all such K; then NðE1 Þ and NðE2 Þ are equal up to square factors. Proof. Let E be an elliptic curve defined over Q: The main result of Monsky [Mo, Theorem 1.5] is that the ‘‘2-Selmer rank’’, s2 ðEðKÞÞ ¼ rankZ EðKÞþ dimF2 V 2 ðEðKÞÞ has the same parity as ords¼1 LðE=K; sÞ: But then if V 2 ðEðKÞÞ is finite it follows that dimF2 V 2 ðEðKÞÞ is even and hence rankZ EðKÞ ords¼1 LðE=K; sÞ ðmod 2Þ: Therefore the finiteness of V 2 ðEi ðKÞÞ; together with the assumption rankZ ðE1 ðKÞÞ rankZ ðE2 ðKÞÞ ðmod 2Þ implies that ðDÞ
ðDÞ
ords¼1 LðE1 =Q; sÞ ords¼1 LðE2 =Q; sÞ ðmod 2Þ;
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and so Theorem 1 gives the conclusion that NðE1 Þ and NðE2 Þ are equal up to square factors. & Corollary 1. Suppose that E1 and E2 are semi-stable elliptic curves defined over Q such that ords¼1 LðE1 =K; sÞ ords¼1 LðE2 =K; sÞ ðmod 2Þ for all extensions K=Q; with ½K : Qp2; then NðE1 Þ ¼ NðE2 Þ: In particular, there are only a finite number of isogeny classes of such elliptic curves. Proof. For semi-stable curves over Q; the results of Wiles [W] and Taylor–Wiles [TW] imply the modularity of the Ei : Then Theorem 1 shows that NðE1 Þ ¼ NðE2 Þ since they are both square-free. & Corollary 2. Suppose that E1 and E2 are semi-stable elliptic curves defined over Q such that rankZ ðE1 ðKÞÞ rankZ ðE2 ðKÞÞ ðmod 2Þ for all extensions K=Q; with ½K : Qp2: If in addition, we assume that V 2 ðEi ðKÞÞ is finite for all such K; then NðE1 Þ ¼ NðE2 Þ: In particular, there are only a finite number of isogeny classes of such elliptic curves. Proof. Corollary 2 follows from Theorem 2 as above.
&
We next construct some examples of nonisogenous elliptic curves to indicate that we cannot expect this approach to yield much more information on Zarhin’s conjecture. Specifically, we will give examples of elliptic curves E1 and E2 ; defined over Q; with the same conductor N ¼ NðEi Þ; and which have the following property: rankZ ðE1 ðKÞÞ rankZ ðE2 ðKÞÞ ðmod 2Þ for all Galois extensions K=Q in which 2 is unramified, provided that the 2-primary parts of their Tate–Shafarevich groups, V 2 ðEi ðKÞÞ; are finite. (Conjecturally they will have ranks of the same parity for all finite extensions K=Q:) Furthermore, the elliptic curves E1 ; and E2 also have the property that: ords¼1 LðE1 =K; sÞ ords¼1 LðE2 =K; sÞ ðmod 2Þ for all solvable extensions K=Q: (Conjecturally they have analytic ranks of the same parity for all finite extensions K=Q:) The change of parity of both the algebraic rank and the analytic rank in quadratic extensions was studied by Kramer [K] and Kramer–Tunnell [K-Tu]. Theorem 3. Let E1 and E2 be semi-stable elliptic curves defined over a number field k such that E1 and E2 have the same reduction type (good reduction, split multiplicative reduction, or nonsplit multiplicative reduction) at all primes p of k: Suppose that E1
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and E2 have models over k with equal discriminants. Let K=k be a quadratic extension. (1) (Kramer) Suppose that the prime 2 is not ramified in k and that the 2-primary parts of the Tate–Shafarevich groups, V 2 ðEi ðKÞÞ; are finite, then rankZ ðE1 ðKÞÞ rankZ ðE2 ðKÞÞ ðmod 2Þ: (2) (Kramer–Tunnell) The elliptic curves E1 =K and E2 =K have the same local efactors. Proof. The statement ð1Þ is proved by Kramer [K, Section 4, Theorem 1 and Corollary 2]. Assuming that V 2 ðEi ðKÞÞ is finite, he computes the parity of rankZ ðEi ðKÞÞ only in terms of primes of K at which Ei has split multiplicative reduction and the quadratic character of K=k: But since these are the same for E1 and E2 under our hypotheses, the first conclusion follows. Statement (2) is a consequence of Theorem 4.4, Section 4 of Kramer–Tunnell [K-Tu]. Here they show that for a semi-stable elliptic curve E; the local e-factors of E=K at a prime P; can be calculated in terms of the quadratic character of K=k at P evaluated on the discriminant of E; and a local norm index which is computed in Kramer [K, pp. 127–128], also in terms of the discriminant of E and the reduction type of E at P: Since the discriminants of E1 and E2 were assumed to be equal, and they had the same reduction type at all primes P of K; their local e-factors over K agree. & We first consider the algebraic rank, rankZ ðEi ðKÞÞ for Galois extensions K=Q unramified at 2. We need the following: Proposition 1. Suppose that V is a finite dimensional Q-vector space which admits an action of a finite group G of odd order. Then dim V dim V G ðmod 2Þ: Proof. This is true for cyclic groups of odd prime order as the only irreducible representations are the trivial representation and the augmentation representation. The result follows by solvability of groups of odd order. Alternatively, it is an exercise in Serre [S, pp. 109–110] that every nontrivial irreducible character of a group of odd order is complex, and hence can be paired with its conjugate. It then is clear that the parity of the degree of the representation r; determined by the action of G on V is equal to the parity of the multiplicity of the trivial representation w0 in r: & Corollary 3. If E is an elliptic curve defined over a field k and K=k is a Galois extension of odd degree then rankZ ðEðKÞÞ rankZ ðEðkÞÞ ðmod 2Þ as long as both are finite.
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Proof. Setting V ¼ EðKÞ#Q and G ¼ GalðK=kÞ the result follows from Proposition 1. & Therefore we have: Theorem 4. Suppose that E1 and E2 are semi-stable elliptic curves defined over Q with the same conductor N: Assume that E1 and E2 both have the same multiplicative reduction type (split or nonsplit) at all primes dividing N: If rankZ ðE1 ðQÞÞ rankZ ðE2 ðQÞÞ ðmod 2Þ then rankZ ðE1 ðKÞÞ rankZ ðE2 ðKÞÞ ðmod 2Þ for all Galois extensions K=Q unramified at 2 provided that the groups V 2 ðEi ðKÞÞ are finite. Proof. Since nonsplit reduction becomes split over a field k if and only if the completion of k contains the unramified quadratic extension of Qp ; the reduction type depends only on k; not on the curve. Thus E1 and E2 have the same reduction type over every k: Let K=Q be a Galois extension unramified at 2; and let G ¼ GalðK=QÞ: If K=Q has odd degree, then Corollary 3 implies that the ranks of E1 ðKÞ and E2 ðKÞ have the same parity provided that the ranks of E1 ðQÞ and E2 ðQÞ have the same parity. If K=Q has even degree, then G contains an element of order 2; and let k0 denote its fixed field. Then k0 =Q is an extension unramified at 2, and it follows from Theorem 3 (1) that the ranks of E1 ðKÞ and E2 ðKÞ have the same parity provided that the groups V 2 ðEi ðKÞÞ are finite. & Now consider the analytic rank for solvable extensions. Proposition 2. Let k be a number field, and suppose that K=k is a Galois extension of odd degree, with GalðK=kÞ ¼ G: Let LðE=k; sÞ be the L-function of E=k; and LðE=K; sÞ be that for K=k: If LðE=k; sÞ is automorphic, then so is LðE=K; sÞ; and furthermore ords¼1 LðE=K; sÞ ords¼1 LðE=k; sÞ ðmod 2Þ: Proof. Since G has odd order it is a solvable group. Hence, by induction, we may reduce the result to the case that G is a cyclic group of prime order. Then the Base Change Theorem of Arthur–Clozel [A-C] asserts that the automorphy of LðE=k; sÞ implies that of LðE=K; sÞ: Furthermore, in this case Y LðE=K; sÞ ¼ LðE=k; w; sÞ: wAGˆ
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Since LðE=k; sÞ is real for real s; it follows that the signs of the functional equations for LðE=k; w; sÞ and LðE=k; w% ; sÞ are complex conjugates, and so have product equal to 1: Since jGj is odd, we see that the sign of the functional equation for LðE=K; sÞ is the same as that for LðE=k; sÞ and hence ords¼1 LðE=KsÞ ords¼1 LðE=k; sÞ ðmod 2Þ:
&
Theorem 5. Suppose that E1 and E2 are semi-stable elliptic curves defined over Q with the same minimal discriminant D and the same conductor N: Assume that E1 and E2 both have split multiplicative reduction at all primes dividing N: If ords¼1 LðE1 =Q; sÞ ords¼1 LðE2 =Q; sÞ ðmod 2Þ then ords¼1 LðE1 =K; sÞ ords¼1 LðE2 =K; sÞ ðmod 2Þ for all solvable extensions K=Q: Proof. Since the curves Ei are semi-stable over Q; Wiles [W] and Taylor–Wiles [T-W] imply that they are modular, and hence their L-functions, LðEi =Q; sÞ; are automorphic. Therefore, by Proposition 3, the L-functions, LðEi =K; sÞ; are automorphic for any solvable extension K=Q: Also, since E1 and E2 have the same discriminant, and since both have split multiplicative reduction at all primes dividing their common conductor, N; the same is true for any finite extension k=Q: Therefore by Theorem 3(2) [K-Tu, Theorem 4.4, Section 4], we see that E1 ; and E2 have the same e-factors over K; for every quadratic extension K=k: If in addition, K=Q is solvable, it follows from their automorphy that LðEi =K; sÞ have the same signs in their functional equations (see [J]). Therefore applying either Proposition 3 or Theorem 3(2) successively, it follows that if ords¼1 LðE1 =Q; sÞ ords¼1 LðE2 =Q; sÞ ðmod 2Þ then ords¼1 LðE1 =K; sÞ ords¼1 LðE2 =K; sÞ ðmod 2Þ for all solvable extensions K=Q: Example (Computed by J. Fearnley). N ¼ 307: There are 4 nonisogenous curves (307A; 307B; 307C; and 307D in Cremona’s tables [Cr]) of conductor 307 and discriminant 307: All of them have rank 0 and split multiplicative reduction at p ¼ 307: By Theorems 4 and 5, it follows that the above curves have analytic ranks of the same parity for all solvable extensions K of Q; and that their algebraic ranks have the same parity for all Galois extensions K=Q in which 2 is unramified provided that the 2-primary parts of their Tate–Shafarevich groups, V 2 ðEi ðKÞÞ; are finite. Note that these curves do not provide counter-examples to Zarhin’s question. Twisting the first by 7 gives a curve of rank 2 (by Cremona’s Mrank program) and
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Lð1Þ ¼ 0; by Kolyvagin [Ko], while the rest have rank 0 and nonvanishing Lfunction at s ¼ 1: N ¼ 1187: There are 3 nonisogenous curves of conductor 1187 and discriminant 1187 of rank 0 and split multiplicative reduction at p ¼ 1187: Twisting the first by 43 gives a curve of rank 2 and Lð1Þ ¼ 0; while the rest have rank 0 and nonvanishing L-function at s ¼ 1:
References [A-C]
[B-C-D-T] [Cr] [J] [K] [Ko] [K-Tu] [Mo] [S] [Sh] [T-W] [W]
J. Arthur, L. Clozel, Simple Algebras, Base Change and the Advanced Theory of the Trace Formula, Vol. 120, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1990. C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001) 843–939. J. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, UK, 1992. H. Jacquet, Principal L-functions, Proceedings of Symposia in Pure Mathematics, Vol. 33, Part 2, American Mathematical Society, Providence, RI, 1979, pp. 63–86. K. Kramer, Arithmetic of elliptic curves upon quadratic extension, Trans. Amer. Math. Soc. 264 (1981) 121–135. V. Kolyavagin, Finiteness of EðQÞ and V E=Q for a subclass of Weil curves, Math. USSRIzv. 1989, 523–542. K. Kramer, J. Tunnell, Elliptic curves and local e-factors, Comp. Math. 46 (3) (1982) 307–352. P. Monsky, Generalizing the Birch–Stephens theorem, Math. Z. 221 (1996) 415–420. J.-P. Serre, Linear Representations of Finite Groups, Springer, New York, 1986. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, Iwanami Shoten and Princeton University Press, Princeton, NJ, 1971. R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. (2) 141 (3) (1995) 553–572. A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. (2) 141 (3) (1995) 443–551.
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Determining L-functions by twists H. Kisilevsky1 Department of Mathematics and Statistics and CICMA Concordia University, 1455 de Maisonneuve Blvd. West Montre´al, Quebec, Canada H3G 1M8 Received 5 February 2003; revised 6 May 2003 Communicated by D. Goss
Abstract Suppose that L1 ðsÞ and L2 ðsÞ are two L-functions whose twists by a set of Dirichlet characters simultaneously vanish (vanish mod p) at a critical point. We examine the extent to which this property determines the L-functions in the cases of L-functions of elliptic curves, of number fields, and of curves over finite fields. r 2003 Elsevier Inc. All rights reserved.
1. Introduction There has been considerable interest in the study of special values of twists of Lfunctions (and their derivatives) by Hecke characters. The vanishing or nonvanishing of such values have profound arithmetic significance, especially when twisting by Dirichlet characters. In this article, we investigate the following question: Is an L-function, LðsÞ; characterized by the set of Dirichlet characters for which the corresponding twisted L-function vanishes at a critical point (or vanishes modulo a set of primes in a suitable sense)? We consider three classes of L-functions: Dirichlet L-functions, Lfunctions of abelian varieties over finite fields, and L-functions associated to modular forms. In Section 2, we consider the vanishing of the central values of twists of modular L-functions by quadratic Dirichlet characters. In Section 3, we study Dirichlet Lfunctions and in Section 4 the L-functions of curves over finite fields. In the latter
1
E-mail address: [email protected]. This work was supported in part by grants from NSERC and FCAR.
0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-314X(03)00158-6
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two cases, the L-functions do not vanish, and we consider instead the vanishing modulo primes of an ‘‘algebraic part’’ defined below. This question is related to a conjecture of Y. Zarhin. Zarhin’s conjecture asserts that: If X1 and X2 are two abelian varieties defined over a number field k; with the property that rankZ ðX1 ðKÞÞ ¼ rankZ ðX2 ðKÞÞ for all finite extensions K=k; then X1 and X2 are isogenous over k: (Here, by rankZ ðX ðKÞÞ we mean the number of free generators of the Mordell–Weil group X ðKÞ:) In view of Faltings’ theorem the analytic form of this conjecture is the statement: If ords¼1 LðX1 =K; sÞ ¼ ords¼1 LðX2 =K; sÞ for all finite extensions K=k; then LðX1 =k; sÞ ¼ LðX2 =k; sÞ; where LðXi =K; sÞ is the L-function of Xi viewed as a variety over K: Restricting to abelian extensions one may modify the conjecture as follows: If ords¼1 LðX1 =k; s; wÞ ¼ ords¼1 LðX2 =k; s; wÞ for all (Hecke) characters w of finite order, then LðX1 =k; sÞ ¼ LðX2 =k; sÞ; where LðXi =k; s; wÞ is the L-function of Xi =k twisted by the character w: The work of Rohrlich [Ro] provides a corresponding (conjectural) algebraic interpretation in terms of the ranks of the ‘‘w-components’’ of the Mordell–Weil groups Xi ðkab Þ: Specifically, if X is an abelian variety defined over k; then Galðkab =kÞ acts on X ðkab Þ: If we set V ¼ C#Z X ðkab Þ; then V is a representation space for Galðkab =kÞ and decomposes V ¼ "w V ðwÞ into finite dimensional eigenspaces V ðwÞ on which Galðkab =kÞ acts via the character w: On the other hand, to each such character of Galðkab =kÞ; one associates the L-function, LðX =k; s; wÞ; afforded by the tensor product of w with the c-adic representation of X : Rohrlich then shows, that the Birch and Swinnerton-Dyer conjecture together with the Deligne–Gross conjecture imply that dimC V ðwÞ ¼ ords¼1 LðX =k; s; wÞ: The analytic conjectures have the advantage that they can be asked for a wider class of L-functions and that they have analogues even in the case of non-vanishing. In [Ki] we considered Zarhin’s conjecture in the case of elliptic curves defined over Q; and proved the following two statements:
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Suppose that E1 and E2 are elliptic curves defined over Q such that ords¼1 LðE1 =K; sÞ ords¼1 LðE2 =K; sÞ ðmod 2Þ for all extensions K=Q with ½K : Qp2; then NðE1 Þ and NðE2 Þ are equal up to square factors (where NðEÞ is the conductor of E). In particular, if E1 and E2 are semi-stable, then there are only finitely many isogeny classes of such curves. Suppose that E1 and E2 are elliptic curves defined over Q such that rankZ ðE1 ðKÞÞ rankZ ðE2 ðKÞÞ ðmod 2Þ for all extensions K=Q with ½K : Qp2 and suppose that the 2-primary part of their Tate–Shafarevich groups, IIIðEi ðKÞÞ2 ; are finite for all such K; then NðE1 Þ and NðE2 Þ are equal up to square factors.
2. Elliptic curves and modular L-functions Let E be an elliptic curve defined over Q: We denote the L-function of E=Q by LðE=Q; sÞ ¼ LðE; sÞ; and denote its twist by the Dirichlet character w by Lw ðE; sÞ ¼ LðE; s; wÞ: Theorem 2.1. Suppose that E1 and E2 are elliptic curves defined over Q such that LðE1 ; 1; wÞ ¼ 03LðE2 ; 1; wÞ ¼ 0 for every quadratic Dirichlet character w: Then their conductors, NðE1 Þ and NðE2 Þ; are equal up to square factors. Proof. The elliptic curves E1 and E2 are modular by the recent work of Wiles and Taylor–Wiles, and Breuil–Conrad–Diamond–Taylor [Wi,Ta–Wi,B–C–D–Ta]. Their L-functions have an analytic continuation to the entire plane, and that they satisfy a functional equation. Applying Waldspurger’s result [Wa], we see that there exists a quadratic character w1 of conductor D1 prime to NðE1 Þ and NðE2 Þ; such that the twisted L-function LðE1 ; 1; w1 Þa0: The hypothesis then implies that LðE2 ; 1; w1 Þa0 and so the L-functions LðEi ; s; w1 Þ both have þ1 as signs in their functional equations and conductors NðE1 ÞD21 and NðE2 ÞD21 respectively. If NðE1 Þ and NðE2 Þ; were not equal up to square factors, there would exist a prime p which appears in the factorization of NðE1 Þ (say) to an odd power and to an even power in NðE2 Þ: Using the extension of Waldspurger’s theorem of Friedberg–Hoffstein [Fr–Ho], we can choose a character w2 of conductor D2 prime to NðE1 Þ; NðE2 Þ and D1 which satisfies: w2 ðlÞ ¼ 1 for all primes lap dividing NðE1 ÞNðE2 ÞD1 ; and such that w2 ð1Þ ¼ þ1; w2 ðpÞ ¼ 1 and for which LðE2 ; 1; w1 w2 Þa0: This exists as the sign of the functional equation for LðE2 ; s; w1 w2 Þ is w2 ðNðE2 ÞD21 Þ ¼ þ1 times that of LðE2 ; s; w1 Þ and hence is þ1: Then the sign of the functional equation of LðE1 ; s; w1 w2 Þ is
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w2 ðNðE1 ÞD21 Þ times that of LðE1 ; s; w1 Þ and hence is 1: But this implies that LðE1 ; 1; w1 w2 Þ ¼ 0 contradicting the hypothesis of Theorem 2.1. & Virtually the same argument will carry over to the case of L-functions attached to normalized newforms of weight 2k for G0 ðMÞ: Theorem 2.2. Suppose that fi AS2k ðMi ; w0 Þ; i ¼ 1; 2; are normalized newforms of weight 2k; and trivial nebentypus. Let Lðs; fi Þ be their associated L-functions. Suppose that Lw ðk; f1 Þ ¼ 03Lw ðk; f2 Þ ¼ 0 for all quadratic Dirichlet characters, w: Then M1 and M2 are equal up to square factors. pffiffiffiffiffiffi s Proof. Let Lðs; fi Þ ¼ Mi =2p GðsÞLðs; fi Þ: For w a quadratic Dirichlet character of conductor Dw relatively prime to Mi ; let pffiffiffiffiffiffi s Lw ðs; fi Þ ¼ Dw Mi =2p GðsÞLw ðs; fi Þ: Then, since we have assumed that fi is a normalized newform, we have fi ¼ fi ; and since w is a quadratic character, it follows (see [Mi, Section 4.3] or [Sh, Theorem 3.66]), that the twist, fi;w ; of fi belongs to S2k ðMi D2w ; w0 Þ; and that Lw ðs; fi Þ satisfies the functional equation Lw ð2k s; fi Þ ¼ ew Lw ðs; fi Þ with ew ¼ 71: Furthermore, since the conductor Dw of w is relatively prime to Mi ; it follows that ew ¼ wðMi Þe; where e is the sign of the functional equation for Lðs; fi Þ: The rest of the argument proceeds as in the proof of Theorem 2.1. & Remark. The Friedberg–Hoffstein result has been further strengthened by Ono– Skinner [O–Sk2, Corollary 3]). They show that among all quadratic twists (with conductors divisible by exactly k primes, and satisfying finitely many local sign conditions) of an L-function associated to a newform, a positive proportion will not vanish at the centre of the critical strip. Corollary 2.3. Let E1 and E2 be semi-stable elliptic curves defined over Q such that LðE1 ; 1; wÞ ¼ 03LðE2 ; 1; wÞ ¼ 0 for every quadratic Dirichlet character w: Then their conductors, NðE1 Þ and NðE2 Þ; are equal.
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Proof. By Theorem 2.1, NðE1 Þ ¼ NðE2 Þ up to square factors. Since E1 and E2 be semi-stable, they have square-free conductors, so it follows that NðE1 Þ ¼ NðE2 Þ: &
3. Dirichlet L-functions Let c be a primitive quadratic (Dirichlet) character of conductor Nc : Let Kð¼ pffiffiffiffiffiffiffi QðcÞÞ be the quadratic field corresponding to c: Then K ¼ Q Dc where Dc ¼ cð1ÞNc is the discriminant of K; and LðsÞ ¼ Lðs; cÞ is the Dirichlet L-function associated to K: Then if c is not the trivial character w0 ; we have Lð1; cÞa0; and we let the ‘‘algebraic part’’ L ð1; cÞ of Lð1; cÞ be the class number hðKÞð¼ hðcÞ ¼ hðDc ÞÞ of K: The trivial character corresponds to Qðw0 Þ ¼ Q; and L ð1; w0 Þ ¼ 1: For another quadratic character w; the ‘‘twist’’ Lw ðs; cÞ ¼ Lðs; c wÞ of Lðs; cÞ is the Dirichlet L-function associated to the primitive quadratic character c w: Denote by KðwÞð¼ Qðc wÞÞ the extension of Q corresponding to the primitive Dirichlet pffiffiffiffiffiffiffiffiffiffiffiffi character c w: Then KðwÞ ¼ Q Dc Dw and L ð1; c wÞ is the class number hðQðc wÞÞ ¼ hðKðwÞÞ: Suppose now that L1 and L2 are Dirichlet L-functions, quadratic pffiffiffiffiffiffi for the pffiffiffiffiffiffi characters c1 and c2 respectively. Let K1 ¼ Q D1 and K2 ¼ Q D2 be the associated quadratic fields. Suppose that L1 and L2 satisfy the following: There is a set S of primes such that for all pAS and for all quadratic Dirichlet characters w (including the trivial character w0 ), we have L ð1; c1 wÞ 0 ðmod pÞ3L ð1; c2 wÞ 0 ðmod pÞ: Note that 2AS implies that hðK1 ðwÞÞ is odd if and only if hðK2 ðwÞÞ is odd. We show below that if 2AS; then L1 ¼ L2 : Theorem 3.1. Let K1 and K2 be quadratic number fields such that, for all quadratic Dirichlet characters w; the class number hðK1 ðwÞÞ is odd if and only if hðK2 ðwÞÞ is odd. Then K1 ¼ K2 : Proof. This follows from the Gauss’ genus theory for quadratic fields. We note that for an imaginary quadratic field, F ; the class number hðF Þ is odd if and only if the conductor NðF Þ of F is a prime power, and for F a real quadratic field, hðF Þ is odd if and only if either the conductor NðF Þ is a prime power, or NðF Þ is a product of two distinct prime powers, at least one of which is congruent to 3 ðmod 4Þ (cf. [Na, Theorem 8.8 and pffiffiffiffiffi Remark 20, p. 483]). As above, we write Ki ¼ Q Di for i ¼ 1; 2: Since twisting both K1 and K2 by any quadratic character doesn’t disturb the hypotheses, we can assume that K1 ¼ Q: Since K1 ¼ Q and hðK1 Þ ¼ 1 is odd, the hypothesis implies that hðK2 Þ is also odd. pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi If K2 aK1 ; then we must have either K2 ¼ Q 1 or Q p with p a prime and
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pffiffiffi pffiffiffiffiffiffiffiffiffi p 2; 3 ðmod 4Þ; if K2 is an imaginary quadratic field, or K2 ¼ Q p or Q p1 p2 with p; p1 ; p2 prime and at least one of p1 ; p2 ðmod 4Þ: Choosing a prime lap; p1 ; p2 with l 1 ðmod 4Þ and twisting by the real quadratic character w corresponding to pffiffi pffiffi the field Q l we have K1 ðwÞ ¼ Q l and hðK1 ðwÞÞ is odd. But then K2 aK1 as above implies that hðK2 ðwÞÞ is even. It then follows that K1 ¼ K2 : & The case that 2eS poses interesting questions. In particular, consider the case that S ¼ fpg consists of a single odd prime p: For a fixed quadratic field K; one knows [Ha,Jo,O–Sk] that there are an infinite number of quadratic twists KðwÞ; of K; for which p does not divide the class number hðKðwÞÞ: It is an old result [An–Ch] that there are also an infinite number of quadratic twists Kðw0 Þ; of K; for which p does divide the class number hðKðw0 ÞÞ: Given two quadratic fields K1 ; and K2 ; it is not known whether one can find a common quadratic twist w such that p divides hðK1 ðwÞÞ but doesnot divide hðK2 ðwÞÞ: pffiffiffiffiffiffiffiffiffi Specifically, suppose that p is a fixed odd prime. If Ki ¼ Q Di are distinct imaginary quadratic fields, does there exist an integer N such the class number pffiffiffiffiffiffiffiffiffiffiffiffiffi pthat ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi of Q D1 N is divisible by p; while the class number of Q D2 N is prime to p? If instead of quadratic twists, one allows twists by Dirichlet characters of arbitrary order then a suggestion of Ralph Greenberg yields the following result. Theorem 3.2. Let K1 and K2 be imaginary quadratic fields and let S be a set of primes of Dirichlet density, dðSÞ43=4: Suppose that their class numbers satisfy hðK1 LÞ 0 ðmod pÞ3hðK2 LÞ 0 ðmod pÞ for all cyclotomic extensions L=Q and for all pAS: Then K1 ¼ K2 : Proof. Suppose that K1 aK2 : Let T be the set of (odd) primes in Z relatively prime to the class numbers hðK1 Þ; and hðK2 Þ; unramified in K1 K2 ; and with decomposition group equal to GalðK1 K2 =K1 ÞCGalðK1 K2 =QÞ: Then T has density dðTÞ ¼ 1=4: Since dðSÞ43=4; we must have S-Ta|; and we can choose a prime lAS-T: But any prime lAT splits in K1 and is inert in K2 : It follows from Iwasawa theory (see [W, Chapter 13]) that the l invariant for the cyclotomic Zl extension of K1 is at least one, whereas all the class numbers in the cyclotomic Zl extension of K2 are prime to l: It follows that hðK1 Ln Þ 0 ðmod lÞ for some nX0; and that hðK2 Ln Þ is prime to l for all nX0; where Ln is the (unique) subfield of degree l n of the cyclotomic field Qðzl nþ1 Þ: This is a contradiction and hence K1 ¼ K2 : & If we take a number field k in place of the rational field, Q; then the set S must be taken sufficiently large in order to distinguish extensions of k by the class numbers of their twists as the following construction shows. For any finite set S; there is a number field k such that the p-class tower is infinite for every prime pAS (see for example [Roq]). Then every finite extension L=k has class number hðLÞ; divisible by p; for every prime pAS: For such a field, k; it is clear
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that, given any two quadratic extensions K1 =k; and K2 =k; the class numbers hðKi LÞ are divisible by p for all pAS; and all finite extensions L=k: In particular, hðK1 ðwÞÞ hðK2 ðwÞÞ 0 ðmod pÞ for all pAS; and all (quadratic) characters w:
4. L-functions of function fields over finite fields Let X =Fq be a complete, non-singular, projective curve of genus g defined over the finite field Fq and let JðX Þ denote its Jacobian variety. If K ¼ KðX Þ is the function field of X ; then K is a function field in one variable over Fq : Let LX ðTÞ ¼ LK ðTÞ ¼
2g Y
ð1 ai TÞAZ½T
i¼1
be the numerator of the z-function of X (of K), so that the ai are algebraic integers satisfying ai ai ¼ q: Let Kn ¼ Fqn K be the constant field extension of degree n of K so that the class number of hðKn Þ of Kn satisfies hðKn Þ ¼ jJðX ÞðFqn Þj ¼
2g Y
ð1 ani Þ:
i¼1
Consider the situation of two such curves, X and X 0 defined over Fq ; with function fields K; and K 0 and Jacobians JðX Þ; and JðX 0 Þ respectively. We ask the analagous question to that considered in Section 3 for X ; and X 0 viz. Suppose that S is a set of primes, and that K and K 0 are two function fields over Fq such that for all nX1 we have: hðKn Þ 0 ðmod pÞ3hðKn0 Þ 0 ðmod pÞ for all primes pAS: then what can be said about LK and LK 0 ? Note that LK is the characteristic polynomial of the Frobenius endomorphism acting on the l-adic Tate module of JðX Þ: In general, since the abelian varieties JðX Þ; and JðX 0 Þ may have the same simple factors but with differing multiplicities, we cannot expect that LK ¼ LK 0 : Theorem 4.1. Let fa1 ; a2 ; y; at ; b1 ; b2 ; y; bs g be algebraic numbers, and let RX1 be an integer. Suppose that Pj
t Y
ð1 ani Þ3P j
i¼1
s Y
ð1 bnj Þ
j¼1
% and for all n 0 ðmod RÞ: Then for each integer i; 1pipt; for all prime ideals PCQ there exists an integer j; 1pjps and integers Ni ; Nj AZ such that N
j i aN i ¼ bj :
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Proof. Let E ¼ Qða1 ; a2 ; y; at ; b1 ; b2 ; y; bs Þ and let jlðEÞj denote the order of the group of roots of unity in E: For each pair of indices ði; jÞ; 1pipt; 1pjps; let Aij ¼ /ai ; bj SCE be the multiplicative subgroup of E generated by ai and bj : Then E =Aij has a finite torsion subgroup of order Tij (say). Also, let Ti (respectively Tj ) denote the order of the torsion subgroup of E =/ai S (respectively E =/bj S). Fix an ai Afa1 ; y:at g; and a prime l4maxðTi ; Tj ; Tij ; jlðEÞj; R; sÞ: Let zl denote a primitive lth root of unity, and zR a primitive Rth root of unity. Let M ¼ 1=l
1=l
Eðzl ; zR ; ai Þ; and L ¼ Mðb1 ; y; b1=l s Þ: Then L=M is a Kummer extension, of r degree ½L : M ¼ l ; rps; and with Galois group GalðL=MÞCðZ=lZÞr : 1=l For each j; 1pjps; let Lj ¼ Mðbj Þ so that MDLj DL; and let Hj ¼ GalðL=Lj Þ: 1=l
Suppose that bj eM for all j; 1pjps: Then jHj j ¼ l r1 for all j; 1pjps; and so [ s Hj psl r1 ol r j¼1 since l4s: Therefore there exists sAGalðL=MÞ but seHj for all j; 1pjps: By Tchebatorev’s density theorem, we may choose infinitely many prime ideals p of M; relatively prime to l; ai ; bj for all i; and j and such that p is (totally) split for M=Q and such that the Artin symbol, ðL=M p Þ ¼ sp ¼ s: Completing at such a prime p; we 1=l
see that Mp CQp ; p 1 ðmod lRÞ and that ai AQp : It then follows that ai ¼ xl AQp ; ðp1Þ=l
so that ai
1 ðmodpÞ: On the other hand, by the choice of p; we see that ðp1Þ=l
1=l bj eMp
for all j; 1pjps; and therefore bj c1 ðmodpÞ for all j: But then we Qt Q ðp1Þ=l ðp1Þ=l have ðp 1Þ=l 0 ðmod RÞ; pj i¼1 ð1 ai Þ and p[ sj¼1 ð1 bj Þ: This 1=l
contradicts the hypothesis of Theorem 3, and therefore we must have bj AM ¼ 1=l
Eðzl ; zR ; ai Þ for some j: Note that for an abelian extension M 0 =E 0 ; and for a prime l not dividing jlðE 0 Þj; an element xAE 0 is an lth power in M 0 if and only if it is already an lth power in E 0 (see [Du–Ki] or [Sc]). By choice of the prime l; bj is not an lth power in E; and so the above remark implies that bj is not an lth power in Eðzl ; zR Þ: Therefore 1=l
1=l
Eðzl ; zR ; bj Þ ¼ Eðzl ; zR ; ai Þ; and then Kummer theory implies that bj ¼ aki xl for some xAEðzl ; zR Þ and some k; 1pkpl 1: But then bj =aki is an lth power in Eðzl ; zR Þ; and so also in E: Now the group Aij ¼ /ai ; bj S is a two generator subgroup of E ; and therefore either Aij CZ Z or Aij CZ ðfiniteÞ or Aij DmðEÞ: If we had Aij CZ Z; then Aij =Alij CZ=lZ Z=lZ: But by our choice of the prime l; Alij ¼ Aij -ðE Þl ; and since bj =aki is an lth power in E; it follows that Aij =Alij I / Z=lZ Z=lZ: Hence there exist a gAE ; and a root of unity zAlðEÞ; such that ai ¼ gai zci and bj ¼ gbj zdj for some integers ai ; bj ; ci and dj : But then taking Ni ¼ bj jlðEÞj; and Nj ¼ ai jlðEÞj; we have N
j i aN i ¼ bj : &
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We would like to point out that the case t ¼ 1 ¼ s was proved by CorralesRodriga´n˜ez and Schoof [Co–Sco] answering a question of Erdo¨s. In fact, they were able to obtain the stronger result that a ¼ b unless they were both roots of unity, or both were units with a ¼ b1 : They also obtained an elliptic analogue of this result. In general, however, one cannot expect such an improvement as the following example shows. If N and M are any positive integers, and zN and zM are primitive Nth, and Mth roots of unity respectively, then M Y
ð1 ziM xN Þ ¼
i¼1
N Y
ð1 zNj xM Þ ¼ 1 xNM :
j¼1
If we take ai ¼ ziM xN ; and bj ¼ zNj xM ; then the hypothesis of Theorem 4.1 is satisfied N and we have aM i ¼ bj ; but smaller powers will not in general suffice. Theorem 4.2. Suppose that RX1 is an integer, and that K and K 0 are two function fields over Fq such that for all n 0 ðmod RÞ: hðKn Þ 0 ðmod pÞ3hðKn0 Þ 0 ðmod pÞ for all primes p: Then there exists an integer N such that LKN ðTÞ and LKN0 ðTÞ have the same zeros. Q Q2g0 n n 0 % Proof. Let hðKn Þ ¼ 2g i¼1 ð1 ai Þ and hðKn Þ ¼ j¼1 ð1 bj Þ: If PCQ is any prime ideal dividing the rational prime p; then Pj
2g Y
ð1 ani Þ3 hðKn Þ 0 ðmod pÞ
i¼1
3 hðKn0 Þ 0 ðmod pÞ 0
2g Y 3 Pj ð1 bnj Þ: j¼1
Applying Theorem 4.1, we see that for each integer i; 1pipt; there exists an integer Nj i j; 1pjps and integers Ni ; Nj AZ such that aN i ¼ bj : But since ai ; and bj are both algebraic integers such that jai j ¼ jbj j ¼ q41; we must have Ni ¼ Nj : Letting N be a common multiple of all the Ni we see that LKN ðTÞ and LKN0 ðTÞ have the same zeros. & Theorem 4.3. Let A and B be abelian varieties defined over Fq ; simple over F% q ; and let RX1 be an integer. If for all integers n 0 ðmod RÞ; and for all primes p; jAðFqn Þj 0 ðmod pÞ3jBðFqn Þj 0 ðmod pÞ; then A and B are isogeneous over a finite extension of Fq :
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Proof. Let l be a prime number, relatively prime to q; and let fA ðTÞ; and fB ðTÞ denote the characteristic polynomials of Frobenius acting on their respective l-adic Q A Q2gB Tate modules. Then we can write fA ðTÞ ¼ 2g i¼1 ð1 ai TÞ and fB ðTÞ ¼ j¼1 ð1 bj TÞ with jAðFqn Þj ¼
2gA Y
ð1 ani Þ
i¼1
and jBðFqn Þj ¼
2gB Y
ð1 bnj Þ:
j¼1
Theorem 4.1 then implies that there exists an integer N such that fA;N ðTÞ ¼ Q2gB Q2gA N N i¼1 ð1 ai TÞ and fB;N ðTÞ ¼ j¼1 ð1 bj TÞ have the same zeroes. But then fA;N ðTÞ divides ð fB;N ðTÞÞM for some integer MX1; and so Tate’s theorem [T] implies that A is FqN -isogeneous to an abelian subvariety of BM defined over FqN : But since A and B were assumed to be simple over Fq ; we see that A and B are isogenous over FqN : & Theorem 4.3 can be thought of as an analogue of Zarhin’s conjecture over finite fields. Ernst Kani has pointed out the following interesting instance of Theorem 4.2 in which only one prime (the characteristic) yields the result. Let E and E 0 be elliptic curves defined over Fq ; and let K; and K 0 be their respective function fields. Suppose that E is supersingular and that q ¼ p f : Then since hðKn Þ ¼ jEðFqn Þj is prime to p for all n; hðKn Þ 0 ðmod pÞ3hðKn0 Þ 0 ðmod pÞ implies that E 0 is also supersingular and then their z-functions become equal after an extension of degree at most 6.
5. Problems We conclude this article with a number of questions which arising from the discussion of the previous sections. One can pose the following problems: Question 1. Let S be a set of primes. Suppose that K1 and K2 are fields of degree ½K1 : Q ¼ ½K2 : Q ¼ d: Suppose that for every (cyclotomic) extension L=Q; the class
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numbers hðKi LÞ satisfy: hðK1 LÞ 0 ðmod pÞ3hðK2 LÞ 0 ðmod pÞ for all pAS: If S is sufficiently large (e.g. S is the set of all primes), does this imply that K1 ¼ K2 ? Since we think of the class number as the algebraic part of an L-function at the critical point s ¼ 1; we can also pose the analagous question at other critical points. Appealing to Lichtenbaum’s conjecture [Li, Question 4.2], we can replace the class numbers by the orders of the K-groups K2n ðOKi L Þ and again ask if the prime factors of these group orders characterize the fields Ki as L ranges over all finite (cyclotomic) extensions of Q: (Here OKi L denotes the ring of integers in the number field Ki L). Question 2. If we replace the Dedekind z-functions by the z-functions of curves (varieties) over finite fields, we can ask for analogues at of Theorems 4.1 and 4.2 at other values of s: Given 0; 1arAQ; and a (sufficiently large) set I of integers n; and a set S of primes, suppose that t Y i¼1
ðr ani Þ 0 ðmod pÞ3
s Y
ðr bnj Þ 0 ðmod pÞ for all primes pAS
j¼1
for all nAI; and all pAS: What conditions on I and S allow us to conclude that fa1 ; a2 ; y; at g ¼ fb1 ; b2 ; y; bs g? As an example, the following problem of elementary number theory is a special case (taking s ¼ t ¼ 1 above): If A and B are positive integers such that for all primes p; and all integers nX1; pjAn 23pjBn 2; does this imply that A ¼ B? We prove this assuming the ABC conjecture. The ABC conjecture asserts that given coprime positive integers A; B; C with A þ B ¼ C then for any e40; there is a constant cðeÞ40; such that Q CocðeÞradðABCÞ1þe ; where for a positive integer N; radðNÞ ¼ pjN p is the product of the distinct prime factors of N: We will say that the sequence fNn g has finite support if the sequence radðNn Þ is bounded. Otherwise, we say that fNn g has infinite support. Lemma 5.1. Assume the ABC conjecture. Let a; N be positive integers. Then an infinite subsequence of the integers aN n þ b has finite support if and only if either N ¼ 1; or b ¼ 0: Proof. Clearly if either N ¼ 1; or b ¼ 0; then aN n þ b has finite support.
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Suppose that there is some infinite set of integers I; and some bound M40; such that radðaN n þ bÞpM for all nAI: Take A ¼ aN n ; B ¼ b; and C ¼ aN n þ b: Then the ABC conjecture (after adjusting for common factors) implies that given e40; there is a constant cðeÞ such that if ABCa0 aN n þ bpcðeÞðradðABCÞÞ1þe pcðeÞðabNMÞ1þe for all nAI: But then N n1e is bounded, and since n is unbounded, we must have N ¼ 1: & Proposition 5.2. Assume the ABC conjecture. Let a40; b; c40; dAZ be integers. Suppose that A and B are positive integers such that p j aAn þ b3p j cBn þ d for all primes p and for an infinite sequence I of positive integers n: Then either both of the sequences aAn þ b and cBn þ d have finite support, or A ¼ B and dðaAn þ bÞ ¼ bðcBn þ dÞ: Proof. It is clear that the hypothesis implies that one of the sequences has finite support if and only if the other does. Therefore we may assume that they both have infinite support. If bd ¼ 0 then one and hence both of the sequences are finitely supported. Hence we may assume that abcda0: Suppose that AoB; and choose e so that A1þe oB: Consider the equation ðcBn þ dÞ d ¼ cBn : Then the ABC conjecture (after removing common factors) implies that there is a constant cðeÞ independent of n such that Bn pcðeÞðradðcBdðcBn þ dÞÞÞð1þeÞ : But radðcBn þ dÞ ¼ radðaAn þ bÞp2aAn for n large and therefore Bn pcðeÞðjcBd2ajAjn Þð1þeÞ : Taking nth roots and letting n-N; we obtain BpAð1þeÞ : But this is a contradiction, and therefore A ¼ B: Thus we have p j aAn þ b3pjcAn þ d for an infinite set of exponents n: But for any prime p; with pjaAn þ b it follows that pjad bc; and since we assumed that the sequence faAn þ bg had infinite support, Lemma 1 implies that ad ¼ bc: Therefore dðaAn þ bÞ ¼ bðcBn þ dÞ: &
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Hence we have the following: Corollary 5.3. Assume the ABC conjecture. Suppose that A and B are positive integers such that radðAn 2Þ ¼ radðBn 2Þ; for an infinite sequence of integers n; then A ¼ B:
References [An–Ch] [B–C–D–Ta] [Co–Sco] [Du–Ki] [Fr–Ho] [Ha] [Jo] [Ki] [Li]
[Mi] [Na] [O–Sk] [O–Sk2] [Ro]
[Roq] [Sc] [Sh] [T] [Ta–Wi]
N.C. Ankeny, S. Chowla, On the divisibility of the class number of quadratic fields, Pacific J. Math. 5 (1955) 321–324. C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001) 843–939. C. Corrales–Rodriga´n˜ez, R. Schoof, The support problem and its elliptic analogue, J. Number Theory 64 (2) (1997) 276–290. D. Dummit, H. Kisilevsky, Abelian extensions generated by division points, J. Number Theory 29 (1) (1988) 21–30. S. Friedberg, J. Hoffstein, Nonvanishing theorems for automorphic L-functions on GLð2Þ; Ann. Math. 142 (1995) 385–423. P. Hartung, Proof of the existence of infinitely many imaginary quadratic fields whose class number is not divisible by 3, J. Number Theory 6 (1974) 276–278. N. Jochnowitz, Congruences between modular forms of half integral weights and implications for class numbers and elliptic curves, preprint, 61pp. H. Kisilevsky, Rank determines semi-stable conductor, J. Number Theory, to appear (doi: 10.1016/S0022-314X(03)00157-4). S. Lichtenbaum, Values of zeta functions, e´tale cohomology, and algebraic K-theory, in: H. Bass (Ed.) Algebraic K-theory II, Lecture Notes in Mathematics, Vol. 342, Springer, Berlin, 1973, pp. 489–501. T. Miyake, Modular Forms, Springer, New York, 1989. W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 2nd Edition, Springer and PWN–Polish Scientific Publishers, New York and Warsaw, 1990. K. Ono, C. Skinner, Fourier coefficients of half-integral weight modular forms modulo, Ann. Math. (2) 147 (1998) 453–470 Corrigendum: Ann. Math. (2) 148 (1998) 361. K. Ono, C. Skinner, Non-vanishing of quadratic twists of modular L-functions, Invent. Math. 134 (1998) 651–660. D. Rohrlich, The vanishing of certain Rankin–Selberg convolutions, in: R. Murty (Ed.) Automorphic Forms and Analytic Number Theory, Univ. Montre´al, Montre´al, Que., 1989, pp. 123–133. P. Roquet, On class field towers, in: J.W.S. Cassels, A. Fro¨lich (Eds.) Algebraic Number Theory, Brighton 1965, Academic Press, New York, 1967, pp. 231–249. A. Schinzel, Abelian polynomials, power residues and exponential congruences, Acta Arith. 32 (1977) 245–274. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, Iwanami Shoten, Japan and Princeton University Press, Princeton, NJ, 1971. J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966) 134– 144. R. Taylor, A. Wiles, Ring–theoretic properties of certain Hecke algebras, Ann. Math. (2) 141 (3) (1995) 553–572.
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ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 301–314
http://www.elsevier.com/locate/jnt
On the discrepancy of ð0; 1Þ-sequences Friedrich Pillichshammer Institut fu¨r Analysis, Universita¨t Linz, Altenbergerstrasse 69, A-4040 Linz, Austria Received 27 February 2003; revised 24 April 2003 Communicated by M. Pohst
Abstract We give bounds for the Lp -discrepancy, pAN; of the van der Corput sequence in base 2. Further, we give a best possible upper bound for the star discrepancy of ð0; 1Þ-sequences and show that this bound is attained for the van der Corput sequence. Finally, we give a ð0; 1Þsequence with essentially smaller star discrepancy than for the van der Corput sequence. r 2003 Elsevier Inc. All rights reserved. MSC: 11K06; 11K31; 11K38 Keywords: L p -discrepancy; Star discrepancy; van der Corput sequence; ð0; 1Þ-Sequence
1. Introduction For a sequence x0 ; x1 ; y of points in the one-dimensional unit interval ½0; 1Þ the discrepancy function DN ; NX1; is defined as DN ðaÞ :¼ AN ð½0; aÞÞ Na for 0pap1; where AN ð½0; aÞÞ denotes the number of indices i; 0pipN 1; with xi A½0; aÞ: Now the Lp -discrepancy LpN ; pX1; of the sequence is defined as the Lp -norm of the discrepancy function DN divided by N and is a measure for the irregularity of distribution of the first N points of the sequence in ½0; 1Þ (see for example [4] or [8]), i.e., for 1ppoN we set LpN
¼
LpN ðx0 ; x1 ; yÞ
1 :¼ N
Z
1
p
jDN ðaÞj da
0
E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.08.002
1p :
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For p ¼ N we get the usual star discrepancy of the sequence DN ¼ DN ðx0 ; x1 ; yÞ :¼
1 sup jDN ðaÞj: N 0pap1
In this paper we consider the discrepancy of a very special class of sequences in ½0; 1Þ; the so-called digital ð0; 1Þ-sequences in base 2 introduced by Niederreiter in [11] (in fact, Niederreiter introduced the much more general concept of ðt; sÞ-sequences in base bX2; where s is the dimension and t denotes the quality parameter of the sequence, see [11] or [12]). A digital ð0; 1Þ-sequence in base 2 is constructed in the following way: choose a N N matrix C over Z2 such that every left upper m m submatrix CðmÞ has full rank over Z2 : For nX0 let n ¼ n0 þ n 1 2 þ n 2 22 þ ? be the base 2 representation of n: Then multiply the vector of digits n ¼ ðn0 ; n1 ; yÞT with the matrix C; Cn ¼: ðy1 ; y2 ; yÞT AZN 2 ; and set xn :¼
y1 y2 þ þ?
2 22
Every sequence constructed in this way is called digital ð0; 1Þ-sequence in base 2. It was shown by Niederreiter [11] that for the star discrepancy of any (not necessarily digital) ð0; 1Þ-sequence in base 2 we have log N þ Oð1Þ NDN p 2 log 2 and hence lim sup sup N-N
NDN 1 ¼ 0; 7213y; p log N 2 log 2
where the supremum is extended over all ð0; 1Þ-sequences in base 2. The by far most famous sequence in the class of digital ð0; 1Þ-sequences is the van der Corput sequence in base 2 which is generated by the N N matrix 0 1 1 0 0 y B0 1 0 y C B C I :¼ B C: @0 0 1 y A ^
^
&
(It is easy to see that the van der Corput sequence is the sequence ðf2 ðnÞÞnX0 where f2 denotes the radical-inverse function, see [10].)
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It was shown by Haber [7] that for the star discrepancy of the van der Corput sequence in base 2 we have log N NDN p þ Oð1Þ 3 log 2 and that the constant 1=ð3 log 2Þ is best possible. In [2] Be´jian and Faure improved Haber’s result to NDN p
log N þ1 3 log 2
and they showed that log N 4 log 3 ¼ 0; 9727y lim sup NDN ¼ þ 3 log 2 9 3 log 2 N-N
ð1Þ
holds. (By a remark in [8]—see also [10]—these results were already proved by Tijdeman (unpublished), with the difference that Tijdeman only had ‘‘X’’ in equality (1).) Concerning the L2 -discrepancy of the van der Corput sequence in base 2 from Faure [6] we know that
ðNL2N Þ2 p
log N 6 log 2
2 11 2 log 3 log N 1 þ þ þ 3 log 2 36 log 2 3
and from Proinov and Atanassov [14] and Chaix and Faure [3] we know, that for 1ppp2 we have lim sup N-N
NLpN 1 ¼ log N 6 log 2
(this formula was already proved by Haber [7] for the case p ¼ 2; see also [6]). In Section 2 of this paper we prove a formula for the Lp -discrepancy of the van der Corput sequence for all integers p; 1ppoN; from which we obtain the order of the Lp -discrepancy and the exact value of the constant at the ‘‘leading term’’ (Theorem 1). From this result we easily deduce the value of lim sup N-N
NLpN log N
(Corollary 1) for all integers pX1: In Section 3 we give a general upper bound for the star discrepancy of digital ð0; 1Þ-sequences in base 2 and show the surprising result, that the van der Corput sequence is the worst distributed digital ð0; 1Þ-sequence in base 2 with respect to star discrepancy (Theorem 2). Further, we construct a digital ð0; 1Þ-sequence in base 2 with much smaller star discrepancy than for the van der Corput sequence (Theorem 3).
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2. The Lp discrepancy of the van der Corput sequence We have Theorem 1. Let p be a positive integer. For the Lp -discrepancy of the van der Corput sequence in base 2 for all NX1 we have log N p p p þOððlog NÞp1 Þ; ðNLN Þ p 6 log 2 where the constant in the O-notation only depends on p: If N is of the form mþ1 ! 2mþ1 1 1 ; N¼ 2 3 we have ðNLpN Þp X
p 1 m 1 1 m þ ð1Þ : 2p 3 9 9 2m
From this result we immediately obtain Corollary 1. Let p be a positive integer. For the Lp -discrepancy of the van der Corput sequence in base 2 we have lim sup N-N
NLpN 1 ¼ 0; 2404y : ¼ log N 6 log 2
3. Bounds for the star discrepancy of ð0; 1Þ-sequences We have Theorem 2. For the star discrepancy D˜ N of any digital ð0; 1Þ-sequence in base 2 for all NX1 we have log N N D˜ N pNDN p þ 1; 3 log 2 where DN denotes the star discrepancy of the van der Corput sequence in base 2. This result shows that the van der Corput sequence in base 2 is the worst distributed ð0; 1Þ-sequence in base 2 with respect to star discrepancy. Now we obtain
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Corollary 2. We have lim sup max N-N
NDN 1 ¼ 0; 4808y; ¼ log N 3 log 2
where the maximum is extended over all digital ð0; 1Þ-sequences in base 2. The value 1=ð3 log 2Þ will be attained for example for the van der Corput sequence in base 2. Finally, we get a digital ð0; 1Þ-sequence in base 2 with much smaller star discrepancy: Theorem 3. For the star discrepancy generated by the matrix 0 1 B B0 B J :¼ B B0 B @0 ^
DN of the digital ð0; 1Þ-sequence in base 2 1
1
1
1
1
1
0 0
1 0
1 1
^
^
y
1
C yC C yC C C yA &
we have 0; 2885y ¼
1 NDN 5099 1 p lim sup ¼ 0; 3265y : p 5 log 2 22528 log 2 N-N log N
Remark 1. In [1] Be´jian proved that for any sequence in ½0; 1Þ one has lim sup N-N
NDN X0; 06: log N
4. Prerequisites A digital ð0; m; 2Þ-net in base 2 is a point set of N ¼ 2m points x0 ; y; xN1 in ½0; 1Þ2 ; which is generated as follows: choose two m m-matrices C1 ; C2 over Z2 with the property that for every integer k; 0pkpm; the system of the first k rows of C1 together with the first m k rows of C2 is linearly independent over Z2 : Then to ð1Þ ð2Þ construct xn :¼ ðxn ; xn Þ for 0pnp2m 1; represent n in base 2, n ¼ n0 þ n1 2 þ ? þ nm1 2m1 ; multiply Ci ; 1pip2; with the vector of digits, i.e., ðiÞ
T m Ci ðn0 ; y; nm1 ÞT ¼: ðy1 ; y; yðiÞ m Þ AZ2
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and set
xðiÞ n :¼
m yðiÞ X j : j 2 j¼1
For any point set x0 ; y; xN1 of points in the two-dimensional unit cube ½0; 1Þ2 the discrepancy function dN is defined as dN ða; bÞ :¼ AN ð½0; aÞ ½0; bÞÞ Nab for 0pa; bp1; where AN ð½0; aÞ ½0; bÞÞ denotes the number of indices i; 0pipN 1; with xi A½0; aÞ ½0; bÞ: The subsequent Lemma 1 will be the basic tool in our considerations. First we need some notation: for
a¼
a1 am b1 bm þ ? þ m and b ¼ þ ? þ m 2 2 2 2
with ai ; bi Af0; 1g we say in the following a (resp. b) is ‘‘m-bit’’. Lemma 1. For all a; b m-bit, for the discrepancy function d2m ða; bÞ of the digital ð0; m; 2Þ-net in base 2 generated by the m m matrices 0
1
0
B 1 B0 B C1 ¼ Im :¼ B :::::: B B 0 @0 0
0
? 0 ? 0 ? 1 ? 0
0
1
C 0C C C C C 0A 1
and C2 arbitrary (such that the ð0; m; 2Þ-net property is fulfilled) we have
d2m ða; bÞ ¼
m1 X
jj2u bjjeu ;
u¼0
where eu Af1; 0; 1g; 0pupm 1; depends on a; b and C2 and where jj jj is the distance to the nearest integer function (i.e., jjxjj :¼ minðx ½x ; 1 ðx ½x ÞÞ). Proof. See [9, Theorem 1] for the proof and for the exact values of the eu ’s.
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Remark 2. (1) If 0
0
B B0 B C2 ¼ I˜m :¼ B B :::::: B @0 1
0
?
0
?
1 0
? ?
0 1
1
C 1 0C C C; C C 0 0A 0 0
then the digital net is the two-dimensional Hammersley point set with N ¼ 2m points and from [9, Example 2] we get eu ¼ amu "amþ1jðuÞ ; where " denotes addition modulo 2 and 8 > : maxfjpu: amþ1j abj g
where if u ¼ 0; if amþ1j ¼ bj for j ¼ 1; y; u; else:
(2) Note that for all 0pa; bp1 we have for the discrepancy function dH 2m of the Hammersley point set dH 2m ða; bÞX0: Lemma 2. Let 2m1 oNp2m : For the discrepancy function DN of a digital ð0; 1Þsequence in base 2 generated by the N N matrix C we have DN ðaÞ ¼ d2m ðaðmÞ; N=2m Þ þ NðaðmÞ aÞ; where aðmÞ is the smallest m-bit number larger or equal to a and where d2m denotes the discrepancy function of the digital ð0; m; 2Þ-net in base 2 generated by the m m matrices C1 ¼ CðmÞ and C2 ¼ I˜m : Proof. Let x0 ; x1 ; y be the digital ð0; 1Þ-sequence and denote by yk ¼ ðxk ; k=2m Þ; 0pkp2m 1; the points of the digital net with 2m points. For 0pa; bp1 let AN ð½0; aÞÞ denote the number of indices i; 0pipN 1; with xi A½0; aÞ and AN ð½0; aÞ ½0; bÞÞ denote the number of indices k; 0pkp2m 1; with net points yk A½0; aÞ ½0; bÞ: Then we have DN ðaÞ ¼ AN ð½0; aÞÞ Na ¼ AN ð½0; aðmÞÞ Na ¼ AN ð½0; aðmÞÞ ½0; N=2m ÞÞ 2m
N aðmÞ þ NðaðmÞ aÞ 2m
¼ d2m ðaðmÞ; N=2m Þ þ NðaðmÞ aÞ:
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Remark 3. From Lemma 2 together with Remark 2 it follows that for the discrepancy function DN of the van der Corput sequence we have DN ðaÞX0 for all 0pap1 and all NX1: This property was already proved by Faure [5] for van der Corput sequences in arbitrary bases. Lemma 3. Let a and b be m-bit and define jðuÞ; 0pupm 1; as in Remark 2. Let 1pkpm 1 be an integer and v1 ; y; vk Af0; 1; y; m 1g with vi avj for 1piajpk: Then we have 2m k X Y 2m a¼1
(Here and in the following Proof. See [13, Lemma 2].
ðamvi "amþ1jðvi Þ Þ ¼ 2mk :
i¼1
P2 m
2m a¼1
means summation over all a40 m-bit.)
&
Lemma 4. Let p be a positive integer and let dH 2m be the discrepancy function of the Hammersley point set with 2m points. Let 2m1 oNp2m : Then we have 2m 1 X log N p H m m p d m ðl=2 ; N=2 Þ p þOððlog NÞp1 Þ: 2m l¼1 2 6 log 2 If N is of the form mþ1 ! 2mþ1 1 N¼ 1 ; 2 3 we have p 2m 1 X 1 m 1 m 1 H m m p þ ð1Þ d m ðl=2 ; N=2 Þ X p : 2m l¼1 2 2 3 9 9 2m
Proof. By Lemma 1 and Remark 2 we have m
2 1 X dHm ðl=2m ; N=2m Þp 2m l¼1 2 2m 1 X ¼ m dHm ða; N=2m Þp 2 2m a¼1 2
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! p 2m m 1 Y X 1 X ui m ¼ m jj2 N=2 jjðamui "amþ1jðui Þ Þ 2 2m a¼1 i¼1 u ¼0 i ! ! p p m1 2m X X Y Y 1 ui m ¼ m jj2 N=2 jj ðamui "amþ1jðui Þ Þ : 2 u ;y;u ¼0 i¼1 2m a¼1 i¼1 p
1
Now for ðu1 ; y; up ÞAf0; 1; y; m 1gp let kðu1 ; y; up Þ be the number of different ui ’s. Then we get with Lemma 3 p 2m X Y ðamui "amþ1jðui Þ Þ ¼ 2mkðu1 ;y;up Þ : 2m a¼1 i¼1
Therefore m
2 1 X dHm ðl=2m ; N=2m Þp 2m l¼1 2
1 ¼ m 2 1 ¼ p 2
m1 X
p Y
u1 ;y;up ¼0
i¼1
!
jj2ui N=2m jj 2mkðu1 ;y;up Þ
m1 X
p Y
u1 ;y;up ¼0 kðu1 ;y;up Þ¼p
i¼1
! ui
m
jj2 N=2 jj
þ
m1 X
p Y
u1 ;y;up ¼0 kðu1 ;y;up Þop
i¼1
! jj2 N=2 jj 2kðu1 ;y;up Þ ui
m
¼: S1 þ S2 : Now we have ! p m1 Y 1 X jj2ui N=2m jj ¼ S1 p p 2 u ;y;u ¼0 i¼1 1 p p 1 m 1 m 1 þ ð1Þ p 2 3 9 9 2m p log N ¼ þOððlog NÞp1 Þ: 6 log 2
m1 1X jj2u N=2m jj 2 u¼0
!p
Here we used [9, Theorem 3] which states that
max
b m-bit
m1 X u¼0
jj2u bjj ¼
m 1 1 þ ð1Þm : 3 9 9 2m
ð2Þ
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Since jjxjjp1=2 for all xAR we get p1 m 1 X 1 1 X 1 S2 p p 2kðu1 ;y;up Þ ¼ p 2 u ;y;u ¼0 2 r¼0 2r 1
p
kðu1 ;y;up Þop
m1 X
1:
u1 ;y;up ¼0 kðu1 ;y;up Þ¼r
From [13] (Proof of Lemma 4) we know that m 1 X m 1 ¼ r! Sðp; rÞ; r u1 ;y;up ¼0 kðu1 ;y;up Þ¼r
where Sðp; rÞ denotes the Stirling numbers of the second kind. Therefore we get p1 m 1 X 1 S2 p p r! Sðp; rÞ r 2 r¼0 2 r p
1 ðmðm 1Þyðm p þ 2ÞÞcðpÞ ¼ Oððlog NÞp1 Þ; 2p
P 1 where cðpÞ ¼ p1 r¼0 2r Sðp; rÞ and the first part of the result follows. From the first part of the proof we have ! p 2m m 1 Y X 1 X p H d m ðl=2m ; N=2m Þ ¼ jj2ui N=2m jj 2kðu1 ;y;up Þ 2m l¼0 2 u1 ;y;up ¼0 i¼1 !p 1 X 1 m u m X p jj2 N=2 jj : 2 u¼0 Again from [9, Theorem 3] we know that the maximum in equation (2) is attained if N=2m is of the form mþ1 ! m N 2 1 N 1 1 1 1 ¼ or m ¼ : 2m 3 2 2 3 2 The result follows.
&
For the proof of Theorem 2 we need Lemma 5. Let 2m1 oNp2m : For the star discrepancy DN of the van der Corput sequence in base 2 we have NDN ¼
m1 X u¼0
jj2u N=2m jj þ
N ; 2m
where jj jj is the distance to the nearest integer function. Proof. See [2].
&
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Remark 4. We remark here that in [6] Faure also gave an exact formula for the L2 discrepancy of the van der Corput sequence in base 2 in terms of the distance to the nearest integer function. Further in [14] Proinov and Atanassov showed that L1N ¼ 1 1 1 2 DN ; where LN denotes the L -discrepancy of the van der Corput sequence and hence we also have a formula for the L1 -discrepancy of the van der Corput sequence in base 2 in terms of the distance to the nearest integer function. 5. The proofs of the results First we give the proof of Theorem 1. Proof of Theorem 1. Let 2m1 oNp2m : By Lemma 2 and Remark 3 we have Z 1 jDN ðaÞjp da 0
¼
Z
1
0
¼
Z
0
Z
1
1
p m ðdH 2m ðaðmÞ; N=2 Þ þ NðaðmÞ aÞÞ da p X p k¼0
k
pk m k da dH 2m ðaðmÞ; N=2 Þ ðNðaðmÞ aÞÞ p1 Z X p
1
m k dH 2m ðaðmÞ; N=2 Þ da k 0 0 k¼0 m p1 2 2m X p 1 X 1 X H m m p m m k ¼ m d m ðl=2 ; N=2 Þ þ dH 2m ðl=2 ; N=2 Þ m 2 l¼1 2 2 k l¼1 k¼0 log N p p þOððlog NÞp1 Þ; 6 log 2
p
m p dH 2m ðaðmÞ; N=2 Þ
da þ
where we used NðaðmÞ aÞp2m 2m ¼ 1 and Lemma 4. The result follows. On the other hand since by Remark 3 DN ðaÞX0 for all 0pap1 we have Z 1 Z 1 p m p jDN ðaÞj daX dH 2m ðaðmÞ; N=2 Þ da 0
0
¼
2m 1 X dHm ðl=2m ; N=2m Þp : 2m l¼1 2
Now an application of Lemma 4 will finish the proof of Theorem 1.
&
Proof of Theorem 2. Let 2m1 oNp2m : By Lemma 2 we have for the discrepancy function DN of any digital ð0; 1Þ-sequence generated by the matrix C for all 0pap1 jDN ðaÞjp max jd2m ðb; N=2m Þj þ N=2m ; b m-bit
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where d2m is the discrepancy function of the digital ð0; m; 2Þ-net generated by the m m matrices C1 ¼ CðmÞ and C2 ¼ I˜m : But this is the same net (in changed order) as the net generated by the m m matrices 0
1
B B0 B C˜ 1 ¼ B B :::::: B @0 0
0 ?
0
1 ?
0
0 ? 0 ?
1 0
0
1
C 0C C C C C 0A 1
and C˜ 2 ¼ C2 C11 : Thus we get with Lemma 1 and Lemma 5 jDN ðaÞjp
m 1 X
jj2u N=2m jj þ
u¼0
N ¼ NDN ; 2m
where DN denotes the star discrepancy of the van der Corput sequence in base 2. The result follows. & Proof of Corollary 2. The corollary follows from Theorem 2 together with the fact that lim sup N-N
NDN 1 ¼ log N 3 log 2
for the star discrepancy DN of the van der Corput sequence in base 2, see [7] or [5]. & Proof of Theorem 3. Let 2m1 oNp2m : By Lemma 2 we have for the discrepancy function DN of our special sequence for all 0pap1 jDN ðaÞjpjd2m ðaðmÞ; N=2m Þj þ N=2m ; where d2m is the discrepancy function of the digital ð0; m; 2Þ-net generated by the m m matrices 0
1
B B0 B C1 ¼ B B :::::: B @0 0
1 y 1 1 y 1 0 y 1 0 y 0
1
1
0
0
0
C B 1C 0 B0 C B C and C2 ¼ B :::::: C B C B 1A 1 @0 1 1 0
? 0 ? 1 ? 0 ? 0
1
1
C 0C C C: C C 0A 0
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But this is the same matrices 0 1 B B1 B ˜ C1 ¼ B B :::::: B @1 1
313
net (in changed order) as the net generated by the m m 1 y 1 1 y 1 1 y 0 0 y 0
1 0 1 1 C B 0C B0 C B C and C˜ 2 ¼ B :::::: C B C B 0A @0 0 0
0 1
? 0 ? 0
0 0
? 1 ? 0
1 0 C 0C C C: C C 0A 1
Denote the star discrepancy of this net by D˜ 2m : Then we obtain jDN ðaÞjp2m D˜ 2m þ 1: By Larcher and Pillichshammer [9, Theorem 6] we have lim sup m-N
2m D˜ 2m 5099 p 11 2048 m
and the upper bound follows. To prove the lower bound we find by Lemma 2 for all 0pap1 the inequality jDN ðaÞjXjd2m ðaðmÞ; N=2m Þj jNðaðmÞ aÞj; where d2m is as above and where 2m1 oNp2m : Therefore we have NDN Xjd2m ðaðmÞ; N=2m Þj 1: By Larcher and Pillichshammer [9] (Proof of Theorem 6) we have m max d2m ða; bÞX : 5
a;b m-bit
Since Np2m the result follows.
&
Acknowledgments The author would like to thank Gerhard Larcher and Gunther Leobacher for their support. The author is supported by the Austrian Research Foundation (FWF), Project 58305.
References [1] R. Be´jian, Minoration de la discre´pance d’une suite quelconque sur T; Acta Arith. 41 (1982) 185–202.
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[2] R. Be´jian, H. Faure, Discre´pance de la suite de van der Corput, C. R. Acad. Sci. Paris Se´r. A 285 (1977) 313–316. [3] H. Chaix, H. Faure, Discre´pance et diaphonie en dimension un, Acta Arith. 63 (1993) 103–141. [4] M. Drmota, R.F. Tichy, Sequences, Discrepancies and Applications, in: Lecture Notes in Mathematics, Vol. 1651, Springer, Berlin, 1997. [5] H. Faure, Discre´pances de suites associe´es a un syste`me de nume´ration (en dimension un), Bull. Soc. Math. France 109 (1981) 143–182. [6] H. Faure, Discre´pance quadratique de la suite de van der Corput et de sa syme´trique, Acta Arith. 55 (1990) 333–350. [7] S. Haber, On a sequence of points of interest for numerical quadrature, J. Res. Nat. Bur. Standards Sect. B70 (1966) 127–136. [8] L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974. [9] G. Larcher, F. Pillichshammer, Sums of distances to the nearest integer and the discrepancy of digital nets, Acta Arith. 106 (2003) 379–408. [10] H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978) 957–1041. [11] H. Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math. 104 (1987) 273–337. [12] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, Vol. 63, in CBMSNSF Series in Applied Mathematics. SIAM, Philadelphia, 1992. [13] F. Pillichshammer, On the Lp -discrepancy of the Hammersley point set, Monatsh. Math. 136 (2002) 67–79. [14] P.D. Proinov, E.Y. Atanassov, On the distribution of the van der Corput generalized sequences, C. R. Acad. Sci. Paris Se´r. I Math. 307 (1988) 895–900.
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 315–326
http://www.elsevier.com/locate/jnt
Higher Heegner points on elliptic curves over function fields Florian Breuer Mathematics Division, National Center for Theoretical Sciences, National Tsing-Hua University, Hsinchu 300, Taiwan, ROC Received 19 March 2003; revised 15 July 2003 Communicated by D. Goss
Abstract Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E; defined over a ZN p -tower of finite extensions of k; and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of Cornut and Vatsal. r 2003 Elsevier Inc. All rights reserved. MSC: 11G05; 11R58 Keywords: Elliptic curves; Heegner points; Drinfeld modular curves
1. Introduction Heegner points are a way of constructing explicit points of infinite order on modular elliptic curves, and have been used to great effect over number fields. More recently, people have produced analogous constructions on elliptic curves over global function fields. The first such construction is due to Brown [2], but there seem to be a number of errors in his paper. Ru¨ck and Tipp [15] have proved an analogue of the celebrated Gross–Zagier formula in a special case, and Longhi [11] and Pa´l [14] have (independently) constructed Heegner points of infinite order on elliptic curves, using the p-adic approach due to Bertolini and Darmon. Ulmer has
Department of Mathematics, University of Stellenbosch, Stellenbosch 7600, South Africa. E-mail addresses: [email protected], fl[email protected].
0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.09.001
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announced a more general version of the Gross–Zagier formula, which, combined with his non-vanishing result for L-functions, yields the full Birch and SwinnertonDyer conjecture for elliptic curves over function fields when their analytic rank is p1 and the characteristic is p43: For a nice survey of all this, see [16]. In this paper we prove a function field version of a result of Cornut [3] and Vatsal [17] on higher Heegner points on modular elliptic curves, which had been conjectured by Mazur [12]. Their result has some important consequences, including Nekova´rˇ ’s celebrated results concerning the parity of ranks of Selmer groups. See [3] for a discussion. Cornut actually found two proofs, the second of which [4] uses a known case of the Andre´-Oort conjecture due to Moonen, and is much simpler. We use the function field analogue of a special case of the Andre´-Oort Conjecture, proved in [1], and then follow closely Cornut’s second proof. Let k be a global function field with field of constants Fq ; where q is a power of the odd prime p: Let E be an elliptic curve defined over k with non-constant j-invariant (we say E is non-isotrivial). Then, replacing k by a finite extension if necessary, we can choose a place N of k such that E has multiplicative reduction at N: Let kN denote the completion of k at N; and set CN ¼ k#%N the completion of an algebraic closure of kN : Furthermore, we let A be the ring of functions in k regular outside N: It is a Dedekind domain with finite class number h ¼ jPicðAÞj ¼ degðNÞhk ; where hk ¼ jPic0 ðkÞj denotes the class number of k: By a Drinfeld module we will always mean a Drinfeld A-module of rank 2; defined over a subfield of CN (in particular, we deal only with the case of ‘‘generic’’ characteristic). The conductor of E may be written as n N; where n is an ideal in A: Then, by the work of Drinfeld (and Weil, Grothendieck, Jacquet-Langlands, Deligne and Zarhin) we have a modular parametrization p : X0 ðnÞ-E;
ð1:1Þ
defined over k; where X0 ðnÞ is the Drinfeld modular curve parametrizing isomorphism classes of pairs ðF; F0 Þ of Drinfeld modules linked by a cyclic isogeny of degree n: Now let K be an imaginary quadratic extension of k (i.e. such that N does not split in K=k) with the property that all primes dividing n split in K (this is known as the Heegner hypothesis). There exist infinitely many such fields. Denote by OK the integral closure of A in K; it contains an ideal N such that OK =NDA=n: Let pCA be a prime not dividing n; and let On ¼ A þ pn OK be the order of conductor pn in OK : We set Nn ¼ N-On ; so On =Nn DA=n: Then On and N 1 n ; viewed as rank 2 lattices 1
in CN ; correspond to a pair of Drinfeld modules ðFOn ; FNn Þ; linked by a cyclic isogeny of degree n: Hence they define a Heegner point xn AX0 ðnÞ; which is in fact 1 defined over the ring classSfield K½pn of On ; as EndðFOn ÞDEndðFNn ÞDOn : We now set K½pN ¼ nX1 K½pn : Then G ¼ GalðK½pN =KÞDZN p G0 ; where N Zp denotes the product of countably many copies of Zp (p being the characteristic of k), and G0 ¼ Gtors is a finite group (Proposition 2.1). This is in marked contrast
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to the number field case, where the analogue of G contains only one copy of Zp ; and where p plays the role of p; and may be chosen. G0 corresponds to a subfield H½pN satisfying GalðH½pN =KÞDZN p : Denote by X TrG0 : EðK½pN Þ-EðH½pN Þ; x/ xs sAG0
the G0 -trace on E: We define the Heegner point yn ¼ TrG0 ðpðxn ÞÞAEðH½pN Þ: Our aim is to prove Theorem 1. Suppose k ¼ Fq ðTÞ and degðNÞ ¼ 1: Let ICN be an infinite subset. Then the group generated by fyn j nAIg in EðH½pN Þ has infinite rank. Remark. We have tried to avoid the hypothesis on k and N as far as possible in this paper. It is used twice, firstly in the proof of Proposition 4.1 (but which should still hold for general k), and at the very end of the proof of Theorem 1, where we invoke an analogue of the Andre´-Oort conjecture which is currently only known in this case. Once a more general case of this conjecture has been proved—which is the object of current efforts—Theorem 1 should become true for general k and N: The layout of this paper is as follows. In Section 2 we describe the group GalðK½pN =KÞ; and show that EðK½pN Þ has finite torsion. In Section 3 we describe map (1.1) in more detail, and construct a family of new modular parametrizations in Section 4 by means of degeneracy maps between Drinfeld modular curves. Then in Section 5 we describe a canonical factorization of cyclic isogenies between CM Drinfeld modules, which we will use in Section 6 to characterize the geometric action of Galois on Heegner points. Finally, we deduce Theorem 1 from the Andre´-Oort conjecture in Section 7.
2. The class field tower Proposition 2.1. G ¼ GalðK½pN =KÞDZN p G0 ; where G0 ¼ Gtors is finite. Proof. Firstly, notice that G ¼ GalðK½pN =KÞ ¼ lim GalðK½pn =KÞD lim PicðOn Þ: ’ n
’
ð2:1Þ
n
Secondly, we have an exact sequence (see e.g. [13, Section I.12]) 1-OK =On -ðOK =pn OK Þ =ðOn =pn OK Þ -PicðOn Þ-PicðOK Þ-1: As OK =On and PicðOK Þ are bounded, it remains to examine the behavior of ðOK =pn OK Þ =ðOn =pn OK Þ as n-N: We insert it into the following diagram, with
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exact rows and columns:
Notice that pn OK ¼ pn On : As ðOK =pOK Þ =ðOn =pOn Þ is bounded, we are lead to studying the (multiplicative) group Hn :¼ ð1 þ pOK Þ=ð1 þ pOn Þ: By computing the cardinality of various groups in the diagram, one finds that Hn is a p-group, of order jpjn 1 : Let xA1 þ pOK ; we will examine its order in Hn : Let s s ¼ Jlogp ðn þ 1Þn; then we find that xp A1 þ pnþ1 OK C1 þ pOn : It follows that Hn is annihilated by ps ; hence the number of generators of Hn is at least logp ðjpjÞðn 1Þ=J logp ðn þ 1Þn-N as n-N: On the other hand, suppose the order of x in Hn is T r r bounded independently of n; say xp A1 þ pOn for all n: But then xp A N n¼1 ð1 þ pffiffiffiffi pOn ÞCA; and so xAA to begin with (recall that K ¼ kð DÞ for some square-free DAA; and that p is odd), so xAA-ð1 þ pOK ÞC1 þ pOn : We have shown that lim Hn DZN p and the proposition now follows. & ’
The following result will be crucial. Lemma 2.2. Etors ðK½pN Þ is finite. Proof. Let l[pn be a prime of k which is inert and principal in K: Then E has good reduction at l; and l splits completely in K½pn ; hence the residue field of K½pn at l is just Fl ¼ OK =l for every nX0: It follows that reduction mod l induces an injection of ˜ l Þ; which is finite. the prime-to-p part of Etors ðK½pN Þ into EðF sep Let K denote the separable closure of K: We complete the proof by showing that EðK sep Þ-E½pN is finite, for any non-isotrivial elliptic curve E=K: Indeed, if E½pn CEðKÞ; then j ¼ jðEÞ is a pn th power in K; as can be seen by factoring the multiplication by pn -map into ½pn ¼ f 3g; with kerðgÞ ¼ E½pn and f the pn th power Frobenius. Now, let n be such that j is not a pn th power in K: Then j is a pn th power in K 0 ¼ KðE½pn Þ; and it follows that K 0 =K is not separable. Alternatively, one may apply a general criterion of Voloch for abelian varieties ([18, Section 4]). &
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3. Modular parametrizations As the literature already contains excellent expositions of the theory of Drinfeld modular curves and parametrization (1.1), such as [7;9;10;11, Sections 1.4 and 1.5], we will not attempt a detailed account. Instead, we only recall here some of the results and notations that we will need, and refer the reader to [10] for the details. Every projective A-module of rank 2 is isomorphic to Ya ¼ A aCk2 for an ideal aCA; so in particular their isomorphism classes correspond to PicðAÞ: The group GL2 ðkÞ acts on k2 from the right. Let x ¼ ½a APicðAÞ and let Gx ¼ StabGL2 ðkÞ ðYa Þ be the stabilizer of Ya : Denote by O ¼ CN kN the Drinfeld upper half-plane, and choose zAO: We map k2 into O by sending ða; bÞ to az þ b: Then the image of any projective rank 2 A-module under this map is a lattice in CN ; i.e. a discrete projective A-submodule of CN of rank 2: GL2 ðkÞ acts on O from the left by fractional linear transformations, and on the lattices this action corresponds to the right action on k2 : Let M0 ðnÞ be the coarse moduli scheme for the moduli problem ‘‘pairs of Drinfeld modules linked by a cyclic isogeny of degree n’’. This is equivalent to the problem ‘‘pairs of lattices L1 Cn L2 ’’, where the Cn notation means that L2 =L1 DA=n as Amodules. We denote by Y0 ðnÞ ¼ M0 ðnÞ k the base extension to k; and by X0 ðnÞ the smooth projective model of Y0 ðnÞ; which may be obtained from Y0 ðnÞ by including finitely many cusps. The curve X0 ðnÞ has h ¼ jPicðAÞj irreducible components, each defined over the Hilbert class field H of ðk; AÞ; and which we denote by Xx for x ¼ ½a APicðAÞ: The group GalðH=kÞ permutes the components by ða; H=kÞX½b ¼ X½a 1 b : Each component Xx has an analytic parametrization Xx ðCN ÞDGx ðnÞ\O ; Gx ðnÞ ¼
a c
where O ¼ O,P1 ðkÞ;
and
b AGx j c 0 mod n : d
Let E=k be a non-isotrivial elliptic curve with conductor n N; where nCA: Then there is a surjective morphism defined over k p : X0 ðnÞ-E:
ð3:1Þ
We may suppose that E is maximal in its k-isogeny class with respect to map (3.1), we call it the strong Weil curve. Then this map can be given explicitly on the CN valued points, which we will describe next. Let T be the Bruhat-Tits tree for PGL2 ðkN Þ; and denote by HðT; ZÞ the group of % harmonic cochains on T with values in Z; and define the subgroup H! ðT; ZÞGx ðnÞ of % cochains invariant under Gx ðnÞ and with compact support on Gx ðnÞ\T: Then these cochains correspond to certain automorphic forms on GL2 : Moreover, to E one associates, for each x ¼ ½a APicðAÞ; a primitive Hecke newform jx AH! ðT; ZÞGx ðnÞ : %
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To jx we associate furthermore a holomorphic theta function ux : O-CN with multiplier cx ; (i.e. ux ðazÞ ¼ cx ðaÞux ðzÞ for all aAGx ðnÞ). Let Dx ¼ fcx ðaÞ j aAGx ðnÞg; which is a multiplicative lattice in CN : Then E; which has multiplicative reduction at N; is isomorphic over kN to the Tate curve CN =Dx : We have the explicit parametrization
4. Degeneracy maps We next define degeneracy maps between Drinfeld modular curves. Let mCA be an ideal coprime to n: A generic point of X0 ðmnÞ can be written as ðF; F=CÞ; where F is a Drinfeld module, and CDA=mn an A-submodule of Ftors : For any divisor djmn we denote by C½d DA=d the d-torsion submodule of C: Now, for every djm we define the dth degeneracy map dd : X0 ðmnÞ- X0 ðnÞ ðF; F=CÞ/ ðF=C½d ; F=C½dn Þ; which maps the ½a -component of X0 ðmnÞ to the ½d 1 a -component of X0 ðnÞ: In this way, we may define the Hecke correspondence Tm CX0 ðnÞ2 as the image of X0 ðmnÞ under the map d1 dm : More generally, let tðmÞ be the number of divisors of m; then we define the full Q degeneracy map d : X0 ðmnÞ-X0 ðnÞtðmÞ as the product d ¼ djm dd : Composing with p; we obtain a new parametrization of E by X0 ðmnÞ: d
p0 : X0 ðmnÞ ! X0 ðnÞtðmÞ z /ðzd Þdjm
p
! E tðmÞ
S
!E X /ðpðzd ÞÞdjm / djm pðzd Þ:
ð4:1Þ
Proposition 4.1. The morphism p0 : X0 ðmnÞ-E is defined over k: Suppose that k ¼ Fq ðTÞ and degðNÞ ¼ 1: Then p0 : X0 ðmnÞ-E is surjective. Proof. It is clear that GalðH=kÞ leaves p0 invariant, and hence p0 is defined over k:
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Now we suppose that k ¼ Fq ðTÞ and degðNÞ ¼ 1: Then, after replacing T by another generator if necessary, we may take A ¼ Fq ½T : In particular, PicðAÞ is trivial, and the modular curves X0 ðmnÞ and X0 ðnÞ are irreducible and defined over k: Analytically, the map p0 is given by p0 ð½z Þ ¼
Y
ACN =D ¼ EðCN Þ;
uðdzÞ mod D
djm
where ½z denotes the class of zAO in Y0 ðmnÞ; u ¼ uj is the theta function associated to the newform j; and dAA is the monic generator of the ideal d; for each djm: We Q need to show that the map u0 ðzÞ ¼ djm uðdzÞ is not constant. Denote by OO ðOÞ the ring of rigid holomorphic functions on O; then there is an exact sequence r
1-CN -OO ðOÞ ! HðT; ZÞ-0: % Furthermore, for any f AGL2 ðkÞ we have rðu3f Þ ¼ j3f AHðT; ZÞ; so if u0 is % constant, then we get 0 ¼ rðuÞ ¼ j þ
X djm;da1
j3d
AHðT; ZÞ; %
ð4:2Þ
where d denotes the matrix d0 01 : Now we examine the Fourier coefficients of these terms. From Proposition 2.10 of [8] follows that the ‘‘first’’ Fourier coefficient of j3d is cðj3d; ð1ÞÞ ¼ 0 if deFq : On the other hand, by a result of Atkin–Lehner (see [6, Section 1.4]) follows that every newform ja0 satisfies cðj; ð1ÞÞa0). Thus (4.2) implies that j ¼ 0; a contradiction. & Remark. In the case of general k and N; Proposition 4.1 should still hold. Indeed, for each component Xx of X0 ðmnÞ; one has a similar analytical description of p0 and an equality of form (4.2). But these now involve a combination of different theta functions uy ; one for each component Xy of X0 ðnÞ such that y ¼ x½d 1 in PicðAÞ for some djm: This may be simplified by considering only those m for which every djm is principal (in our application in Section 7 this amounts to choosing K such that only principal primes of k ramify in K), but one must still calculate the Fourier coefficients in this case.
5. Canonical factorizations of isogenies In this section, we will describe the following canonical factorization of cyclic isogenies between CM Drinfeld modules.
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Proposition 5.1. Let f : C1 -C2 be a cyclic isogeny of degree d between Drinfeld modules with complex multiplication by orders EndðFi Þ of conductor ci in K; for i ¼ 1; 2: Then we have a commutative diagram of cyclic isogenies
Here EndðC01 Þ ¼ EndðC02 Þ ¼ Oc is an order of conductor c in K; and f ; f1 ; f2 ; f 0 are of degree d; d1 ; d2 ; d0 ; respectively. This data is summarized in the right-hand diagram. Furthermore, we have d ¼ d1 d2 d0 ; c ¼ c1 =d1 ¼ c2 =d2 and c þ d0 ¼ A: Lastly, there is an ideal DCOK with OK =DDA=d0 such that f 0 corresponds to D-Oc : Proof. We follow closely the appendix of [4]. We will use the following notation. Let aCb be lattices (all lattices here have rank 2 and are contained in KCCN ), then we write aCm b if b=aDA=m: Also, the conductor cðaÞ of the lattice a is the conductor of the order EndðaÞ in OK : Suppose aCd b: Following Section 9 of [4] word for word, one arrives at an inclusion of lattices aCd1 Oc aCd0 d2 Oc bCd2 b:
ð5:1Þ
Here Oc is an order (of conductor c), which is maximal with respect to the property Oc aCb: Let c1 ¼ cðaÞ; c2 ¼ cðbÞ; then d1 ¼ c1 =c; d2 ¼ c2 =c and d0 ¼ d=d1 d2 : To continue the argument, we need the following folklore result. Lemma 5.2. Let b be a lattice of conductor c; and let qCA be a prime dividing c: Then there exactly jqj þ 1 sublattices aCq b: One of them (given by a ¼ qOc=q b) has conductor cðaÞ ¼ c=q and the other jqj lattices have conductor cðaÞ ¼ cq: We want to show that c and d0 are coprime. Let qCA be a prime. If qjc; then by Lemma 5.2, a0 ¼ Oc=q a is the unique Oc -stable lattice satisfying qa0 Cq Oc a; or, equivalently, Oc aCq a0 : Now suppose q divides d0 : Then d2 Oc b=Oc aDA=d0 has a unique A-submodule isomorphic to A=q; namely its q-torsion submodule: ðd2 Oc b=Oc aÞ½q ¼ ðd2 Oc b-q 1 Oc aÞ=Oc a; hence a00 ¼ d2 Oc b-q 1 Oc a is also Oc -stable and satisfies Oc aCq a00 ; and is furthermore contained in b: Thus, if q divides both c and d0 ; then Oc=q a ¼ a0 ¼ a00 Cb; which contradicts the maximality of Oc : So c and d0 are coprime. 1 Lastly, we set D ¼ d 1 2 OK b a: Then we see easily that DCd0 OK : Also, Dc ¼ D-Oc is invertible in Oc and we have d2 Oc b ¼ D 1 c Oc a: Now, using the equivalence between lattices and Drinfeld modules, Proposition 5.1 follows. &
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6. Geometric action Let Fn ¼ FOn : We say an element sAG is ‘‘geometric’’ if there exists a cyclic isogeny fd : Fn -Fsn of fixed degree d for infinitely many nAN: In particular, if d þ n ¼ A; then each ðxn ; xsn Þ lies in the Hecke correspondence Td CX0 ðnÞ2 : Let p1 ; y; pg be the primes ap of k which ramify in K=k; and let P1 ; y; Pg be the primes of K lying above them. Denote by si ¼ ðPi ; K½pN =KÞ the Frobenius elements, and by G1 ¼ /s1 ; y; sg S the group they generate. Suppose ½pi has order ei in PicðAÞ: Then si has order 2ei in G; and it follows that G1 CG0 : Let m ¼ 2e 1
1 1 ?pg g ; which is coprime to n (recall that every prime in n splits in K=k). p2e 1 Then theQelements of G1 are Q in a one-to-one correspondence with the divisors d of m; via d ¼ gi¼1 pni i /sd ¼ gi¼1 sni i :
Proposition 6.1. G1 is the subgroup of geometric elements of G0 : More precisely, (1) Let s ¼ sd AG1 for some djm: Then ðxn ; xsn ÞATd CX0 ðnÞ2 for all nAN: In particular, ðxsn ÞsAG1 CX0 ðnÞjG1 j lies in the image of X0 ðmnÞ under the full degeneracy map d described in Section 4. (2) Conversely, let sAG0 and suppose that there exist cyclic isogenies fn : Fn -Fsn of fixed degree Ds for infinitely many nAN: Then sAG1 : Proof. From (2.1) follows that each sAG can be written in the form s ¼ ðs1 ; s2 ; yÞ; where for each nAN sn AGalðK½pn =KÞ corresponds to an invertible ideal An ¼ A-On in On ; for some ACOK : For each mXn the theory of complex multiplication gives an isogeny fm : Fn -Fsn m with kerð fm ÞDOm =Am : Now let aCA lie under A ¼ An OK : If sAG1 ; then a is a product of primes which ramify in K=k: Thus each kerð fm ÞDOm =Am DA=a is cyclic. It follows that ðFn ; Fsn Þ lies on the curve Y0 ðaÞ: Furthermore, a is prime to n; hence s is compatible with n-isogenies, and it follows that the pair of Heegner points ðxn ; xsn Þ lies on the Hecke correspondence Ta CX0 ðnÞ2 : This proves part (1) of Proposition 6.1. To prove part (2), we let ICN be an infinite subset, sAG; and we suppose that there exists a cyclic isogeny fn : Fn -Fsn of degree d for all nAI: One potential source of trouble is the fact that p might divide d: Write d ¼ pt d0 with p[d0 : We apply Proposition 5.1 to the case where C1 ¼ Fn and C2 ¼ Fsn ; and nAI satisfies nXt=2: Then we see that t ¼ 2r is even, and we have a commutative diagram (with the relevant conductors and degrees shown on the right)
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s 0s s We claim that C02 ¼ C0s 1 : Indeed, we have maps f1 : C2 ¼ C1 -C1 ; and f2 : 0 C2 -C2 ; both of which correspond to lattice inclusions of the form L1 ; L2 Cpr L; where cðL1 Þ ¼ cðL2 Þ ¼ pn r ¼ cðLÞ=pr : It follows from Lemma 5.2 (and induction on r) that L1 ¼ L2 ; which proves the claim. 0 0 Now we restrict our attention to f 0 : C01 -C0s 1 : As f has degree d ; which is prime 0 to p; it follows that f corresponds, from Proposition 5.1, to an invertible ideal Dn COK such that OK =Dn DA=d0 for all n: This leaves only finitely many possibilities for Dn ; hence, by restricting I; we may assume Dn ¼ DCOK for all nAI: It now follows that ðD; K½pN =KÞ ¼ s: In particular, if sAG0 ; then s has finite order, and s there exists an integer s such that Ds is principal in OK : Moreover, T D -On is s principal for each nAI; and so D is generated by an element dA nAI On ¼ A: Denote by z/z% the non-trivial element of GalðK=kÞ: Then for every prime P of OK % also divides D: But as OK =DDA=d0 is cyclic, this is only dividing D we see that P possible if P is ramified in K=k; and hence sAG1 : This concludes the proof of Proposition 6.1. &
Remark. We have also shown that the geometric elements of G are the elements of the form ðD; K½pN =KÞ for some DCOK with OK =D cyclic. Thus they form a countable subgroup of G:
7. Proof of Theorem 1 Lemma 2.2 says that Etors ðK½pN Þ is finite, so in particular, all these torsion points are defined over a finite extension of k: Now Theorem 1 will follow if the fields of definition of the yn ’s grow with n: We prove this as follows. Let m ¼ p1 p2 ?pg CA as in Section 6, and recall the full degeneracy map d : X0 ðmnÞ-X0 ðnÞjG1 j from Section 4. From Proposition 6.1 follows that the point ðxsn ÞsAG1 lies in the image of d; and we denote by x0n AX0 ðmnÞðK½pN Þ its preimage, 1
1
which is given by ðFOn ; FNn Mn Þ; where Mn ¼ P1 ?Pg -On : We combine d with p to obtain the new modular parametrization p0 : X0 ðmnÞ-E of Section 4. Let RCG0 be a set of representatives for G0 =G1 : Notice that yn ¼ TrG0 ðpðxn ÞÞ ¼ TrR
X
! pðxn Þ
s
¼
X
p0 ðx0n Þs :
sAR
sAG1
For each mAN we choose ym AGalðK½pN =K½pm Þ such that none of the elements ym s is geometric, for sAR: Then we consider the composite map p0
r
fm : N ! X0 ðmnÞjRj X0 ðmnÞjRj ! E jRj E jRj S
@
!E E !E
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0ym s n/ðx0s ÞsAR / ðp0 ðx0n Þs ; p0 ðx0n Þym s ÞsAR n ; xn
/ ðyn ; yynm Þ/yn yynm : We will show that r is dominant, hence fm : N-E; fm ðnÞ ¼ yn yynm has finite fibres. In particular, fm 1 ð0Þ is finite, and the proof will be complete. Here the Andre´-Oort conjecture (see e.g. [5]) enters the picture, for which we state the following characteristic-p analogue (see [1]). Conjecture 7.1 (Andre´-Oort). Let X ¼ X1 ? Xn be a product of Drinfeld modular curves, and let ZCX be an irreducible algebraic subvariety for which every projection Z-Xi is dominant. Suppose Z contains a Zariski-dense set of CM points. Then Z is a ‘‘modular’’ subvariety, which ‘ means the following. Q There exist m g1 ; y; gn AGL2 ðkÞ and a partition f1; y; ng ¼ m S such that Z ¼ j¼1 j j¼1 Zj ; and Q each Zj ðCN ÞC iASj Xi ðCN Þ is the image of O under the map z/ð½gi ðzÞ ÞiASj : Let ICN be an infinite subset. We suppose for the moment that Conjecture 7.1 holds. In our case X ¼ X0 ðmnÞ2jRj and we take Z to be a positive-dimensional irreducible component of the Zariski-closure of rðIÞ in X : Then Conjecture 7.1 implies that either Z ¼ X ; or there exist some si asj AR,ym R such that the projection of Z onto the factor X0 ðmnÞ2 indexed by ðsi ; sj Þ is contained in some Hecke correspondence Td : But the latter case is impossible, as this would mean that si s 1 j is geometric, contrary to the definitions of R and ym : Thus Z ¼ X ; and we see that r is dominant. The result follows. More precisely, we have shown: Theorem 2. Suppose Conjecture 7.1 holds in the case where X ¼ X0 ðmnÞm and ZCX contains a Zariski-dense set of CM points w ¼ ðw1 ; y; wm Þ for which Endðw1 Þ ¼ ? ¼ Endðwm Þ ¼ On for some nAN: Suppose further that the map p0 : X0 ðmnÞ-E is surjective. Then for every infinite subset ICN the group generated by fyn j nAIg in EðH½pN Þ has infinite rank. & In the special case where k ¼ Fq ðTÞ; A ¼ Fq ½T and q is odd, Conjecture 7.1 is known [1], and p0 is surjective (Proposition 4.1), and so Theorem 1 follows. & Remark. Theorems 1 and 2 have two very easy mild generalizations: Firstly, fix an ideal cCA prime to n: Then we can construct Heegner points corresponding to the orders Oc;n ¼ A þ cpn OK ; for which similar results hold. Secondly, let w : G0 -f71g be a character, and let Kw ½pN be the subfield of K½pN corresponding to kerðwÞ: Note that Kw ½pN ¼ H½pN if w is trivial. Then we may replace the G0 -trace on E by the ðG0 ; wÞ-trace X wðsÞxs ; TrG0 ;w : EðK½pN Þ-EðKw ½pN Þ; x/ sAG0
and obtain a similar result. We leave the details to the dedicated reader.
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Acknowledgments The author would like to thank Ernst-Ulrich Gekeler for help with Proposition 4.1, and Christophe Cornut for showing him that the yn ’s actually generate a group of infinite rank.
References [1] F. Breuer, The Andre´-Oort conjecture for products of Drinfeld modular curves, preprint available at http://arxiv.org/abs/math.NT/0303038. [2] M.L. Brown, On a conjecture of Tate for elliptic surfaces over finite fields, Proc. London Math. Soc. (3) 69 (1994) 489–514. [3] C. Cornut, Mazurs’s Conjecture on higher Heegner points, Invent. Math. 148 (3) (2002) 495–523. [4] C. Cornut, Non-trivialite´ des points de Heegner, C. R. Acad. Sci. Paris, Ser. I 334 (12) (2002) 1039–1042. [5] S.J. Edixhoven, Special points on products of modular curves, preprint available at http://arxiv.org/ abs/math.NT/0302138. [6] E.-U. Gekeler, Automorphe Formen u¨ber Fq ðTÞ mit kleinem Fu¨hrer, Abh. Math. Sem. Univ. Hamburg 55 (1985) 111–146. [7] E.-U. Gekeler, Drinfeld Modular Curves, Lecture Notes in Mathematics, Vol. 1231, Springer, Berlin, Heidelberg, 1986. [8] E.-U. Gekeler, Improper Eisenstein series on bruhat-Tits trees, Manuscripta Math. 86 (1995) 367–391. [9] E.-U. Gekeler, Analytical construction of Weil curves over function fields, J. Th. Nomb. Bordeaux 7 (1995) 27–49. [10] E.-U. Gekeler, M. Reversat, Jacobians of Drinfeld modular curves, J. Reine Angew. Math. 476 (1996) 27–93. [11] I. Longhi, Non-Archimedean integration and elliptic curves over function fields, J. Number Theory 94 (2002) 375–404. [12] B. Mazur, Modular curves and arithmetic, Proceedings of the International Congress of Mathematicians, Warszawa, 1983. [13] J. Neukirch, Algebraische Zahlentheorie, Springer, Berlin, Heidelberg, 1992. [14] A. Pa´l, Drinfeld modular curves, Heegner points and interpolation of special values, Thesis, Columbia University, 2000. [15] H.-G. Ru¨ck, U. Tipp, Heegner points and L-series of automorphic cusp forms of Drinfeld type, Doc. Math. 5 (2000) 365–444 (electronic). [16] D. Ulmer, Elliptic curves and analogies between number fields and function fields, in: Heegner points and L-series, MSRI Publications 48, to appear. Preprint available at http://arxiv.org/abs/math.NT/ 0305320. [17] V. Vatsal, Uniform distribution of Heegner points, Invent. Math. 148 (1) (2002) 1–46. [18] J.F. Voloch, Diophantine approximation on abelian varieties in characteristic p; Amer. J. Math. 117 (4) (1995) 1089–1095.
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http://www.elsevier.com/locate/jnt
On an additive representation function R. Balasubramanian and Gyan Prakash The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India Received 21 March 2003; revised 23 July 2003 Communicated by D. Goss
Abstract Let A be an infinite subset of natural numbers, nAN and X a positive real number. Let rðnÞ denotes the number of solution of the equation n ¼ a1 þ a2 where a1 pa2 and a1 ; a2 AA: Also let jAðX Þj denotes the number of natural numbers which are less than or equal to X and belong to A: For those A which satisfy the condition that for all sufficiently large natural numbers n we have rðnÞa1; we improve the lower bound of jAðX Þj given by Nicolas et. al. [NRS98]. The bound which we obtain is essentially best possible. r 2003 Elsevier Inc. All rights reserved. Keywords: Addition of sequence; Additive representation function
Let N denote the set of all natural numbers. If A is an infinite subset of N then we set AðxÞ ¼ fapx : aAAg: Let rðA; nÞ denote the number of solutions of the equation n ¼ ai þ a j ;
where ai paj ; ai ; aj AA:
Here and in what follows A will always denote an infinite subset of N such that there exists a natural number n0 ðAÞ such that rðA; nÞa1 for nXn0 ðAÞ:
Corresponding author. Fax: +91-44-22541934. E-mail addresses: [email protected] (R. Balasubramanian), [email protected] (G. Prakash).
0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.09.004
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Also a0 ðAÞ shall denote the least natural number such that a0 ðAÞAA and a0 ðAÞXn0 ðAÞ: Regarding such sets, Nicolas et al. [NRS98] proved the following theorem: Theorem. If A is an infinite subset of N such that rðA; nÞa1 for all sufficiently large natural numbers n; then ln ln x 3=2 1 lim sup jAðxÞj X : ln x 20 They also gave an example of a set A such that rðA; nÞa1 for all sufficiently large natural numbers n and jAðxÞj5ðln xÞ2 : In this paper we shall show the following: Theorem 1. There exists an absolute constant c40 with the following property: for any infinite subset A of N such that rðA; nÞa1 for all sufficiently large natural numbers n; then jAðxÞjXc
2
ln x ln ln x
for all x sufficiently large:
Theorem 1 follows from Proposition 2 by noting that if Y is sufficiently large then 1
for some positive absolute constant c the interval ½ðY Þ2 ; Y Þ contains at least cðlnlnlnYY Þ disjoint intervals of the form ½b; bðln Y Þ11 Þ: Apart from the arguments used in proving Lemma 4, the rest of arguments used in this paper are as in [NRS98]. Lemma 4 improves inequality (1) of Proposition 1 and this improves result of Proposition 1 and gives Proposition 2. As remarked above Theorem 1 is an immediate corollary of Proposition 2. The sequence BY constructed in Lemma 3 is a slight modification of analogous sequence constructed in [NRS98] (see [NRS98, p. 304]). Lemma 1. For all real numbers x4a0 ðAÞ the interval ðx; 2x contains an element of the set A: Proof. Let a be the largest element of A not exceeding x: Then aXa0 ðAÞ so that the integer n ¼ a þ a is 4n0 ðAÞ: It now follows that there is a pair ðc; dÞ; with cpd; of elements of A distinct from the pair ða; aÞ such that n ¼ c þ d: Since dXc this implies that d4a whence d4x by the choice of a: Clearly we have dpn ¼ a þ ap2x: In summary, we have verified that the element d of A lies in ðx; 2x : & l-good interval: An interval I ¼ ½k; k þ l is defined to be l-good if I-A ¼ fk þ lg; that is it is of length l; the last element is in A and no other element is in A:
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Lemma 2. Let Y be a sufficiently large real number and jAðY Þjpðln Y Þ2 : Then for any real number b such that 1pbp2ðlnYY Þ10 there exists b good interval in ½bðln Y Þ5 ; 2bðln Y Þ10 : Proof. We consider interval C ¼ ½bðln Y Þ5 ; bðln Y Þ10 : Then the length of C is at least 12 bðln Y Þ10 for all Y sufficiently large, but jC-AjpjAðY Þjpðln Y Þ2 o12 ðln Y Þ10 : Therefore, there exists in C a closed interval I of length b and void of A: Moving I to right till it hits A; we get a b-good interval I 0 : Using Lemma 1 it follows that I 0 C½bðln Y Þ5 ; 2bðln Y Þ10 : & Lemma 3. Let Y be a sufficiently large real number and jAðY Þjpðln Y Þ2 : Then there pffiffiffiffi exists an increasing sequence fb1 ; b2 ; y; bm g ¼ BY of elements of A not exceeding Y and satisfying the following properties: (I) For each 1pipm 1; biþ1 Xbi ðln Y Þ5 : (II) For each 1pipm 1; ½biþ1 bi ; biþ1 Þ does not contain an element of the set A: (III) The number of terms m of sequence BY is at least clnlnlnYY where c is a positive absolute constant. Proof. We shall define BY ¼ fb1 ; b2 ; y; bi ; yg recursively. We set b1 ¼ a0 ðAÞ: pffiffiffi Suppose b1 ; b2 ; y; bi have been determined and bi p12 ðln YYÞ10 then applying Lemma 2 we choose the smallest aA½bi ðln Y Þ5 ; 2bi ðln Y Þ10 such that ½a bi ; aÞ does not contain any element of A: We set biþ1 to be a: The recursion is terminated if pffiffiffi bi 412 ðln YYÞ10 : Let BY be a sequence constructed in manner described above. Clearly, (I) and (II) hold for each 1pipm 1: Further for each i we have that biþ1 p2bi ðln Y Þ10 obi ðln Y Þ11 whence by induction bm oa0 ðAÞðln Y Þ11m : Since recursion terminates at bm we have pffiffiffi bm 412 ðln YYÞ10 : These remarks imply (III). & In what follows, BY will denote the sequence constructed as in the proof of Lemma 3. Proposition 1. Let Y be a sufficiently large real number and jAðY Þjpðln Y Þ2 : Let b be pffiffiffiffi any real number such that ½b; bðln Y Þ11 ÞC Y ; Y : Then the number of elements of A
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contained in the interval ½b; bðln Y Þ11 Þ is 4cðlnlnlnYY Þ2 where c is a positive absolute constant. Proof. Lemma 2 implies that there is an element a of the set A lying in the interval ½b; 3bðln Y Þ10 Þ such that the interval ½a b; aÞ does not contain any element of A: We choose one such a: Let S denote the set of elements of A in the interval ½b; bðln Y Þ11 Þ and s denote the cardinality of S: Let S1 and S2 denote the sets of elements of A in the intervals ½b; aÞ and ½a; bðln Y Þ11 Þ; respectively, and let s1 and s2 denote the cardinalities of S1 and S2 ; respectively. We then have s ¼ s1 þ s2 : For each i; 1pipm; let ni ¼ a þ bi ; where fbi g ¼ BY is the sequence supplied by Lemma 3. Since each ni XY 1=2 ; we see that when Y is sufficiently large, each ni is Xn0 ðAÞ: For each i we then choose a pair ðci ; di Þ; with di Xci ; of elements of A distinct from the pair ða; bi Þ such that ni ¼ ci þ di : For each i we then have either di oa or di 4a: Let P1 denote the set of those pairs ðci ; di Þ with di oa and P2 the set of those pairs ðci ; di Þ with di 4a: Let p1 and p2 denote the cardinalities of P1 and P2 ; respectively. We then have p1 þ p2 ¼ m: If ðci ; di Þ is in P1 we have ci pdi oa b and hence that di Xci ¼ a þ bi di Xa ða bÞ ¼ b: In other words, ci and di are elements of S1 : It follows that S1 S1 contains P1 : Consequently, we have that s21 Xp1 or that 1=2
ð1Þ
s1 Xp1 : If ðci ; di Þ is in P2 we have aodi : Further, we have that 1
ci þ di ¼ a þ bi p3bðln Y Þ10 þ Y 2 p3bðln Y Þ10 þ bpbðln Y Þ11
ð2Þ
and hence that di pbðln Y Þ11 : It follows that the mapping f that associates ðci ; di Þ to di maps P2 into S2 : Let us verify that f is injective. Suppose to the contrary that ðci ; di Þ and ðcj ; dj Þ are elements of P2 such that di ¼ dj and ioj: Then cj Xcj ci ¼ bj bi Xbj bj 1 :
ð3Þ
Also cj obj because cj þ dj ¼ a þ bj and dj 4a: It follows that the element cj of A lies in the interval ½bj bj 1 ; bj Þ contradicting (ii) of Lemma 3. The injectivity of f implies that s2 Xp2 : In summary we have verified that 1=2
1=2
1=2
s ¼ s1 þ s2 Xp1 þ p2 Xp1 þ p2 Xðp1 þ p2 Þ1=2 Xm1=2 from which the proposition follows on recalling (III) of Lemma 3.
ð4Þ &
Corollary. There exists an absolute constant c40 with the following property: For any infinite subset A of N such that rðA; nÞa1 for all sufficiently large natural numbers n;
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we have:
ln Y jAðY ÞjXc ln ln Y
3 2
:
Proof. The corollary follows from Proposition 1 on noting that if Y is sufficiently 1
large then for some positive absolute constant c the interval ½ðY Þ2 ; Y contains at least c lnlnlnYY disjoint intervals of the form ½b; bðln Y Þ11 Þ: & Result in Proposition 1 can be improved and we have Proposition 2. Rest of arguments being the same, Proposition 2 follows by improving inequality (1) in Proposition 1 using Lemma 4. We shall first just state Lemma 4 and deduce Proposition 2. Later we shall prove Lemma 4 which require a few other lemmas. Lemma 4. With notations and assumptions as in Proposition 1 we have j½b; aÞ-AjXcjP1 j; where c is a positive absolute constant. Proposition 2. Let Y be a sufficiently large real number and jAðY Þjpðln Y Þ2 : Let b be pffiffiffiffi any real number X1 such that ½b; bðln Y Þ11 ÞC Y ; Y : Then the number of elements of A contained in the interval ½b; bðln Y Þ11 Þ is 4cðlnlnlnYY Þ where c is an positive absolute constant. Proof. Notice that assumptions of Propositions 1 are satisfied here. Then arguing as in proof of Proposition 1 and using Lemma 4 in place of inequality (1) Proposition follows. & Lemmas 5 and 7 are required for proving Lemma 4. Lemma 5. P Let BY ¼ fb1 ; b2 ; y; bm g be a sequence as constructed in Lemma 3. Suppose ni¼1 xi bi ¼ 0 where 1pnpm and xi Af1; 1; 0; 2; 2g for all 1pipn: Then xi ¼ 0 for all i: P Proof. Suppose it is not true and there exist sequence fxi g such that ni¼1 xi bi ¼ 0 where 1pnpm and xi is not zero for some i: Without loss of generality we may assume that xn a0: Then, xn bn ¼
n 1 X
xi bi :
i¼1
As nomojAðY Þjoðln Y Þ2 so bn pjxn bn jo2ðln Y Þ2 bn 1 : But by construction of BY ; bn Xðln Y Þ5 bn 1 : Hence there is a contradiction. &
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Let us recall some definitions from graph theory which we need for our purpose. A graph G consists of a finite nonempty set V ¼ V ðGÞ of vertices together with a prescribed set X of unordered pairs of elements of V : Each pair x ¼ fu; vg is an edge of G and is said to join u and v: Notice that a graph thus defined is a finite undirected graph without multiple edges but may have loops. A walk of a graph G is an alternating sequence of vertices and edges v1 ; x1 ; v2 ; y; vn 1 ; xn 1 ; vn ; beginning and ending with vertices, in which each edge joins two vertices immediately preceding and following it. It is closed if v1 ¼ vn : It is a trail if all the edges are distinct. By an even closed trail we shall mean a trail which is closed and have even number of edges. A cycle is a closed trail in which all the vertices are distinct. Two trails which define same subgraph are considered equivalent and are not distinguished. Lemma 6. Let G be a graph with no loops and no even closed trails. Then any two distinct closed trails in G are disjoint, that is, if T1 and T2 are two distinct closed trails in G; and V ðT1 Þ and V ðT2 Þ denote the set of vertices in T1 and T2 ; respectively, then V ðT1 Þ-V ðT2 Þ ¼ |: Proof. Suppose it is not true. Then there exist two distinct closed trails T1 and T2 in G such that V ðT1 Þ-V ðT2 Þ ¼ Vc (say) a|: As T1 and T2 are two distinct trails so there is an edge in at least one of them which is not common to both of them. Let say x is one such edge and without loss of generality we may assume it is in T1 : Suppose T1 ¼ v1 ; x1 ; v2 ; x2 ; v3 ; y; vi ; xi ; viþ1 ; y; vn 1 ; xn ; vn : As Vc a| so we may assume that v1 ¼ vn AVc : Then if we choose vl AV ðT2 Þ nearest to left of x and vr AV ðT2 Þ nearest to right, in sequence for T1 thus considered, then only vertices which T ¼ vl ; xl ; vlþ1 ; y; vr 1 ; xr 1 ; vr share with T2 are vl and vr : (It is possible that vl is same as vr :) Also then by choice of x; vl ; vr the trail T does not have any common edge with T2 : As vl ; vr AV ðT2 Þ so there is a trail T 00 in T2 starting from vl and ending with vr : Now by choice of T we have that Tu ¼ T,T 00 is a closed trail. Also again by choice of T we have that Tr ¼ ðT2 \T 00 Þ,T is another closed trail. (Notice that it may be so that ðT2 \T 00 Þ is empty but that does not affect our arguments.) Now it is clear that either Tu or Tr has an even number of edges depending on whether number of edges of T and T 00 have same parity or different parity. But this is contrary to the assumption that G has no even closed trail. & Lemma 7. Let G be a graph with n vertices and having no loops. Further assume that G has no even closed trail. Then number of edges in G; say eðGÞ; is at most 2n: Proof. It is clearly enough to prove lemma in case when G is connected. From previous lemma no closed trail in G has a proper closed sub-trail. This implies that any closed trail is a cycle and any two cycles are disjoint. So G cannot have more than n cycles. Now, we shall show that dðGÞ ¼ eðGÞ number of vertices is at most n and this proves the lemma. If we shrink all cycles in G to get new graph G 0 then G 0 has no cycle and is connected. So G0 is a tree. But then dðG0 Þ ¼ 1: Also as cycles in G are disjoint so dðGÞ ¼ dðG 0 Þþ number of cycles in G: This implies that dðGÞpn 1: &
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Now we shall prove Lemma 4. Lemma 4. With notations and assumptions as in Proposition 1 we have j½b; aÞ-AjXcjP1 j; where c is a positive absolute constant. Proof. From Proposition 1 we recall that the set P1 consists of pairs ðcj ; dj Þ of elements of the set A such that cj pdj oa: Also for each pair ðcj ; dj Þ belonging to the set P1 there is exactly one term bj of the sequence BY such that cj þ dj ¼ a þ bj : Let S1 denote the set of elements of A lying in interval ½b; aÞ; that is, S1 ¼ ½b; aÞ-A: Then it was shown in Proposition 1 that P1 CS1 S1 : We shall construct a graph G associated to the set P1 : As P1 CS1 S1 we define f1 : P1 -S1 and f2 : P1 -S1 by f1 ðci ; di Þ ¼ ci ; f2 ðci ; di Þ ¼ di : The set of vertices of graph G; let say V ; consists of those elements v of S1 such that either v belongs to image of f1 or of f2 : Then we have following upper bound on number of vertices of G: jV j ¼ npjImage of f1 j þ jImage of f2 jp2jS1 j ¼ 2j½b; aÞ-Aj:
ð5Þ
The set of edges of G (say X ) consists of those unordered pair fv1 ; v2 g of V such that either ðv1 ; v2 Þ or ðv2 ; v1 ÞAP1 : In other words two vertices v1 and v2 are joined by an edge if and only if either ðv1 ; v2 Þ or ðv2 ; v1 ÞAP1 : The graph G thus constructed satisfy following properties: (I) There is a natural one–one correspondence between edges of G and elements of P1 : (II) If x is an edge in G joining vertices v1 and v2 then there is a term bx in the sequence BY such that v1 þ v2 ¼ a þ bx : (III) For two distinct edges x and y; the corresponding bx and by given as above are distinct. All these properties are easily verified using definition of G and P1 : So (I) in particular implies that number of edges in G is same as number of elements in P1 : Then to prove the lemma it is enough to show that number of edges in G ¼ eðGÞpcn for some positive absolute constant c: ð6Þ Now G can have at most n loops. So if we remove all loops from G to get another graph G1 then to show (6) it is enough to show that eðG1 Þpcn for some positive absolute constant c: We claim that G1 does not have any even closed trail. Then using claim and Lemma 7 we have (6).
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Suppose claim is not true and G1 has an even closed trail T ¼ v1 ; x1 ; v2 ; x2 ; v3 ; y; vi ; xi ; viþ1 ; y; v2m 1 ; x2m 1 ; v2m ; x2m ; v1 ; where vi is a vertex of G and xi is an edge joining vertices immediately preceding and following it. Also by definition of trail we have, for 1pi; jp2m and iaj; xi axj : Then using property (II) of G we have vi þ viþ1 ¼ a þ bi ;
where 1pip2m 1;
v2m þ v1 ¼ a þ b2m ; where bi Afbi g for all 1pip2m: Further using property (III) of G it follows that for 1pi; jp2m and iaj we have bi abj : Now we have 2m 1 X
ð 1Þi ðvi þ viþ1 Þ ¼
i¼1
2m 1 X
ð 1Þi ða þ bi Þ;
ð7Þ
i¼1
v2m þ v1 ¼ a þ b2m :
ð8Þ
Adding (7) and (8) we get 0¼
2m X
ð 1Þi bi
i¼1
which is a contradiction to Lemma 5. &
References [NRS98] J.-L. Nicolas, I.Z. Ruzsa, A. Sa´rko¨zy, On the parity of additive representation functions, J. Number Theory 73 (1998) 292–317.
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On the p-adic Riemann hypothesis for the zeta function of divisors Daqing Wana,b,,1 and C. Douglas Haessigb,2 b
a Institute of Mathematics, Chinese Academy of Sciences, Beijing, PR China Department of Mathematics, University of California, Irvine, CA 92697-3875, USA
Received 31 March 2003; revised 2 June 2003 Communicated by D. Goss
Abstract In this paper, we continue the investigation of the zeta function of divisors, as introduced by the first author in Wan (in: D. Jungnickel, H. Niederreiter (Eds.), Finite Fields and Applications, Springer, Berlin, 2001, pp. 437–461; Manuscripta Math. 74 (1992) 413), for a projective variety over a finite field. Assuming that the set of effective divisors in the divisor class group forms a finitely generated monoid, then there are four conjectures about this zeta function: p-adic meromorphic continuation, rank and pole relation, p-adic Riemann hypothesis, and simplicity of zeros and poles. This paper proves all four conjectures when the Chow the group of divisors is of rank one. Also, an example with higher rank is provided where all four conjectures hold. r 2003 Elsevier Inc. All rights reserved. Keywords: P-adic Riemann hypothesis; Zeta function of algebraic cycles; Zeta function of divisors; Riemann-Roch problem; Effective cone; Newton polygon
1. Introduction In this paper, we continue the investigation of the zeta function of divisors introduced in [6] for a projective variety X defined over a finite field Fq of q elements of characteristic p: Assume that the set of effective classes in the divisor class group
Corresponding author. Chinese Academy of Sciences, Institute of Mathematics, Beijing, China. E-mail addresses: [email protected] (D. Wan), [email protected] (C.D. Haessig). 1 Partially supported by the NSF and the NNSF of China (10128103). 2 Partially supported by the ARCS Foundation. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.08.008
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forms a finitely generated monoid, then there are four standard conjectures about this zeta function: p-adic meromorphic continuation, rank and order relation, p-adic Riemann hypothesis, and the simplicity of zeros. When the divisor class group is of rank one, the first two conjectures were proved in [6]. The aim of this paper is to prove the remaining two conjectures under the same assumption. We shall also give an example which provides evidence for the validity of all four conjectures in the higher rank case. As initiated in [6], we derive the p-adic meromorphic continuation of these zeta functions via a Riemann–Roch approach. After this, we use the theory of Newton polygons to investigate the zeros, specifically the p-adic Riemann hypothesis and the simplicity conjecture. Let us start by recalling how analytic information about a zeta function leads to arithmetic information. For example, if Nk denotes the number of Fqk -rational points on X ; then we know from Dwork’s theorem that the associated zeta function is rational: Z0 ðX ; TÞ ¼ exp
N X k¼1
! k
Nk T =k
Q ð1 ai TÞ : ¼Q ð1 bj TÞ
Taking the logarithmic derivative of both sides and equating coefficients, we immediately obtain a formula for Nk in terms of the (reciprocal) zeros and poles of the zeta function X X bkj aki : Nk ¼ P k For a curve of genus g defined over Fq ; we have Nk ¼ qk þ 1 2g i¼1 ai where 1 and q are the (reciprocal) poles of the zeta function and the ai are the (reciprocal) zeros. Thus rationality, and more generally, meromorphic continuation implies a nice formula (in terms of the zeros and poles) for the number of objects the zeta function is counting. This is a general principle. For another example, the meromorphic continuation of the Riemann zeta function means there is a formula, in terms of zeros and poles, for the number of primes in an interval ½0; t; which led to the prime number theorem. Similarly, the conjectural meromorphic continuation of the Hasse– Weil zeta function would yield a formula, in terms of zeros and poles, for the number of closed points on an arithmetic scheme with norm at most t; see [8]. The zeta function of algebraic r-cycles introduced in [6] is concerned with a formula for the number of irreducible subvarieties of a fixed dimension r: This zeta function will not be complex analytic in general, unlike the above classical zeta functions. However, the conjectural p-adic meromorphic continuation immediately yields a p-adic formula in terms of the p-adic zeros and poles of this zeta function as we shall see below. Let Fq be a finite field with q elements, q a power of a prime p: Let X be a projective n-dimensional integral scheme defined over Fq : Let 0prpn be an integer. A prime r-cycle of X is an r-dimensional closed integral subscheme of X defined over Fq : An r-cycle on X is a formal finite linear combination of prime r-cycles. An r-cycle P is called effective, denoted ni Pi X0; if each ni X0:
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Each prime r-cycle P has an associated graded coordinate ring "N k¼0 Sk ðPÞ since X is projective. By a theorem of Hilbert–Serre, for all k sufficiently large, we have dimFq Sk ðPÞ equal to a polynomial ar kr þ ðlower termsÞ: Define the degree of P; denoted degðPÞ; as r! times the leading P coefficient P ar : We extend the definition of degree to arbitrary r-cycles by degð ni Pi Þ :¼ ni degðPi Þ: Defining the degree allows us to measure and compare the prime r-cycles. Define the zeta function of algebraic r-cycles on X as Y Zr ðX ; TÞ :¼ ð1 T degðPÞ Þ1 ; P
where the product is taken over all prime r-cycles P in X : Denote the set of all effective r-cycles of degree d on X by Er;d ðX Þ: A theorem of Chow and van der Waerden states that this set has the structure of a projective variety. Since we are over a finite field, Er;d ðX Þ is finite. This means that Zr ðX ; TÞ is a well-defined element of 1 þ TZ½½T; and so, converges p-adically in the open unit disk jTjp o1: Equivalent forms of this zeta function are Zr ðX ; TÞ ¼
N X
#Er;d ðX ÞT d
d¼0
¼
N Y
ð1 T d ÞNd
d¼1
! N X Tk Wk ; ¼ exp k k¼1 P where Nd is the number of prime r-cycles of degree d and Wk :¼ djk dNd is the weighted number of prime r-cycles of degree dividing k; each prime r-cycle of degree d is counted d times. The p-adic meromorphic continuation of Zr ðX ; TÞ would imply the complete p-adic factorization Q ð1 ai TÞ ; Zr ðX ; TÞ ¼ Q ð1 bj TÞ where the products are now infinite with ai -0 and bj -0 in Cp : Again, taking the logarithmic derivative, we obtain a formula for Wd in terms of infinite series: X X bdj adi : Wd ¼ Thus, the p-adic meromorphic continuation implies a well-structured formula for the sequence Wk : By Mo¨bius inversion, this gives a well-structured formula for the sequence Nk as well. If Zr ðX ; TÞ is p-adic meromorphic, we can adjoin all the reciprocal zeros ai ’s and all the reciprocal poles bj ’s to Qp : The resulting field extension of Qp is called the
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splitting field of Zr ðX ; TÞ over Qp : This splitting field is automatically a Galois extension (possibly of infinite degree) over Qp by the Weierstrass factorization of Zr ðX ; TÞ over Qp and the fact that we are in characteristic zero. Now, if r ¼ 0; then Zr ðX ; TÞAQðTÞ is rational and it satisfies the Riemann hypothesis by the Weil Conjectures. If r ¼ n ¼ dimðX Þ; then Zn ðX ; TÞ is trivially rational, its zeros and poles are roots of unity, and thus satisfies the Riemann hypothesis as well. In particular, Zr ðX ; TÞ is well understood if np1: So, we will assume n ¼ dimðX ÞX2 and 1prpn 1: Let CHr ðX Þ be the Chow group of r-cycles on X ; that is, the free abelian group generated by the prime r-cycles on X modulo the rational equivalence. Let EffConer ðX Þ be the set of effective r-cycle classes in CHr ðX Þ: It is conjectured that CHr ðX Þ is a finitely generated abelian group; note that our base field is a finite field. This is known if r ¼ n 1: In general, EffConer ðX Þ may not be a finitely generated monoid. The following conjectures only apply to those X for which EffConer ðX Þ is a finitely generated monoid. It is an interesting but independent question to determine when EffConer ðX Þ is a finitely generated monoid. Assumption. Assume that EffConer ðX Þ is a finitely generated monoid. Then, we have the following conjectures. Conjecture I (p-adic meromorphic continuation). Zr ðX ; TÞ is p-adic meromorphic in T: Conjecture II (Order and rank). ordT¼1 ðZr ðX ; TÞÞ ¼ rank CHr ðX Þ: Conjecture III (p-adic Riemann hypothesis). The splitting field of Zr ðX ; TÞ over Qp is a finite extension of Qp : Conjecture IV (Simplicity of zeros and poles). All zeros and poles, except for finitely many, are simple. A slightly weaker version of the p-adic Riemann hypothesis says that the slopes of the zeros and poles are rational numbers with bounded denominator, that is, the splitting field of Zr ðX ; TÞ over Qp has finite ramification degree. For a characteristic p analogue of this type of Riemann hypothesis, see [2,7]. The above stronger formulation of the p-adic Riemann hypothesis is motivated by Goss’ corresponding formulation for the characteristic p L-function. Note that in the characteristic p case, the infinite primes need to be taken into account; see [3] for the new reformulation. In the present p-adic case, the infinite primes do not enter into the picture, and so the original formulation in [2] does seem to be the right one to look at. As we shall see in this paper, some positive results supporting the above conjectures come from the case when r ¼ dimðX Þ 1; the divisor case. We call ZdimðX Þ1 ðX ; TÞ the zeta function of divisors. We have Theorem 1. Let X be a normal, connected scheme over Fq of dimension nX2: Assume the rank of CHn1 ðX Þ is one. Then all four conjectures listed above are true for Zn1 ðX ; TÞ:
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This extends Theorem 4.1 in [6] which had proven Conjectures I and II under the same rank-one assumption. Evidence to support Conjectures I and II in the higher rank case is also given in [6, Theorem 6.1]. This states that when the effective cone is a finitely generated monoid, then the zeta function of divisors is p-adic meromorphic on the closed unit disk and satisfies Conjecture II. In this paper, we also give an example which satisfies all four conjectures in the higher-rank case. This is the first higher-rank example for which Conjectures III and IV are proved. We have Theorem 2. Consider the zeta function of divisors for the quadric surface xw ¼ zy in P3Fq : Then all four conjectures hold, except possibly the simplicity conjecture in the case p ¼ 2: The method used to investigate the zeta function of algebraic cycles comes from rewriting the zeta function in terms of cycle classes. We then need to estimate the number of effective cycles in a cycle class. In the divisor case, these are linear equivalence classes, and counting effective divisors in a linear equivalence class comes down to knowing the dimension of a certain vector space. See Section 2 for a more precise description of the method. Finally, we use the theory of Newton polygons to study the p-adic Riemann hypothesis and the simplicity of zeros. See Section 3 for a description of this connection.
2. The generalized Riemann–Roch problem A general strategy for investigating the zeta function of divisors comes from a generalized Riemann–Roch problem. In this section, this strategy is recalled and the generalized Riemann–Roch problem is easily solved for the rank-one case. In the case when we are not dealing with codimension one cycles a similar method has been proposed in [6, Section 7], but essentially nothing could be proved. Let X be a normal, connected scheme over Fq of dimension nX2: Let CHn1 ðX Þ denote the Chow group of ðn 1Þ-cycles and define the effective cone, denoted EffConeðX Þ; as the effective divisor classes in CHn1 ðX Þ: The effective cone helps in determining properties of the zeta function by gathering divisors of the same degree; recall, the degree depends only on the divisor class. That is, we may rewrite the zeta function of divisors as X Zn1 ðX ; TÞ ¼ MD T degðDÞ ; DAEffConeðX Þ
where MD is the number of effective divisors linearly equivalent to D: The numbers MD are studied using the following Fq -vector space for investigating. Denote the function field of X by Fq ðX Þ: Let D be a divisor on X : Define LðDÞ :¼ f f AFq ðX Þj f ¼ 0 or divð f Þ þ DX0g:
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Note, LðDÞ is a finite-dimensional Fq -vector space; we will denote its dimension by lðDÞ: Proposition 1. Consider the projective space obtained from LðDÞ\f0g modulo the equivalence: f Bg if there is a lAFq so that f ¼ lg: This space is in one-to-one correspondence with the set MD : See [6, Lemma 4.2] for a proof. Thus, we may conclude that Zn1 ðX ; TÞ ¼
X DAEffConeðX Þ
qlðDÞ 1 degðDÞ T : q1
As demonstrated in Section 6 of [6], when the effective cone is finitely generated, by using a simplicial decomposition D of the effective cone, one may reduce the zeta function to functions of the form ZD ðX ; TÞ ¼
X a1 ;y;at X0
qlðEþa1 D1 þ?þat Dt Þ 1 degðEþa1 D1 þ?þat Dt Þ T : q1
This decomposition focuses our attention on the behaviour of lðE þ a1 D1 þ ? þ at Dt Þ as a1 þ ? þ at tends to infinity. Simply understanding lðkDÞ as k tends to infinity is difficult; this is known as the Riemann–Roch problem. See [4, Problem II.7.6]. Zariski worked on this problem for surfaces in [9]; also see [1]. We state this as a general question which may be considered a generalized Riemann–Roch problem. Question. For effective divisors E and D1 ; y; Dt ; what is the behaviour of lðE þ a1 D1 þ ? þ at Dt Þ as a1 þ ? þ at tends to infinity? We will need to understand this behaviour when CHn1 ðX Þ is of rank one. In this case, every divisor of positive degree is ample and the following proposition provides us with the needed understanding. Proposition 2. Let D be an ample divisor and D0 an arbitrary divisor. Then for all k sufficiently large, lðD0 þ kDÞ is a polynomial in k of degree n ¼ dimðX Þ; with leading coefficient Dn =n! where Dn is the self-intersection number of D: Proof. From Riemann–Roch, we know X iX0
ð1Þi dim H i ðX ; D0 þ kDÞ ¼
Dn n k þ ðlower terms in kÞ: n!
Since D is ample, H i ðX ; D0 þ kDÞ ¼ 0 for all iX1 and k sufficiently large. The proposition follows since H 0 ðX ; D0 þ kDÞ ¼ LðD0 þ kDÞ: &
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3. p-adic Riemann hypothesis In this section, we discuss a method used to investigate the p-adic Riemann hypothesis using Newton polygons. Once we know that Zr ðX ; TÞ is p-adic meromorphic, we may write it as a quotient of p-adic entire functions, that is, PN i i¼0 ai T Zr ðX ; TÞ ¼ PN j j¼0 bj T P with ai ; bj AZp and a0 ¼ b0 ¼ 1: The Newton polygon of iX0 ai T i is the convex hull of the points ði; ordq ðai ÞÞ: Since this series defines an entire function, the Newton polygon grows faster than any linear function; in particular, there are no infinitely long line segments. Newton polygons are exciting because they encode arithmetic P information about the zeros of the series ai T i : Specifically, if there is a line segment of horizontal length h and slope s; then there will be exactly h reciprocal zeros of this series having q-adic order s: Now, if the horizontal lengths of the line segments creating the Newton polygon are bounded (in particular, the slopes have bounded denominator), then by the padic Weierstrass preparation theorem, we may write X Y ai T i ¼ fj ðTÞ; iX0
j
where each fj A1 þ TZp ½T; degð fj Þ is uniformly bounded and the reciprocal zeros of fj ðTÞ approach zero as j grows. Let us now relate this back to the p-adic Riemann hypothesis. Recall, the p-adic P P Riemann hypothesis states that the splitting field of iX0 ai T i and jX0 bj T j over Qp is a finite extension of Qp : This finitness is a consequence of the above bounded degree factorization and the following finitness lemma. Lemma 1. Let d be a positive integer. Let EðdÞ denote the set of all f AQp ½T with degð f Þpd: Let K be the field extension obtained by adjoining to Qp every zero of every f AEðdÞ: Then ½K : Qp oN: The proof of the lemma follows easily from the fact that for each positive integer n; there are only finitely many extensions of Qp of degree n in a fixed algebraic closure of Qp ; see p. 132 of [5].
4. The rank-one case In this section, we prove that when CHn1 ðX Þ is of rank one, then all four conjectures hold true for Zn1 ðX ; TÞ:
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Theorem 1. Let X be a normal, connected projective scheme over Fq of dimension nX2: Assume CHn1 ðX Þ is of rank one. Then all four conjectures hold true. That is, Zn1 ðX ; TÞ is p-adic meromorphic, ordT¼1 Zn1 ðX ; TÞ ¼ 1; the p-adic Riemann hypothesis holds, and almost all zeros and poles are simple. Remark. Conjectures I and II of this theorem were already proven in [6]. The proof below of this theorem refines the arguments given there. It will also prove another result from [6] about the special value of Zn1 ðX ; TÞ at T ¼ 1: We state this as a corollary. Corollary 1. Let h be the number of torsion elements in CHn1 ðX Þ: With m :¼ minfdegðDÞjD a divisor of X ; degðDÞ40g; we have Res Zn1 ðX ; TÞjT¼1 ¼
h : mðq 1Þ
Proof. First, notice that the degree map deg : CHn1 ðX Þ-Z sends any torsion element to zero. Also, the image of deg is Zm for some positive integer m: Choose DACHn1 ðX Þ such that degðDÞ ¼ m: Then, CHn1 ðX Þ ¼ ZD"fD1 ; y; Dh g where CHn1 ðX Þtors ¼ fD1 ; y; Dh g: By construction, if m does not divide d; then there are no effective divisors of degree d: That is, all divisors must have degree d ¼ km for some k: Let Md be the number of effective divisors of degree d :¼ km: Then, using Proposition 1, Md ¼ #Er;d ðX Þ ¼
X
1¼
D0 X0 degðD0 Þ ¼ km ¼
h X
X
i¼1
D0 X0
1
D0 BDi þ kD
h X qlðDi þkDÞ 1 : q1 i¼1
Now, D is ample since X is projective and CHn1 ðX Þ has rank one. So, we may bring in Proposition 2. This tells us that for each torsion element Di ; for all k sufficiently large, lðDi þ kDÞ :¼ dimFq LðDi þ kDÞ
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will agree with a degree nX2 polynomial in k with leading coefficient Dn =n!: Consequently, we see Zn1 ðX ; TÞ ¼
N X
Md T d
d¼0 N X h X qlðDi þkDÞ 1 mk T ¼ q1 k¼0 i¼1
¼
N 1 X h qlðDi þkDÞ T mk : q 1 ðq 1Þð1 T mÞ i¼1 k¼0 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} h X
rational
p-adic entire
From this, we deduce that Zn1 ðX ; TÞ is p-adic meromorphic and satisfies the order and rank conjecture. Further, we see that the residue at T ¼ 1 is as mentioned in the corollary. Now, substitute T m with T into the above and rewrite Zn1 ðX ; TÞ as h X 1 ¼ ðq 1Þð1 TÞ i¼1
¼
"
N X
! q
lðDi þkDÞ
T
k
#
ð1 TÞ 1
k¼0
N h X X 1 Tk ½qlðDi þkDÞ qlðDi þðk1ÞDÞ : ðq 1Þð1 TÞ k¼0 i¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Ak
Define Ak as the finite sum indicated above. To get at the p-adic Riemann Hypothesis, it suffices to show the horizontal length of each side of the Newton P k polygon of the p-adic entire function N k¼0 Ak T is uniformly bounded. See Section 3 for an explanation of this. Since the lðDi þ kDÞ are polynomials of the same degree for large k; without loss, we may assume that for all k sufficiently large, lðD1 þ kDÞplðD2 þ kDÞp?plðDh þ kDÞ: For convenience, with each 1piph; write ci ðkÞ :¼ lðDi þ kDÞ lðD1 þ kDÞX0 and ai ðkÞ :¼ lðDi þ kDÞ lðD1 þ ðk 1ÞDÞX0: Note that ai ðkÞ tends to infinity as k-N since it is a polynomial in k of degree n 1X1 with the positive leading coefficient Dn =ðn 1Þ!: Let m be the integer such
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that for all k sufficiently large, bounded ðconstantÞ ci ðkÞ ¼ unbounded
if 1pipm; if m þ 1piph:
Now, Ak ¼
h X
½qlðDi þkDÞ qlðDi þðk1ÞDÞ
i¼1
¼
h X
½qlðD1 þðk1ÞDÞþai ðkÞ qlðD1 þðk1ÞDÞþci ðk1Þ ;
i¼1
which we may write as Ak ¼
h X i¼1
q
ai ðkÞ
m X
q
ci ðk1Þ
i¼1
h X
! q
ci ðk1Þ
qlðD1 þðk1ÞDÞ :
i¼mþ1
Let bk :¼ ordq
m X
qci ðk1Þ :
i¼1
By construction, for all k sufficiently large, the bk are constant. Now, since ai ðkÞ and cmþ1 ðkÞ; y; ch ðkÞ grow as an unbounded polynomial, we see from the above that for all k sufficiently large, hk :¼ ordq Ak ¼ bk þ lðD1 þ ðk 1ÞDÞ: Define Dh :¼ ðhkþ2 hkþ1 Þ ðhkþ1 hk Þ: If we can show Dh is strictly positive, then Fig. 1 accurately displays the Newton polygon; this implies that the horizontal length of each side (except for the first
Fig. 1. All large slopes are in Z:
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finitely many sides) of the Newton polygon will have length 1: Now, recall that lðD1 þ kDÞ is a degree nX2 polynomial with initial coefficient Dn =n!: Also, recall how we defined ai ðkÞ: Then for nX2; Dh is strictly positive for all k sufficiently large since Dh ¼ a1 ðk þ 1Þ a1 ðkÞ ¼ nðn 1Þ
Dn n2 k þ ðpoly of degreepn 3Þ: n!
Since all except perhaps finitely many line segments comprising the Newton polygon have horizontal length one, we see that almost all the zeroes are in Qp and are simple zeros. Changing back to the variable T m and using Lemma 1 of Section 3, we see that adjoining the zeroes and poles of Zn1 ðX ; TÞ to Qp gives a finite extension of Qp : Furthermore, almost all zeroes are simple since T m a and T m b clearly have no common roots if a and b have different absolute values. This finishes the proof of the theorem. &
5. The quadric surface example In this section, since the Riemann–Roch problem is easily and explicitly solved for the quadric surface P1 P1 ; we will obtain an explicit description of the zeta function of divisors for this quadric surface. Proving the conjectures from this description, while technical, uses only freshman calculus (for details, see Section 6). The method used to determine the Riemann–Roch problem on this surface and calculate the degree of its subvarieties explicitly is by the surface’s intersection theory. Let us recall the details of this. Intersection theory of P1 P1 : Let ½x : y : z : w be the coordinates for P3Fq and let X be the quadric surface defined by xw ¼ yz: Knowing that this surface is the image of the Segre embedding of P1 Fq P1 in P3 helps us determine its intersection theory. Denote the coordinates of the two projective lines by ½u0 : u1 and ½v0 : v1 : Then, the Segre embedding is the map ½u0 : u1 ½v0 : v1 /½u0 v0 : u0 v1 : u1 v0 : u1 v1 ¼ ½x : y : z : w: Consider the prime divisors D1 :¼ ðx ¼ y ¼ 0Þ
and
D2 :¼ ðx ¼ z ¼ 0Þ:
It is well known that CH1 ðX Þ; which equals PicðX Þ in this case, equals ZD1 "ZD2 : Notice, when considering D1 and D2 in P1 P1 ; that D1 ¼ ðu0 ¼ 0Þ P1
and
D2 ¼ P1 ðv0 ¼ 0Þ:
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This helps determine their intersection product: D21 ¼ D22 ¼ 0 and ðD1 D2 Þ ¼ 1: To see this, notice that D1 and D2 meet (transversally) at only one point, thus, their product is one. If we let p1 : P1 P1 -P1 be the projection morphism on the first coordinate, we see that D1 is the fibre of the point u0 ¼ 0: Since fibres are disjoint from one another and linearly equivalent, their self-intersection is zero. Thus D21 ¼ D22 ¼ 0: Next, notice that for the hyperplane ðx ¼ 0Þ in P3 ; we have ðx ¼ 0Þ-X equal to the union of the two divisors D1 ¼ ðx ¼ y ¼ 0Þ and D2 ¼ ðx ¼ z ¼ 0Þ: This observation shows that H :¼ D1 þ D2 is a hyperplane section. From this, we may compute the degree of a divisor D on X by degðDÞ :¼ ðD HÞ: In the first section of the paper, we defined the degree via the Hilbert polynomial. The one given here is equivalent. Notes to take away from the intersection theory: For any effective divisor D on this quadric surface, it will be linearly equivalent to aD1 þ bD2 for some a; bAZX0 : Thus, from the above, we see that degðDÞ ¼ a þ bX0: Also, using Proposition 1, we may calculate the number of effective divisors linearly equivalent to D by using the dimension of LðaD1 þ bD2 Þ: Since this vector space is parameterized by bihomogeneous polynomials of bidegree ða; bÞ; we see that EffConeðX Þ ¼ ZX0 D1 "ZX0 D2 since lðaD1 þ bD2 Þ ¼ ða þ 1Þðb þ 1Þ when a; bX0 and 0 otherwise. So, we may write Z1 ðX ; TÞ ¼ ¼
X
qlðDÞ 1 degðDÞ T q1 DAEffConeðX Þ X qlðaD1 þbD2 Þ 1 T degðaD1 þbD2 Þ q 1 a;bX0
X qðaþ1Þðbþ1Þ 1 T aþb q 1 a;bX0 " # X 1 1 ðaþ1Þðbþ1Þ aþb ¼ þ q T : q 1 ð1 TÞ2 a;bX0 ¼
After rewriting the sum over a and b (which will be done in Section 6), we have the following. Lemma 2. The zeta function of divisors of the surface xw ¼ yz may be written as " # 2 N X 1 1 qðnþ1Þ T 2n ½1 þ qnþ1 T Z1 ðX ; TÞ ¼ þ : q 1 ð1 TÞ2 n¼0 1 qnþ1 T
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In this form, we may deduce that Z1 ðX ; TÞ is p-adic meromorphic and ordT¼1 Z1 ðX ; TÞ ¼ 2 ¼ rank CH1 ðX Þ: This was essentially known already in [6, Example 6.7]. Also, notice the poles f1; q1 ; q2 ; yg already satisfy the p-adic Riemann hypothesis. Thus, we need only to concentrate on the zeros. We do this via its Newton polygon, as we did in Theorem 1. Lemma 3. For the quadric surface, write the numerator of Z1 ðX ; TÞ as 1 þ where each ai is a p-adic integer. Then 8 3 2 1 if i ¼ 3k; > < ð2Þk þ ð2Þk 3 2 3 ordq ðai Þ ¼ greater than ð2Þk þ ð2Þk þ 1 if i ¼ 3k þ 1 > : 3 2 ð2Þk þ ð52Þk þ 1 if i ¼ 3k þ 2:
P
iX1 ai T
i
;
The proof of this will be given in Section 6. From this explicit description of the Newton polygon, we may prove the following. Theorem 2. The zeta function of divisors for the quadric surface xw ¼ zy satisfies all four conjectures. That is, it is p-adic meromorphic, satisfies the order and rank relation, satisfies the p-adic Riemann hypothesis, and almost all zeros and poles are simple except possibly when pa2: Remark. In the proof, we show the horizontal lengths of the segments of the Newton polygon alternate between one and two. Proof. We mentioned above that Z1 ðX ; TÞ is meromorphic and satisfies the order and rank relation. Further, we have seen that the poles satisfy the p-adic Riemann hypothesis. So, let us do the same with the zeros. P First, write the numerator as 1 þ ai T i : Since the Newton polygon of this power series is obtained as the lower convex hull of the points ði; ordq ðai ÞÞ; we need to determine which points are the vertices. For convenience, define the points P0 ðkÞ :¼ ð3k; 3k2 =2 þ k=2Þ; P1 ðkÞ :¼ ð3k þ 1; 3k2 =2 þ 3k=2 þ 1Þ; P2 ðkÞ :¼ ð3k þ 2; 3k2 =2 þ 5k=2 þ 1Þ: From Lemma 3, we know that P0 ðkÞ ¼ ð3k; ordq ða3k ÞÞ; P1 ðkÞ lies below ð3k þ 1; ordq ða3kþ1 ÞÞ; P2 ðkÞ ¼ ð3k þ 2; ordq ða3kþ2 ÞÞ:
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Fix kX0: Denote the slope between two points Q1 and Q2 by SlopeðQ1 2Q2 Þ: Now, since SlopeðP0 ðkÞ2P2 ðkÞÞ ¼ k þ 12; SlopeðP2 ðkÞ2P0 ðk þ 1ÞÞ ¼ k þ 1; SlopeðP0 ðkÞ2P1 ðkÞÞ ¼ k þ 1; we see that Fig. 2 accurately represents the Newton polygon. For the line segment of horizontal length two, since the midpoint is strictly above this line segment, we see that all the zeros are simple if pa2: This is because the roots of T 2 a are already distinct if p42 and a is a non-zero element in Fp : If p ¼ 2; the simplicity of the zeros does not seem to be obvious and would require some more work. &
6. Proofs of the lemmas for the quadric surface Let X be the quadric surface xw ¼ yz: In Section 5, we saw how we may write Z1 ðX ; TÞ as " # X 1 1 ðaþ1Þðbþ1Þ aþb Z1 ðX ; TÞ ¼ þ q T : q 1 ð1 TÞ2 a;bX0 As promised in Lemma 2, let us see how to rewrite this as " # 2 N X 1 1 qðnþ1Þ T 2n ½1 þ qnþ1 T Z1 ðX ; TÞ ¼ þ : q 1 ð1 TÞ2 n¼0 1 qnþ1 T
Fig. 2. Newton Polygon of xw ¼ yz:
ð2Þ
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Proof of Lemma 2. Let us concentrate on rewriting X qðaþ1Þðbþ1Þ T aþb : a;bX0
For the following, refer to Fig. 3. Since ða þ 1Þðb þ 1Þ is symmetric with respect to the coordinates ða; bÞ; we gather together points on the lines a ¼ 0 and b ¼ 0; next gather all the points on the lines a ¼ 1 and b ¼ 1 that were not on the previous lines; continue this. Double counting the point of intersection of the lines (and then subtracting one off), we obtain " # " # N N X X X 2 ðaþ1Þðbþ1Þ aþb ðnþ1Þ n 2ðnþ1Þ 1þn q T ¼2 q T qþ2 q T q2 T 2 a;bX0
n¼0
"
þ 2
N X
#
n¼1 2
q3ðnþ1Þ T 2þn q3 T 4 þ ?:
n¼2
Finally, after simplifying each geometric series, we obtain " # 2 N N X 1 1 2qðnþ1Þ T 2n X ðnþ1Þ2 2n Z1 ðX ; TÞ ¼ þ q T q 1 ð1 TÞ2 n¼0 1 qnþ1 T n¼0 which we may rewrite as (2). This completes the lemma.
&
Lastly, we need to prove our explicit description of the Newton polygon. Recall, P Lemma 3 said if we write the numerator of Z1 ðX ; TÞ as 1 þ ai T i ; then 8 3 2 1 if i ¼ 3k; > < ð2Þk þ ð2Þk 3 2 3 ordq ðai Þ ¼ greater than ð2Þk þ ð2Þk þ 1 if i ¼ 3k þ 1; > : 3 2 ð2Þk þ ð52Þk þ 1 if i ¼ 3k þ 2:
Fig. 3. Plot of ða þ 1Þðb þ 1Þ on the effective cone ZX0 "ZX0 :
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Proof. For each nAZX0 ; define 2
Fn ðTÞ :¼ qðnþ1Þ T 2n ð1 TÞ2 ½1 þ qnþ1 T
QN
ð1 qkþ1 TÞ : 1 qnþ1 T
k¼0
The numerator of Z1 ðX ; TÞ now takes the form N N Y X ð1 qnþ1 TÞ þ Fn ðTÞ: n¼0
n¼0
Fix mX1: To find the q-adic valuation of the coefficient of T m in the above series, we compute (or estimate) the valuation of each of the terms in the series, compare them, and choose the smallest. Let us start by computing the q-adic valuation of the coefficient of T m in each Fn : Since each Fn has the term T 2n ð1 TÞ2 ¼ T 2n 2T 2nþ1 þ T 2nþ2 ;
ð3Þ
there are three cases to consider: when mX2n þ 2; when m ¼ 2n þ 1; and when m ¼ 2n: Case 1: mX2n þ 2: Considering the many ways to multiply the T’s together in Fn to obtain T m ; we see that using any term in (3) other than T 2nþ2 yields a coefficient of larger valuation. Thus, when mX2n þ 2; the q-adic valuation of the coefficient of T m in Fn ðTÞ is vm ðnÞ :¼ ðn þ 1Þ2 þ ½1 þ 2 þ ? þ ðm ð2n þ 2ÞÞ ¼ ðn þ 1Þ2 þ ð12Þ½m 2n 2½m 2n 1: Note, the derivative v0m ðnÞ ¼ 6n 2m þ 5: Of the Fn with mX2n þ 2; we wish to find the smallest valuation vm ðnÞ: Since vm ðnÞ is a convex parabola in the variable n with vertex m=3 56; we need to consider the different residue classes of m modulo 3: If m ¼ 3k then the vertex is at k 56: The integer closest to this is k 1: Thus, Fk1 gives the smallest valuation v3k ðk 1Þ ¼ 3k2 =2 þ k=2: If m ¼ 3k þ 2; then the vertex is at k 16: Thus, Fk produces the smallest valuation v3kþ2 ðkÞ ¼ 3k2 =2 þ 5k=2 þ 1: Now, if m ¼ 3k þ 1; then the vertex is at k 12: So, we see that both Fk and Fk1 produce the same (smallest) valuation. However, the sign of T m in Fk is the opposite of Fk1 ; and since we are adding the Fn together, they will cancel. So, while the exact value illudes us, we have a lower bound that will do for our purpose: v3kþ1 ðkÞX3k2 =2 þ 3k=2 þ 1: Case 2: m ¼ 2n þ 1: Using the term T 2n in (3) to obtain T m we obtain a coefficient with valuation ðn þ 1Þ2 þ 1: Using the term T 2nþ1 yields a valuation ( ðn þ 1Þ2 if qa2r ; ð4Þ ðn þ 1Þ2 þ 1=r if q ¼ 2r :
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Thus, if qa2; we see that using the term T 2nþ1 yields a strictly smaller valuation than using the term T 2n : We will consider the case q ¼ 2 in a moment. Meanwhile, since n ¼ ðm 1Þ=2; we see that (4) takes the form ðm þ 1Þ2 =4
and
ðm þ 1Þ2 =4 þ 1=r:
However, in both instances, we see that if m ¼ 3k þ j with j ¼ 0; 1; or 2; the valuation is roughly 9k2 =4: So, for large k; this is larger than the valuations found for the Fn when mX2n þ 2; and so, may be disregarded. One may quickly check that this is true for all k; not just the large ones. Now, suppose m ¼ 2n þ 1 and q ¼ 2: Using only T 2n from T 2n ð1 TÞ2 in the definition of Fn ; we get 2
qðnþ1Þ T 2n ð1 qTÞð1 q2 TÞ?ð1 q2n TÞð1 þ q2nþ1 TÞð1 q2nþ2 TÞ?:
ð5Þ
Then, the coefficient of T 2nþ1 ; with q ¼ 2; is 2
2ðnþ1Þ ð2 22 23 ? 22n þ 22nþ1 22nþ2 ?Þ: Consider the same product in (5) but with T 2n replaced by 2T 2nþ1 : The coefficient 2 of T m in this product becomes 2ðnþ1Þ ð2Þ: Adding the two coefficients obtained from both products, we see that the coefficient of T m in Fn is 2
2ðnþ1Þ ð22nþ2 22nþ3 ?Þ: Thus, the 2-adic valuation is ðn þ 1Þ2 þ ð2n þ 2Þ: Since n ¼ ðm 1Þ=2; and setting m ¼ 3k þ j where j ¼ 0; 1 or 2, we see that the valuation is approximately 9k2 =4: So, for large m (which means large k), the valuation is much larger than the Fn with mX2n þ 2: Again, one may check this is true for all k: Case 3: m ¼ 2n: Then we need to consider Fm=2 : The coefficient of T m in this has valuation ðm þ 2Þ2 =4: Again, if m ¼ 3k þ j with j ¼ 0; 1; or 2, then the valuation is roughly 9k2 =4 which will be larger than the Fn with mX2n þ 2: And again, one may check this is true for all k: Q nþ1 TÞ contributes to the Lastly, the smallest valuation the term N n¼0 ð1 q m coefficient of T is mðm þ 1Þ=2: If m ¼ 3k þ j; with j ¼ 0; 1; or 2; then the valuation is roughly 9k2 =2: Again, this is too large compared to the valuations obtained by the Fn when mX2n þ 2: And again(!), one may check this is true for all k: &
Acknowledgments We would like to thank H. Esnault for helpful comments.
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References [1] S.D. Cutkosky, V. Srinivas, On a problem of Zariski on dimensions of linear systems, Ann. of Math. 137 (1993) 531–559. [2] D. Goss, A Riemann hypothesis for characteristic p L-functions, J. Number Theory 82 (2000) 299–322. [3] D. Goss, The impact of the infinite primes on the Riemann hypothesis for characteristic p-valued Lseries, Proceedings of the Conference on Algebra and Algebraic Geometry with Applications in honor of S.S. Abhyankar’s 70th birthday, to appear. [4] R. Hartshorne, Algebraic Geometry, Springer, Berlin, 1977. [5] A. Robert, A Course in p-adic Analysis, GTM 198, 2000. [6] D. Wan, Zeta functions of algebraic cycles over finite fields, Manuscripta Math. 74 (1992) 413–444. [7] D. Wan, On the Riemann hypothesis for the characteristic p zeta function, J. Number Theory 58 (1996) 196–212. [8] D. Wan, Moment zeta functions in arithmetic geometry, preprint. [9] O. Zariski, The theorem of Riemann–Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. 76 (1962) 560–615.
Further reading D. Wan, Pure L-functions from algebraic geometry over finite fields, in: D. Jungnickel, H. Niederreiter (Eds.), Finite Fields and Applications, Springer, Berlin, 2001, pp. 437–461.
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http://www.elsevier.com/locate/jnt
A lower bound for the canonical height on elliptic curves over abelian extensions$ Joseph H. Silverman Mathematics Department, Box 1917, Brown University, Providence, RI 02912, USA Received 16 April 2003 Communicated by S.-W. Zhang
Abstract Let E=K be an elliptic curve defined over a number field, let hˆ be the canonical height on E; and let K ab =K be the maximal abelian extension of K: Extending work of M. Baker (IMRN 29 (2003) 1571–1582), we prove that there is a constant CðE=KÞ40 so that every nontorsion ˆ point PAEðK ab Þ satisfies hðPÞ4CðE=KÞ: r 2003 Elsevier Inc. All rights reserved. MSC: primary: 11G05; secondary: 11G10, 14G25; 14K15 Keywords: Elliptic curve; Canonical height, Lehmer conjecture
1. Introduction The classical Lehmer conjecture [14] asserts that there is an absolute constant C40 so that any algebraic number a that is not a root of unity satisfies hðaÞ4C=½QðaÞ : Q: Recently Amoroso and Dvornicich [1] showed that if a is restricted to lie in Qab ; then the stronger inequality hðaÞ4C is true. The analog of Lehmer’s conjecture for elliptic curves and abelian varieties has also been much studied [3,7,11,13,16,17,26]. Baker [4] has proven the elliptic analog of the Amoroso– Dvornicich result if the elliptic curve either has complex multiplication or has nonintegral j-invariant, but he was unable to handle the general case of integral j-invariant. In this note we extend Baker’s result to include all elliptic curves. $
This research supported by NSF DMS-9970382. E-mail address: [email protected].
0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.07.001
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Theorem 1. Let K=Q be a number field, let E=K be an elliptic curve, and let hˆ : % EðKÞ-R be the canonical height on E: There is a constant CðE=KÞ40 such that every nontorsion point PAEðK ab Þ satisfies ˆ hðPÞ4C: Remark 1. This theorem gives a proof of Baker’s Conjecture 1.10 [4] for elliptic curves. In Baker’s terminology, we will prove that if E does not have complex multiplication, then EðK ab Þ has the ‘‘strong discreteness property.’’ We begin in Section 2 with a brief history of known results on Lehmer’s conjecture. In Section 3 we set notation and recall basic facts about local heights. Section 4 discusses torsion points and quotes a result of Serre that will be needed in the proof of Theorem 1. In Section 5 we take up the unramified case. Here we use the characteristic polynomial of Frobenius at a prime ideal P; evaluated at Frobenius, to annihilate points modulo P: This device replaces the use of a complex multiplication map in [4], allowing us to deal with general elliptic curves. Next, in Section 6, we deal with the ramified case. We simplify the argument in [4] by applying t 1 twice to our point, the first time to move it into the formal group, and the second time to make it a difference of conjugate points in the formal group. Finally, in Section 7, we complete the proof of our main theorem. In Section 8, we indicate how many of the arguments can be generalized to the case of abelian varieties. In particular, we sketch ˆ a proof that hðPÞ4CðA=KÞ for all nontorsion points PAAðK ab Þ if the abelian variety A=K has no complex multiplication abelian subvarieties.
2. Earlier results on Lehmer’s Conjecture For the convenience of the reader, in this section we summarize some of the known results regarding Lehmer’s Conjecture for number fields and for elliptic curves. Detailed information and an extensive bibliography for the former is given on the Lehmer Conjecture Website [15]. 2.1. The classical Lehmer Conjecture % ; assume that a is not a root of unity, and let D ¼ ½QðaÞ : Q: The Let aAQ Lehmer Conjecture says that there is a constant c40 so that hðaÞXcD1 for all such a: Blanksby and Montgomery [5] and Stewart [31] independently, and by rather different methods, proved that hðaÞXcD1 ðlog DÞ1 :
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This was superceded by Dobrowolski’s estimate [7] hðaÞXcD1 ðlog logðDÞ=logðDÞÞ3 ; which is currently the best known general result. In the special case that a1 is not a Galois conjugate of a (one says that a is nonreciprocal), Smyth [30] proved the full Lehmer Conjecture hðaÞXcD1 : See also [1] for an alternative proof of Smyth’s result. 2.2. Lehmer’s Conjecture for elliptic curves % be a Let E=K be an elliptic curve, let hˆ be the canonical height on E; let PAEðKÞ nontorsion point, and let D ¼ ½KðPÞ : K be the degree of the field of definition of P: 1 ˆ : Table 1 sumThe elliptic analog of Lehmer’s Conjecture says that hðPÞXcD ˆ marizes the history of lower bounds for hðPÞ; where c denotes a positive constant that depends on E=K; but not on P: We also note that Pacheco [20, Theorem 2.3] has proven the elliptic Lehmer Conjecture for function fields over finite fields. 2.3. Lehmer’s Conjecture in abelian extensions In the classical setting, Amoroso and Dvornicich [1] recently gave a ‘‘superLehmer’’ lower bound for every aAQab that is not a root of unity: hðaÞXc: Amoroso and Zannier [2] generalize this to relative abelian extensions K ab =K: More generally, they give a Dobrowolski-type bound for aAK% with the degree D ¼ ½KðaÞ : K replaced by the ‘‘nonabelian’’ part of the degree D0 ¼ ½KðaÞ : KðaÞ-K ab : Table 2 gives the history of lower bounds for the canonical height in EðK ab Þ for elliptic curves E=K: In particular, the results in the present paper complete the one case left undone in [4], namely non-CM elliptic curves with integral j-invariant. We observe that the non-CM integral j-invariant case often presents the greatest difficulties when studying algebraic points on elliptic curves, see for example Serre’s
Table 1 % History of lower bounds for hˆ in EðKÞ ˆ hðPÞX
Restriction on E
Reference
cD10 ðlog DÞ6
none CM
Anderson–Masser (1980) [3] Laurent (1983) [13]
none j nonintegral
Masser (1989) [17] Hindry–Silverman (1990) [11]
logðDÞ 3 Þ cD1 ðloglogðDÞ 2 3
cD ðlog DÞ cD2 ðlog DÞ2
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Table 2 History of lower bounds for hˆ in EðK ab Þ ˆ hðPÞX
Restriction on E
Reference
cD2 cD1 ðlog DÞ2 cD2=3 c c
none none j nonintegral j nonintegral or CM none
Silverman (1981) [26] Masser (1989) [17] Hindry–Silverman (1990) [11] Baker (2002) [4] Silverman (2003)
theorem [22] on the image of Galois in EndðEtors Þ: The reason is that if E has CM or if E has nonintegral j-invariant, then there is some control over the action of Galois on torsion. In the former case, there is almost complete control via class field theory, and in the latter case, the Tate curve over Kp gives control for the decomposition group at p: 2.4. Lehmer’s Conjecture for abelian varieties A number of authors have considered the analog of Lehmer’s Conjecture for abelian varieties A=K of dimension gX2: Let L be an ample symmetric line bundle % be a nontorsion point, and let D ¼ ½KðPÞ : K: Masser [16] proves on A; let PAAðKÞ that there is a constant k ¼ kðdim AÞ so that hˆL ðPÞXcDk : If A has complex multiplication, then David and Hindry [6] generalize Laurent’s result to give a Dobrowoski-type estimate 1 ˆ hðPÞXcD ðlog logðDÞ=logðDÞÞk :
Restricting to special types of fields, S. Zhang notes that the equidistribution theorems proven in [32,34] imply that if L is a finite extension of a totally real field, then AðLÞtors is finite and hˆL ðPÞXc for all nontorsion PAAðLÞ: (See [4, Theorem 1.8].) In particular, this is true for AðK ab Þ if K itself is totally real. [The author would like to thank Michael Rosen for pointing out this last fact.]
3. Preliminaries In this section we set notation and recall some basic facts about the decomposition of the canonical height into a sum of local heights.
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3.1. Notation We set the following notation. K=Q a number field. MK the set of absolute values on a number field K extending the usual absolute values on Q: We write vðxÞ ¼ logjxjv ; and for nonarchimedean v; we denote the associated prime ideal by pv : E=K an elliptic curve defined over K: % hˆ the canonical height hˆ : EðKÞ-R on E; see [27, VIII, Section 9]. #l the local canonical height (Ne´ron function) % l# : MK% EðKÞ-R,N; normalized as described in [29, VI.1.1]. Remark 2. With absolute values normalized as above, the product formula reads X ½Kv : Qv vðaÞ ¼ 0 ½K : Q vAMK
for all aAK
We also have the usual formula for finite extensions L=K; X
½Lw : Kv ¼ ½L : K:
wAML ; wjv
Remark 3. If v is nonarchimedean, say corresponding to a prime ideal p of residue characteristic p and ramification index e ¼ eðp=pÞ; then v is discrete and its smallest positive value is inffvðaÞ : aAK and vðaÞ40g ¼
log p : e
This is clear, since if p is a uniformizer in K for p; then locally ðpÞe ¼ ðpÞ; so vðpÞ ¼ vðpÞ=e ¼ ðlog pÞ=e:
3.2. Local height functions We recall some well-known properties of the local canonical height on an elliptic curve. Theorem 2. Let % l# : MK% EðKÞ-R,fNg;
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be the local canonical height (also called a Ne´ron function) for the divisor ðOÞ and normalized as described in [29, VI.1.1]. (a) For any finite extension L=K and any point PAAðLÞ\fOg; ˆ hðPÞ ¼
X ½Lw : Qw l# w ðPÞ: ½L : Q wAM L
(b) There is a constant c1 ðE=KÞX0 so that for any finite extension L=K and any point PAAðLÞ\fOg; X ½Lw : Qw minfl# w ðPÞ; 0gX c1 ðE=KÞ: ½L : Q wAM L
(c) Let vAMK be a finite place of good reduction for E; and choose a minimal Weierstrass equation for E at v; y2 þ a1 xy þ a3 y ¼ x3 þ a2 x2 þ a4 x þ a6 : Let L=K be a finite extension and let wAML lie over v: Then l# w ðPÞ ¼ maxfwðx=yÞ; 0g
for all P ¼ ðx; yÞAEðLÞ\fOg:
Proof. (a) This is [29, Theorem VI.2.1]. (c) Since we are assuming that E has good reduction and that we have chosen a minimal Weierstrass equation, this is [29, Theorem VI.4.1], except that we have written wðx=yÞ instead of 12wðx1 Þ: (Note that since E has good reduction and the equation is minimal at v; the same is true for any extension w of v; and hence wðDÞ ¼ 0 and E0 ðLw Þ ¼ EðLw Þ:) The integrality of the Weierstrass equation implies easily that wðx1 Þo03wðx=yÞo0
and
minf0; 3wðxÞg ¼ minf0; 2wðyÞg;
so 12wðx1 Þ ¼ wðx=yÞ if either is negative. (b) Let SK CMK be the union of the archimedean places of K and the set of places of K at which E has bad reduction, and let SL CML be the set of places of L lying above places of K: From (c), we know that l# w ðPÞX0 for all weSL : Suppose now that vASK is archimedean. We fix an isomorphism EðK% v ÞDEðCÞDC=qZv with 0ojqv jv oep : Then the local height is given by the explicit formula [29, VI.3.4] #lv ðuÞ ¼ 1B2 logjujv logjq1 j v v 2 logjqv jv X logj1 ujv logjð1 qnv uÞð1 qnv =uÞjv ; ð1Þ nX1
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where B2 ðtÞ is the Bernoulli polynomial t2 t þ 16 for 0ptp1; extended periodically to R=Z: It is easy to see from (1) that 1 logjujv l# v ðuÞX B2 logjq1 v j v c2 2 logjqv jv for an absolute constant c2 : The polynomial B2 ðtÞ has a minimum at t ¼ 12; so we find that 1 l# v ðPÞX logjq1 v jv c2 24
for all PAEðK% v Þ\fOg:
Since K and E are fixed, this says that there is a constant c3 ¼ c3 ðE=KÞ so that l# v ðPÞX c3 for all archimedean places vAMK and all points PAEðK% v Þ\fOg: It follows that for any finite extension L=K and any point PAAðLÞ\fOg; X ½Lw : Qw X ½Lw : Qw minfl# w ðPÞ; 0gX ðc3 Þ ¼ c3 : ½L : Q ½L : Q wAM N wAM N L
L
Note that if w lies over v; then L% w DK% v ; so we can compute the local height l# w ðPÞ using the inclusions EðLÞCEðLw ÞCEðK% v Þ: Next suppose that vAMK is a nonarchimedean place at which E=K has split multiplicative reduction. Then the exact same argument gives an analogous lower bound for l# v ; because the local height is given by the same formula (1), see [29, VI.4.2]. Finally, if E=K has additive or nonsplit multiplicative reduction at v; then we may either make a finite extension of K to reduce to one of the previous cases or use explicit formulas in these cases (see, e.g., [25, Table 1]) to obtain the desired lower bound. & Remark 4. Using explicit formulas for the local canonical height, it is not difficult to give an explicit estimate for the constant c1 ðE=KÞ in terms of the j-invariant and minimal discriminant DE=K of E=K: Roughly, one can take c1 to have the form c1 ¼ c01 maxf1; hð jÞ; log NK=Q DE=K g; where c01 is an absolute constant. See [28] for a similar computation of explicit constants associated to local height functions. Remark 5. We have chosen to prove Theorem 2(b) by appealing to explicit formulas for the local height on elliptic curves. It is also possible to prove this result by appealing to the MK -continuity of the local height and using an MK -compactness type of argument. Of course, EðK% v Þ is not even locally compact for nonarchimedean places v; so one must be careful. A detailed development of the correct concept, which is called MK -boundedness, is given in [12].
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4. Torsion points in abelian extensions The following result is a simple consequence of a deep theorem of Serre. Theorem 3 (Serre [22]). Let K=Q be a number field and let E=K be an elliptic curve % ¼ Z: Then without CM, that is, EndðE=KÞ EðK ab Þtors is finite: In particular, EðK ab Þ½c ¼ 0 for all but finitely many primes c: Proof. Serre [22] proves that there is a finite set of primes S so that for all ceS; the Galois representation rc : GK=K % -AutðE½cÞDGL2 ðFc Þ is surjective. Suppose that TAEðK ab Þ½c with TaO for some ceS: The group GL2 ðFc Þ acts transitively on the nonzero vectors in F2c ; so Serre’s theorem tells us that the Galois orbit of T consists of all nonzero elements of E½c: The conjugates of T are all defined over K ab ; so we conclude that E½cCEðK ab Þ: Hence the representation rc factors through the maximal abelian quotient GK ab =K of GK=K % : This contradicts the fact that the image of rc is a nonabelian group, which completes the proof that EðK ab Þ½c ¼ 0 for all ceS: This statement suffices for our later applications. In order to prove the stronger statement that EðK ab Þtors is finite, it remains to show that for any particular c; the c-power torsion in EðK ab Þ is finite. This follows by a similar argument using the fact, also due to Serre [23], that the image of GK=K in AutðTc ðEÞÞ is open, and thus has finite % index. &
5. The unramified case In this section we prove the basic estimate required for the proof of our main result in the case that the extension L=K is unramified at a (small) prime p of K: We begin with the observation that there is an element of the group ab ab ring Z½GK=K % that simultaneously annihilates EðK Þ modulo every prime of K lying above p: Theorem 4. Let K=Q be a number field, let p be a prime of K; and let E=K be an elliptic curve with good reduction at p: Let Fp ðX Þ ¼ detðX Frobp jTc ðEÞÞ ¼ X 2 aX þ q
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be the characteristic polynomial of Frobenius at p acting on the c-adic Tate module for pffiffiffi some prime c prime to p: Thus Fp ðX ÞAZ½X with q ¼ NK=Q p and jajp2 q (see [27, Chapter V]). % Let p% be a prime of K% lying over p and let sAðp; be in the associated % K=KÞCG % K=K Frobenius conjugacy class. % (a) For all PAEðKÞ; Fp ðsÞP O ðmod pÞ: % (This congruence is taking place on the Ne´ron model of E; or more prosaically, on a Weierstrass equation for E that is minimal at p:) % satisfies Fp ðsÞP ¼ O; then P is a torsion point. (b) If PAEðKÞ
Proof. (a) When reduced modulo p; acts as the q-power % the element sAGK=K % ˜ p Þ: Further, the map Fp ð fq Þ annihilates Tc ðE=F ˜ p Þ; since Frobenius map fq AEndðE=F ˜ Fp is the characteristic polynomial of fq acting on Tc ðE=Fp Þ and the Cayley– Hamilton theorem tells us that a linear transformation satisfies its own characteristic equation. However, we have the general fact that the map EndðEÞ-EndðTc ðEÞÞ is injective (cf. [27, V.7.3]), so we conclude that Fp ð fq Þ ¼ 0 as an element of ˜ p Þ: In other words, EndðE=F ˜ F% p Þ: Fp ð fq ÞQ ¼ O for all QAEð Finally, using the fact that the reduction map commutes with the action of Galois, % the point Fp ðsÞP is in the kernel of reduction. In other we see that for any PAEðKÞ; words, FP ðsÞP O ðmod pÞ; % which completes the proof of (a). % satisfy Fp ðsÞP ¼ O: Fix a finite Galois extension L=K with (b) Let PAEðKÞ PAEðLÞ; say of degree m ¼ ½L : K: Then sm ¼ 1 in GL=K ; so in particular, sm P ¼ P: Let r ¼ ResultantðFp ðX Þ; X m 1ÞAZ: The complex roots of X m 1 have absolute value 1 and the complex roots of Fp ðX Þ pffiffiffi have absolute value q; so they have no complex roots in common. It follows that ra0: The resultant of two polynomials in Z½X is a generator for the ideal that they generate, so we can find aðX Þ; bðX ÞAZ½X satisfying aðX ÞFp ðX Þ þ bðX ÞðX m 1Þ ¼ r:
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Substituting X ¼ s gives the identity aðsÞFp ðsÞ þ bðsÞðsm 1Þ ¼ r in the group ring Z½GK=K % : Hence rP ¼ aðsÞðFp ðsÞPÞ þ bðsÞððsm 1ÞPÞ ¼ O; so P is a point of finite order.
&
Corollary 5. Continuing with the notation and assumptions from Theorem 4, let % be a nontorsion point, and let e be the absolute ramification index of p% in the PAEðKÞ field of definition KðPÞ of P: (That is, let e be the ramification index of the prime ideal p-KðPÞ over the prime p:) Then % log p l# p% ðFp ðsÞPÞX : e Proof. To ease notation, let Q ¼ Fp ðsÞP: Theorem 4(a) tells us that Q O ðmod pÞ; % so Q is in the kernel of reduction modulo p: % Further, the assumption that P is nontorsion and Theorem 4(b) imply that QaO: Let v% be the absolute value associated to p: % Then on a minimal Weierstrass equation, we have 3v%ðxðQÞÞ ¼ 2v%ðyðQÞÞo0; so we can apply Theorem 2(c) to conclude that lp% ðQÞ ¼ v%ðxðQÞ=yðQÞÞ40: Finally, we use the fact that for any extension L=K; the minimum positive value of v% restricted to L is ðlog pÞ=e; where e is the ramification index of p% in L: (See Remark 3.) &
6. The ramified case In this section we prove the basic estimate needed to handle the case of ramified extensions. We begin by recalling the proof of the key number field lemma, which is due to Amoroso and Dvornicich. We then apply this lemma to obtain an analogous result for elliptic curves. In [4], this was done by altering the given point to make sure it is nonzero modulo P: We describe an alternative approach in which we force the point to be zero modulo P: This allows us to work entirely within the formal group, where computations are much easier. 6.1. A congruence in ramified abelian extensions The following lemma is modeled after [1, Lemma 2]. See also [2, Lemma 3.2]; [4, Lemma 3.5].
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Lemma 6 (Amoroso-Dvornicich [1]). Let K=Q be a number field, let p be a degree 1 prime in K with residue characteristic p; and let L=K be an abelian extension that is ramified at p: Let P be a prime of L lying over p; and let OL;P denote the localization of L at P: Then there exists an element tAIL=K with ta1 such that tðaÞp ap ðmod pOL;P Þ
for all aAOL;P :
(Note that the strength of this result is that the congruence is modulo p; and not merely modulo P:) Proof. Without loss of generality, we may replace K and L by their completions Kp ¼ Qp and LP ; respectively. Then L is an abelian extension of Qp ; so by the local Kronecker–Weber theorem, there is an integer mX1 so that LCQp ðzm Þ: (Here zm is a primitive mth root of unity.) We take m to be minimal, i.e., m is the conductor of L=Qp : The extension L=Qp is ramified by assumption, which implies that pjm: Let t be a generator for the cyclic group GQp ðzm Þ=Qp ðzm=p Þ : The minimality of m implies that LgQp ðzm=p Þ; so the restriction of t to GL=K is not trivial. Further, since t fixes zpm ; we see that tðzm Þ ¼ ozm
for some primitive pth root of unity o:
Now let aAOL COQp ðzm Þ ¼ Zp ½zm : Thus a ¼ f ðzm Þ for some polynomial f ðX ÞAZp ½X ; and hence ta ¼ tð f ðzm ÞÞ ¼ f ðtzm Þ ¼ f ðozm Þ: Now taking pth powers yields ðtaÞp ¼ f ðozm Þp f ððozm Þp Þ ðmod pZp ½zm Þ ¼ f ðzpm Þ f ðzm Þp ðmod pZp ½zm Þ ¼ ap : Since L-pZp ½zm ¼ pOL ; this completes the proof.
&
6.2. Ramified points in EðK ab Þ In this section we prove the key estimate needed to handle the case of ramified extensions. In particular, by applying ðt 1Þ2 to a point, we can do most of our computations in the formal group, where the multiplication-by-p map is easier to describe.
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Lemma 7. Let E=Kp be an elliptic curve defined over a local field Kp =Qp and assume that E has good reduction at p: Let LP =Kp be a finite Galois extension that is ramified at p; let OP be the ring of integers of LP ; and let tAILP =Kp be in the inertia group of LP =Kp : We denote by E1 ðLP Þ the kernel of reduction and by Eˆ the formal group of E; ˆ [27, Chapter IV]. so there is a natural isomorphism E1 ðLP ÞDEðPÞ (a) Let PAEðLP Þ: Then ðt 1ÞPAE1 ðLP Þ: (b) Fix a minimal Weierstrass equation for E and let z ¼ x=y be a parameter for the ˆ Then formal group E: zð½ pðt 1ÞQÞAððtzðQÞÞp zðQÞp ÞOP þ pOP ˆ for all QAE1 ðLP ÞDEðPÞ: Proof. (a) This is clear, since an element of the inertia group fixes everything modulo P; so tP P ðmod PÞ; and hence the difference tP P lies in the kernel of reduction. (b) This is an immediate consequence of the following elementary lemma about commutative formal groups, applied with x ¼ tzðQÞ and y ¼ zðQÞ: & Lemma 8. Let R be a ring, let pAZ be a prime, and let F ðx; yÞAR½½x; y be a formal group over R: Let iðtÞAR½½t be the inversion series for F and let Mp ðtÞAR½½t be the multiplication-by-p series for F : Then Mp ðF ðx; iðyÞÞÞAðxp yp ÞR½½x; y þ pR½½x; y: Proof. There are power series aðtÞ; bðtÞAR½½t with að0Þ ¼ bð0Þ ¼ 0 so that multiplication-by-p is given by power series of the form Mp ðtÞ ¼ aðtp Þ þ pbðtÞ: See [27, IV.4.4]. Further, the definition of iðtÞ implies that F ðx; iðyÞÞ vanishes when y ¼ x; and hence that F ðx; iðyÞÞ is divisible by x y; say F ðx; iðyÞÞ ¼ ðx yÞGðx; yÞ
with Gðx; yÞAR½½x; y:
Then Mp ðF ðx; iðyÞÞÞA aðF ðx; iðyÞÞp Þ þ pR½½x; y A aððx yÞp Gðx; yÞp Þ þ pR½½x; y
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A ðx yÞp R½½x; y þ pR½½x; y A ðxp yp ÞR½½x; y þ pR½½x; y:
&
Combining Lemmas 6 and 7 gives the following crucial local contribution to the canonical height in the ramified case. The key is the fact that the lower bound does not depend on the ramification degree of the field of definition of the point. Proposition 9. Let K=Q be a number field, let PAEðK ab Þ; and let L ¼ KðPÞ be the field of definition of P: Fix a degree 1 unramified prime p of K; let p ¼ NK=Q p; and suppose that p ramifies in L: Then there exists an element tAIL=K with ta1 such that the point P0 ¼ ½ pððt 1Þ2 PÞ satisfies l# P ðP0 ÞXlog p for all primes P of L lying over p: (Note that if P0 ¼ O; then l# P ðP0 Þ ¼ N by definition, so the result is vacuously true.) Proof. The extension L=K is abelian, so Lemma 6 says that there is a nontrivial element tAIL=K such that ðtaÞp ap ðmod pOL Þ
for all aAOL :
ð2Þ
Let Q ¼ ðt 1ÞP; so P0 ¼ ½ pðt 1ÞQ: Lemma 7(a) says that Q is in the kernel of reduction modulo P: Fix a minimal Weierstrass equation for E at p and let z ¼ x=y: Note that zðQÞ is in the localization OL;P of OL at P: Lemma 7 tells us that zðP0 ÞAððtzðQÞÞp zðQÞp ÞOL;P þ pOL;P : Applying (2) with a ¼ zðQÞ; we find that zðP0 ÞApOL;P : Hence Theorem 2(c) yields l# P ðP0 Þ ¼ vP ðzðP0 ÞÞXvP ðpÞ ¼ log p:
&
7. Proof of the main theorem We are now ready to prove our main result, which we restate for the convenience of the reader. Theorem 10. Let K=Q be a number field, let E=K be an elliptic curve, and let hˆ : % EðKÞ-R be the canonical height on E: There is a constant CðE=KÞ40 such that every
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nontorsion point PAEðK ab Þ satisfies ˆ hðPÞ4C: Proof. Baker [4] has proven Theorem 10 in the case that the elliptic curve has % ¼ Z: (Baker also proves complex multiplication, so we may assume that EndðE=KÞ the theorem in the case that jðEÞ is nonintegral, but our proof will cover all non-CM curves.) We begin by fixing a rational prime p and a prime p of K lying over p with the following properties: 1. EðK ab Þ½ p ¼ fOg: 2. pX expð½K : Qð1 þ c1 ðE=KÞÞÞ; where c1 ðE=KÞ is the constant appearing in Theorem 2(b). 3. E has good reduction at p: 4. p is an unramified prime of degree 1. % ¼ Z and Serre’s Theorem 3 tell us that (1) Our assumption that EndðE=KÞ eliminates only finitely many primes p; and similarly (2) and (3) eliminate only finitely many primes p: Condition (4) describes a set of primes of density one, so we can find a prime satisfying (1)–(4) that depends only on the field K and the curve E=K: Note, however, that we are strongly using here the assumption that E does not have CM, since if E has CM, then (1) may be false for all primes. Let PAEðK ab Þ be a nontorsion point and let L ¼ KðPÞ be its field of definition. The proof of the theorem proceeds by induction on the ramification degree of L=K at p: We begin with the unramified case, say pOL ¼ P1 P2 ?Pd : The extension L=K is abelian, so the Frobenius elements associated to the Pi are all the same. Let s ¼ ðPi ; L=KÞAGL=K be this Frobenius element, and let Fp ðX Þ ¼ X 2 aX þ p be the characteristic polynomial of Frobenius. Then Theorem 4(b) tells us that Fp ðsÞP is a nontorsion point, so Corollary 5 and our assumption that Pi is unramified imply that l# Pi ðFp ðsÞPÞXlog p: Note that this is true for every Pi ; because every Pi has the same sAGL=K as its Frobenius element. Adding over the Pi ; we obtain the estimate X ½LP : Kp l# P ðFp ðsÞPÞXlog p: ½L : K Pjp
ð3Þ
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Next we combine this positive lower bound with the (potentially negative) contribution from the other absolute values. We compute: ˆ p ðsÞPÞ hðF X ½Lw : Qw l# w ðFp ðsÞPÞ ðTheorem 2ðaÞÞ ½L : Q wAML X ½Lw : Qw # lw ðFp ðsÞPÞ X ½L : Q wAM ; wjv ¼
L
p
X
½Lw : Qw minfl# w ðFp ðsÞPÞ; 0g ½L : Q wAML ; w[vp 0 1 X ½Lw : Qw # lw ðFp ðsÞPÞA c1 ðE=KÞ X@ ½L : Q wAM ; wjv þ
L
X
log p c1 ðE=KÞ ½K : Q
X1
ðTheorem 2ðbÞÞ
p
ðfrom ð3ÞÞ
from the choice of p:
ˆ In order to obtain a lower bound for hðPÞ; we use the fact that hˆ is a Galois invariant [27, VIII.5.10] positive semidefinite quadratic form [27, VIII.9.3]. Hence ˆ p ðsÞPÞ ¼ hðs ˆ 2 P ½asP þ ½ pPÞ hðF ˆ ˆ pPÞÞ ˆ 2 PÞ þ hð½asPÞ þ hð½ p 3ðhðs ˆ ˆ ˆ þ p2 hðPÞÞ ¼ 3ðhðPÞ þ a2 hðPÞ pffiffiffi ˆ p 3ð1 þ 4p þ p2 ÞhðPÞ since jajp2 p: This gives the lower bound 1 ˆ hðPÞX ; 3ð1 þ 4p þ p2 Þ and since p was chosen depending only on E=K; independent of the point PAEðK ab Þ; this completes the proof of Theorem 10 in the case that the extension KðPÞ=K is unramified at p: Assume now that L=K is ramified at p and assume by induction that the theorem is proven for all points defined over abelian extensions whose p-ramification index is strictly smaller than ep ðL=KÞ: Let tAIp ðL=KÞ be the nontrivial element in the inertia group described in Proposition 9, so the point P0 ¼ ½ pððt 1Þ2 PÞ
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satisfies l# P ðP0 ÞX log p for all primes Pjp: Summing over these primes, we find that X ½LP : Kp l# P ðP0 ÞXlog p: ½L : K Pjp Note that this is exactly the same inequality that we obtained earlier (cf. (3)), except that it applies to the point P0 ; rather than to the point Fp ðsÞP: Hence as long as we know that P0 aO; then the same computation as was done above for Fp ðsÞP yields the same lower bound for P0 ; namely ˆ 0 ÞX1: hðP In the other direction, we can estimate ˆ 0 Þ ¼ hð½ ˆ 2 P ½2tP þ PÞ ˆ pðt 1Þ2 PÞ ¼ p2 hðt hðP ˆ 2 PÞ þ 4hðtPÞ ˆ ˆ þ hðPÞÞ p 3p2 ðhðt ˆ ¼ 18p2 hðPÞ: 2 ˆ Hence hðPÞX1=ð18p Þ; which completes the proof of Theorem 10 provided P0 aO: Finally, suppose that
P0 ¼ ½ pðt 1Þ2 P ¼ O: Let m be the order of t in GL=K : There are polynomials aðX Þ; bðX ÞAZ½X satisfying aðX ÞðX m 1Þ þ bðX ÞpðX 1Þ2 ¼ mpðX 1Þ: (The existence of such an identity follows immediately from the fact that the resultant of X m1 þ ? þ X þ 1 and X 1 is m: Explicitly, one can take aðX Þ ¼ p P i and bðX Þ ¼ m2 i¼0 ðm 1 iÞX :) We evaluate this identity at X ¼ t and apply it m to the point P: We know that t ¼ 1 and ½ pðt 1Þ2 P ¼ O; so we find that ½mpðt 1ÞP ¼ O: In particular, ðt 1ÞPAEðLÞtors : However, we also know from Lemma 7 that ðt 1ÞP is in the kernel of reduction modulo P; so it follows from general facts about formal groups [27, IV.3.2(b)] that the order of ðt 1ÞP is a power of p: We now recall that p was chosen so that EðK ab Þ½ p ¼ fOg; from which we conclude that ðt 1ÞP ¼ O: Thus P is fixed by t; so P lies in the proper subfield Lt of L: (Note that ta1; so t L aL:) Further, P is a nontorsion point by assumption. Hence by induction we ˆ conclude that hðPÞXCðE=KÞ; which completes the proof of the theorem. &
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8. Generalization to abelian varieties Much of material in this paper can be generalized to the case of abelian varieties, at the usual cost of making the arguments more complicated. In this section we will sketch the changes required to prove the following generalization of Theorem 10. Theorem 11. Let K=Q be a number field, let A=K be an abelian variety, let L be a symmetric ample line bundle on A=K; and let % hˆL : AðKÞ-R be the canonical height on A associated to L: Assume that A=K contains no abelian subvarieties having complex multiplication. Then there is a constant c ¼ cðA=K; LÞ40 such that for all nontorsion points PAAðK ab Þ; hˆL ðPÞXc: Proof (Sketch). Replacing K by a finite extension, we may assume that A splits into a product of geometrically simple abelian varieties, and then by looking at the individual factors, we may assume that A itself is geometrically simple. Next, replacing L by L#n ; we may assume that L is very ample. Fix effective divisors D1 ; y; Dr for L whose intersection consists of only the point O; and then fix canonical local heights (also known at Ne´ron functions) % l# Di : MK% AðKÞ-R,fNg: See [12, Chapters 10 and 11] for standard properties of local and global height functions. In particular, standard properties of (canonical) local height functions give the analog of Theorem 2. Thus [12, Chapter 11, Theorem 1.6] says that for any % we can compute hˆL ðQÞ as nonzero point QAAðKÞ; hˆL ðQÞ ¼
X wAMKðQÞ
½KðQÞv : Qv # lD ðw; QÞ ½KðQÞ : Q i
by choosing an index i with Qe SupportðDi Þ; which gives the analog of Theorem 2(a). Similarly, the analog of Theorem 2(b) follows fairly directly from the MK -positivity of local height functions attached to positive divisors [12, Chapter 10, Proposition 3.1], and the analog of Theorem 2(c) follows from the fact that for places of good reduction, the canonical local height is given as an intersection index on the special fiber of the Ne´ron model of A [12, Chapter 11, Theorem 5.1].
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Next consider Theorem 3, which uses a deep result of Serre to limit the torsion in EðK ab Þ: For the proof of Theorem 11, we really only need to know that AðK ab Þ½ p ¼ 0
for all but finitely many primes p:
ð4Þ
(Indeed, it would suffice to know that AðK ab Þ½ p ¼ 0 for infinitely many primes.) In certain cases, Serre has proven an ‘‘image of Galois’’ theorem for abelian varieties [24]. Fortunately, the weaker statement (4) that we require is proven in full generality by Zarhin [33] (see also [21]). All of these results [21,24,33] rely heavily on the groundbreaking methods used by Faltings in his proof of Tate’s Isogeny Conjecture [8]. Theorem 4 generalizes directly to abelian varieties. The characteristic polynomial Fp ðX Þ of Frobp acting on Tc ðAÞ is a monic polynomial of degree 2g pffiffiffi whose roots have norm q: See [18, Section 18] or [19, Section 21, Application II]. The proof of both parts of Theorem 4 is an easy consequence of these facts and the injectivity of ˜ p% Þ-EndðTc ðAÞÞ: ˜ EndðA=F P Further, if we write Fp ðX Þ ¼ ai X 2gi ; then the triangle inequality yields 2g i=2 q : This allows us to handle the unramified case. jai jp i The ramified case again relies on the Amoroso–Dvornicich Lemma 6, together with an abelian variety version of Lemma 7. And just as in the case of elliptic curves, Lemma 7 for abelian varieties follows from standard facts about commutative formal groups. See [9] or [10] for basic results on formal groups, and in particular a description of the multiplication-by-p map that gives a higher dimensional version of Lemma 8. Having now assembled all of the required pieces, they fit together to give a proof of Theorem 11 using the same argument as given in the proof of Theorem 10. & Remark 6. Combining the methods of this paper, which essentially handles the nonCM case, with the method for CM elliptic curves described in Baker’s article [4], one may be able to prove in full generality the inequality hˆL ðPÞXCðA=K; LÞ for all nontorsion points PAAðK ab Þ on all abelian varieties A=K: (This is essentially Conjecture 1.10 in [4].) The details of this argument will be given in a subsequent publication. Acknowledgments The author would like to thank Matt Baker for making his paper [4] available and for his many helpful comments on the first draft of this paper, Marc Hindry for many pleasant hours spent working together on lower bounds for canonical heights, and Michael Rosen for numerous mathematical discussions.
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References [1] F. Amoroso, R. Dvornicich, A lower bound for the height in abelian extensions, J. Number Theory 80 (2000) 260–272. [2] F. Amoroso, U. Zannier, A relative Dobrowolski lower bound over abelian extensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 (4) (2000) 711–727. [3] M. Anderson, D. Masser, Lower bound for heights on elliptic curves, Math. Zeit. 174 (1980) 23–34. [4] M. Baker, Lower bounds for the canonical height on elliptic curves over abelian extensions, IMRN 29 (2003) 1571–1582. [5] P.E. Blanksby, H.L. Montgomery, Algebraic integers near the unit circle, Acta Arith. 18 (1971) 355–369. [6] S. David, M. Hindry, Minoration de la hauteur de Ne´ron-Tate sur les varie´te´s abe´liennes de type C.M, J. Reine Angew. Math. 529 (2000) 1–74. [7] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979) 391–401. [8] G. Faltings, Endlichkeitssa¨tze fu¨r abelsche Varieta¨ten u¨ber Zahlko¨rpern, Invent. Math. 73 (1983) 349–366. [9] A. Fro¨hlich, Formal Groups, in: Lecture Notes in Mathematics, Vol. 74, Springer, New York, 1968. [10] M. Hazewinkel, Formal Groups and Applications, Academic Press Inc., New York, 1978. [11] M. Hindry, J.H. Silverman, On Lehmer’s conjecture for elliptic curves, in Se´minaire de The´orie des Nombres, Paris (1988–1989), Progress in Mathematics, Vol. 91, Birkha¨user, Boston, MA, 1990, pp. 103–116. [12] S. Lang, Fundamentals of Diophantine Geometry, Springer, New York, 1983. [13] M. Laurent, Minoration de la hauteur de Ne´ron-Tate, Se´minaire de The´orie de Nombres Paris 1981–1982, Vol. 38, Birkha¨user, Boston, Basil, Stuttgart, 1983, pp. 137–152. [14] D.H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. 34 (1933) 461–479. [15] Lehmer Conjecture Website, http://www.math.ucla.edu/~mjm/lc/lc.html. [16] D. Masser, Small values of the quadratic part of the Ne´ron-Tate height on an abelian variety, Compositio Math. 53 (1984) 153–170. [17] D. Masser, Counting points of small height on elliptic curves, Bull. Soc. Math. France 117 (1989) 247–265. [18] J.S. Milne, Abelian varieties, in: G. Cornell, J.H. Silverman (Eds.), Arithmetic Geometry, Springer, Berlin, New York, 1986. [19] D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Tata Institute, Bombay, 1970. [20] A. Pacheco, Lehmer’s conjecture in positive characteristic, preprint, April 2002. [21] W.M. Ruppert, Torsion points of abelian varieties in abelian extensions, unpublished manuscript, http://www.math.uiuc.edu/Algebraic-Number-Theory/0101/tpavae.dvi. [22] J.-P. Serre, Proprie´te´s galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972) 259–331. [23] J.-P. Serre, Abelian l-adic Representations and Elliptic Curves (revised reprint of the 1968 original.) Research Notes in Mathematics, Vol. 7, A K Peters, Ltd., Wellesley, MA, 1998. [24] J.-P. Serre, Oeuvres Colllected Papers, Vol. IV, 1985–1998, p. 136, Re´sume´ des cours de 1985–1986 and 137, Lettre a´ Marie-France Vigne´ras du 10/2/1986, Springer, Berlin, 2000. [25] J.H. Silverman, The Ne´ron–Tate Height on Elliptic Curves, Ph.D. Thesis, Harvard, December 1981. [26] J.H. Silverman, Lower bound for the canonical height on elliptic curves, Duke Math. J. 48 (1981) 633–648. [27] J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 106, Springer, New York, 1986. [28] J.H. Silverman, The difference between the Weil height and the canonical height on elliptic curves, Math. Comp. 55 (1990) 723–743. [29] J.H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 151, Springer, New York, 1994.
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[30] C.J. Smyth, On the product of the conjugates outside the unit circle of an algebraic number, Bull. London Math. Soc. 3 (1971) 169–175. [31] C.L. Stewart, Algebraic integers whose conjugates lie near the unit circle, Bull. Soc. Math. France 106 (2) (1978) 169–176. [32] L. Szpiro, E. Ullmo, S. Zhang, E´quire´partition des petis points, Invent. Math. 127 (1997) 337–347. [33] Yu.G. Zarhin, Endomorphisms and torsion of abelian varieties, Duke Math. J. 54 (1987) 131–145. [34] S. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. 147 (1998) 159–165.
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http://www.elsevier.com/locate/jnt
Kronecker limit formula for Drinfeld modules Satoshi Kondo Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Received 28 May 2003 Communicated by K.A. Ribet
Abstract We define Eisenstein series and theta functions for Drinfeld modules of arbitrary rank, and prove an analog of Kronecker limit formula. r 2003 Elsevier Inc. All rights reserved. MSC: 11G09 Keywords: Limit formula; Drinfeld module
1. Introduction The Kronecker limit formula says that the logarithm of the theta function is equal to the value at zero of the first derivative of the Eisenstein series. In [2], Gross and Rosen treat three cases: (1) the base field is Q and the lattice is a fractional ideal of Z in R; (2) the base field F is quadratic imaginary and the lattice is a projective OF -module of rank one in C; (3) the base field is K (see Section 2.1 for definition) and the lattice is a rank one A-module in C: They define Eisenstein series and theta function for each of the three cases, and prove the limit formula in a uniform manner. Case (3) is particularly important for the study of Drinfeld modules of rank one. In this article, we generalize this to the case of higher-rank Drinfeld modules.
E-mail address: [email protected]. 0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.08.004
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We fix a Drinfeld module of rank dðdANÞ over C: It corresponds to a lattice of d rank d in C: We may then define a function GðxÞ ðxAKN \Ad Þ as an analytic function having a zero at each point in the lattice, and the Eisenstein series Es ðxÞ with complex variable s as a sum over the lattice. Our theorem is Theorem 1. We have
@ Es ðxÞ ¼ log jGðxÞj: @s s¼0
See Theorem 3 for the precise statement. The main tool used in the proof is d Fourier analysis on the compact abelian group KN =Ad : We recall some facts regarding Fourier analysis in Section 2. In Sections 3 and 4, we define Eisenstein series and theta functions. The main theorem is in Section 5. 2. Fourier analysis We use the following notation throughout this paper. 2.1. Notations F X N K A KN q C jj d Od
a finite field an absolutely irreducible projective and smooth curve over F a closed point of X the function field FðX Þ of X over F the ring H 0 ðX \N; OX Þ of functions regular on X \N the completion of K at N with valuation ring ON the number of elements of the residue field at N the completion of the algebraic closure of KN the valuation on C such that the uniformizer of KN takes the value q1 a natural number S the Drinfeld symmetric space ð¼ Pd1 ðCÞ\ ðKN -rational hyperplanes))
2.2. Fourier analysis We recall the facts needed about Fourier analysis on a locally compact abelian group which we will need below. The proofs are found in [3]. Let us fix a nontrivial additive character c of KN ; and put A> ¼ fa0 AKN jcða0 aÞ ¼ 1 for all aAAg: d =A of KN =A is identified with A> by Then the character group KN d A> -KN =A =A;
a0 /ðx/cða0 xÞÞ:
d =Ad and ðA> Þd : We will identify the character group of KN
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d We give KN the Haar measure such that the volume of ON is 1, KN the product d d measure, and KN =A the quotient measure. d =Ad ; and ðm1 ; y; md ÞAðA> Þd ; we define the If f is an integrable function on KN ðm1 ; y; md Þ-th Fourier coefficient of f to be the integral Z Z ˆ 1 ; y; md Þ ¼ % 1 x1 þ ? þ md xd Þ fðm ? f ðx1 ; y; xd Þcðm KN =A
KN =A
dx1 ?dxd ; where the bar denotes complex conjugation. The Fourier inversion formula says X ˆ 1 ; y; md Þcðm1 x1 þ ? þ md xd Þ: fðm f ðx1 ; y; xd Þ ¼ m1 ;y;md AA>
3. Eisenstein series Let us fix t ¼ ðt1 : ? : td ÞAOd with t1 ¼ 1: We define the Eisenstein series to be X Es ðx1 ; y; xd Þ ¼ jðx1 a1 Þt1 þ ? þ ðxd ad Þtd js ; a1 ;y;ad AA
where x1 ; y; xd AKN \A and sAC: The sum is absolutely convergent for Re s4d: We define modified Eisenstein series which are integrable. For zAC such that ðt1 : ? : td : zÞAOdþ1 ; we let X Es ðz; xÞ ¼ jz þ ðx1 a1 Þt1 þ ? þ ðxd ad Þtd js ; a1 ;y;ad AA d where we write x to mean ðx1 ; y; xd ÞAKN : Again, the sum is absolutely convergent for Re s4d: From now on we take z such that
jz þ x1 t1 þ ? þ xd td j ¼ supðjzj; jx1 t1 þ ? þ xd td jÞ: By the result of Goldman and Iwahori [1, Proposition 1.3], without loss of generality, we may assume that jz þ x1 t1 þ ? þ xd td j ¼ supðqa0 þN0 ; jx1 jqa1 þN1 ; y; jxd jqad þNd Þ;
ð1Þ
where ai AQ; Ni AZ ð0pipdÞ; 0pa1 p?pad o1; and 0pa0 o1: Proposition 2. The modified Eisenstein series Es ðz; xÞ can be analytically continued as a meromorphic function to the complex plane.
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Proof (Sketch). The idea is to look at each Fourier coefficient. By direct computation, using the explicit expression of the absolute value (1), we see that each Fourier coefficient is a rational function in qs ; and can be continued to the complex plane as a meromorphic function. The claim follows. & 4. Theta function Using the same tAOd as in Section 3, we define some analytic functions which are periodic modulo the lattice At1 þ ? þ Atd : Let Y y et ðyÞ ¼ y 1 ; a1 t 1 þ ? þ ad t d ða ;y;a ÞAAd \fð0;y;0Þg 1
2 DðtÞ ¼ 4b1
d
Y
3 1 jbj 1 ; et ðu1 t1 þ ? þ ud td Þ5
ðu1 ;y;ud ÞAðb1 A=AÞd \fð0;y;0Þg
where bAA; jbja1: The function D is well defined and independent of b only up to roots of unity. For a more intrinsic definition, see Section 7 (7.11) of [2]. We define theta function to be Gt ðxÞ ¼ et ðx1 t1 þ ? þ xd td ÞDðtÞ: We may often omit the subscript t: Let zAC be as in Section 3. We define the modified theta function to be Gt ðz; xÞ ¼ et ðz þ x1 t1 þ ? þ xd td ÞDðtÞ: 5. Main theorem Theorem 3. Let Es ðxÞ and GðxÞ be as in Sections 3 and 4. Then @ Es ðxÞ ¼ log jGðxÞj: @s s¼0 The following proposition implies the theorem by taking jzj close to zero.
Proposition 4. Let Es ðxÞ and GðxÞ be as above. Then
@ Es ðz; xÞ ¼ log jGðz; xÞj: @s s¼0
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Proof (Sketch). We check that the Fourier coefficients of both sides are equal by straightforward calculation. We note that the computation of the ð0; y; 0Þ-th Fourier coefficient of the logarithm of the theta function is similar to the one carried out in Lemma 4.6 of [2], where the distribution property is the key. The long but elementary calculation is omitted. &
Acknowledgments The author thanks Professor Karuya Kato for encouragement.
References [1] O. Goldman, N. Iwahori, The space of p-adic norms, Acta Math. 109 (1963) 137–177. [2] B. Gross, M. Rosen, Fourier series and the special values of L-functions, Adv. Math. 69 (1) (1988) 1–31. [3] D. Ramakrishnan, R.J. Valenza, Fourier Analysis on Number Fields, Graduate Texts in Mathematics, Vol. 186, Springer, New York, 1999.
ARTICLE IN PRESS
Journal of Number Theory 104 (2004) 378–379
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Corrigendum
Corrigendum to ‘‘Coefficients of half-integral weight modular forms’’ [J. Number Theory 99 (2003) 164–179]$ Communicated by D. Goss
Theorems 1 and 2.3 are not stated correctly. In these two results, the integer M is required to be an odd prime. For completeness, here are the correct statements. Theorem 1. Let f ðzÞ ¼
PN
n¼1
aðnÞqn AS
1 ðN; wÞ-Z½½q lþ2
be a half-integral weight cusp
form, and let w be a real Dirichlet character. If M is an odd prime and there is a positive integer n for which gcdðaðnÞ; MÞ ¼ 1; then at least one of the following is true: (1) If 0proM; then #f0pnpX : aðnÞ r ðmod MÞgbr;M
( pffiffiffiffi X =log X X
if 1proM; if r ¼ 0:
(2) There are finitely many square-free integers, say n1 ; n2 ; y; nt ; for which f ðzÞ
t N X X i¼1
2
aðni m2 Þqni m ðmod MÞ:
m¼1
Moreover, if gcdðM; NÞ ¼ 1; eAf71g and p[NM is a prime with ðnpi ÞAf0; eg for each 1pipt; then ðp 1Þf ðzÞ is an eigenform modulo M of the half-integral weight Hecke operator Tðp2 ; l; wÞ: In particular, we have ! ð 1Þl 2 ðp 1Þf ðzÞ j Tðp ; l; wÞ ewðpÞ ðpl þ pl 1 Þðp 1Þf ðzÞ ðmod MÞ: p
$
doi of original article 10.1016/S0022-314X(02)00061-6. Corresponding author.
0022-314X/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2003.08.007
ARTICLE IN PRESS Corrigendum / Journal of Number Theory 104 (2004) 378–379
379
Theorem 2.3. Suppose that f ðzÞ and M are as in Theorem 1. If there is a positive integer n for which gcdðaðnÞ; MÞ ¼ 1; then at least one of the following is true: (1) If 0proM; then there are infinitely many integers n for which aðnÞ rðmod MÞ: (2) There are finitely many square-free integers, say n1 on2 o?ont ; for which t N X X 2 aðni m2 Þqni m ðmod MÞ: f ðzÞ i¼1
m¼1
Jan H. Bruinier Mathematisches Institut, Universita¨t Ko¨ln, Weyertal 86-90, D-50931, Ko¨ln, Germany E-mail address: [email protected] Ken Ono Department of Mathematics, University of Wisconsin, Madison, WI, USA E-mail address: [email protected]
Journal of Number Theory 104 (2004) 381
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Author Index for Volume 104 A
Li, Charles C.C., 177 Louboutin, Ste´phane R., 118
B
M
Amou, Masaaki, 132
Mozzochi, C.J., 62
Balasubramanian, R., 327 Breuer, Florian, 315 Bruinier, Jan H., 378
N D
Deshouillers, Jean-Marc, 255 Deutsch, Jesse Ira, 263 F Freiman, Gregory A., 255 H Haessig, C. Douglas, 335 Hou, Xiang-Dong, 14 I
Nagaoka, S., 100 O Ono, Ken, 378 P Pillichshammer, Friedrich, 301 Poulakis, Dimitrios, 210 Prakash, Gyan, 327 R Ryan, Dermot, 1 S
Ichikawa, Takashi, 170 K Katsurada, H., 100 Katsurada, Masanori, 132 Keating, Kevin, 14 Kisilevsky, H., 279, 287 Kondo, Satoshi, 373
Silverman, Joseph H., 353 Sun, Zhi-Hong, 380 Szydlo, Michael, 75 V van der Heiden, Gert-Jan, 193 W
L Laurent, Michel, 210 Lev, Vsevolod F., 162
Wan, Daqing, 335 Wenpeng, Zhang, 156
Printed in The Netherlands 0022-314X/$ - see front matter doi:10.1016/S0022-314X(03)00235-X
