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®’® ®z a® ’ô ‰† ?† EÓÅà s¡/§s¦©ª¡ Á—§/Ó ÃA§/¡)ª ãLÃÜà¡s¢L¢¤£ ¥@¦£(§„§)©ª «Å¡ Á Õ v z •x a®ç ´Ý ´ o Ö _ ‚ ® g « ¡ Õ ¾ k Ô C © E ¬ ¾ U E µ x ­  £ ‚ ¾ m Ð ¯ ¨ p µ m ° 4 ¬ § /w$Ì ¹rÚr pÏ4ÖC°‚©m¾ ²Ö Úß:n 9n m¾ m¾7­x¼¸£‚¾p§`Ђ©p¨¯¨¯½†§-½jÖ )Ì Ÿ7” «g³9[ Ö ¹rß EÏ4Ö ‚Ö_ÒG§`¬X­¯3®‚¨¯F § Ã7©p­xQ§`¥'4 ¤g° Uµo­¯£¾ ›8wGÖAÔk´‚X¨¯Ö¤g¤]°‚°Å¹p§`ÃA¨`®‚Ö0³ Ã7» §`§-Á ¨ Ci¾ 90m¾7©p¨ ¼i§º X_» Öm§-§-· «Þ¨¯Ö«]¬X¤g­J°·µ¸Ö ¾ ¹·AdrÚ¨xߧ-¤]­xrm­x£G £ ¾ ¾©²œ
Ö $̾A¹¼¸rÚr C¾ fÕÏ4ÖC¤g³T°‚©m³T¾ §-‚¨[Ö ,Ö l r ‚¾ Úßm¾i0m¾k¼i§ » §-«ÞÖk·Ó¾ d§-­x£ ©²œ Ö Ó¾k¼¸¾fÕ¤g³¸³T§`¨`Ö mI ç ²Ö É©³T´²©¥v¤]­x¤g© UµE­x£¾ 9w$Ì rrßÏXÖk°‚©²¾ ‚Ö Ún k¾ Úrm¾0ÙÖ 1 if A has split multiplicative reduction at p : 0 if A has additive reduction at p. The integers a(n are obtained recursively from the ap as follows:  apr = aaprr; ap ; papr; ifif pp j N N p  anm = an am , if (n; m) = 1. The conjectures made by Serre in [102], which are the subject of this paper, concern representations  : Gal(Q=Q) ! GL(2; F` ). We always require (usually tacitly) that our representations are continuous. The continuity condition just means that the kernel of  is open, so that it corresponds to a nite Galois extension K of Q. The representation  then embeds Gal(K=Q) into GL(2; F` ). Since K is a nite extension of Q, the image of  is contained in GL(2; F) for some nite sub eld F of F` . 1

2

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Q

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Q

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For various technical reasons, the original conjectures of Serre insist that  be irreducible. It is nevertheless fruitful to consider the reducible case as well (see [111]). The conjectures state (in particular) that each continuous irreducible  that satis es a necessary parity condition \arises from" (or is associated with) a suitable modular form mod `. To explain what's going on, let's start with 1 X

1

Y
:=  (n)qn = q (1 ; qi )24 ; n=1 i=1

the unique (normalized) cusp form of weight 12 on SL(2; Z). In [92], Serre conjectured the existence of a \strictly compatible" family of `-adic representations of Gal(Q=Q) whose L-function is the L-function of
, namely

L(
; s) =

1 X

n=1

 (n)n;s =

Y

p

(1 ;  (p)p;s + p11;2s );1 ;

where the product is taken over all prime numbers p. The conjectured `-adic representations were constructed soon after by Deligne [24]. Speci cally, Deligne constructed, for each prime `, a representation `1 : Gal(Q=Q) ! GL(2; Z` );

1.1. Introduction

151

unrami ed outside `, such that for all primes p 6= `, tr(`1 (Frobp )) =  (p); det(`1 (Frobp )) = p11 : On reducing `1 mod `, we obtain a representation ` : Gal(Q=Q) ! GL(2; F` ) with analogous properties. (Equalities are replaced by congruences mod `.) In other words, the ` for
are just like the ` for an elliptic curve E , except that the integers ap are replaced by the corresponding values of the  -function. The determinant of ` is the 11th power of the mod ` cyclotomic character  : Gal(Q=Q) ! F` , i.e., the character giving the action of Gal(Q=Q) on the group of `th roots of unity in Q (see Section 1.5). P More generally, take a weight k  12 and suppose that f = n cn qn is a nonzero weight-k cusp form for SL(2; Z) that satis es f jTn = cn f for all n  1, Tn being the nth Hecke operator on the space of cusp forms of weight k for SL(2; Z) (see Section 1.5). Then the complex numbers cn (n  1) are algebraic integers. Moreover, the eld E := Q(: : : cn : : : ) generated by the cn is a totally real number eld (of nite degree over Q). Thus the cn all lie in the integer ring OE of E . For each ring homomorphism ' : OE ! F` , one nds a representation  = ' : Gal(Q=Q) ! GL(2; F` ); unrami ed outside `, such that tr((Frobp )) = '(cp ); det((Frobp )) = pk;1 for all p 6= `. We have det  = k;1 . Of course, there is no guarantee that  is irreducible. We can (and do) suppose that  is semisimple by replacing it by its semisimpli cation. Then  is determined up to isomorphism by the displayed trace and determinant conditions; this follows from the Cebotarev density theorem and the Brauer-Nesbitt theorem [21], which states that semisimple representations are determined by their characteristic polynomials. It is important to note that k is necessarily an even integer; otherwise the space Sk (SL(2; Z)) of weight-k cusp forms on SL(2; Z) is easily seen to be 0. Thus the determinant k;1 of  is an odd power of . In particular, det  : Gal(Q=Q) ! F` is unrami ed outside ` and takes the value ;1 on complex conjugations c 2 Gal(Q=Q). It's a nice exercise to check that, conversely, all continuous homomorphisms with these properties are odd powers of  (see Exercise 1). In the early 1970s, Serre conjectured that all homomorphisms that are \like " come from cusp forms of some weight on SL(2; Z). Namely, let  : Gal(Q=Q) ! GL(2; F` ) be a continuous, irreducible representation that is (1) unrami ed outside ` and (2) of odd determinant, in the sense that det (c) = ;1 2 F` for complex conjugations c 2 Gal(Q=Q). In a May, 1973 letter to Tate, Serre conjectured that  is of the form '. This means that there is a weight k  12, an eigenform f 2 Sk (SL(2; Z)), and a homomorphism ' : OE ! F` (where OE is the ring of integers of the eld generated by the coecients of f ) so that ' and  are isomorphic. To investigate Serre's conjecture, it is fruitful to consider the operation  7! 

 on representations. This \twisting" operation preserves the set of representations that comePfrom modular forms. Indeed, let  = q dqd be the classical dierential P n operator an q 7! nan qn . According to Serre and Swinnerton-Dyer [61, 94,

152

RIBET AND STEIN, SERRE'S CONJECTURES

112], if f is a mod ` form of weight k, then f is a mod ` form of weight k + ` + 1. Then if  is associated to f ,   is associated with f . According to a result of Atkin, Serre and Tate (see [97, Th. 3] and Section 2.1), if  comes from an eigenform in some space Sk (SL(2; Z)), then a suitable twist  i of f comes from a form of weight  ` + 1. Serre's conjecture thus has the following consequence: each two-dimensional irreducible odd representation of Gal(Q=Q) over F` that is unrami ed outside ` has a twist (by a power of ) coming from an eigenform on SL(2; Z) of weight at most ` + 1. In particular, suppose that ` < 11. Then the spaces Sk (SL(2; Z)) with k  ` + 1 are all 0; as a result, they contain no nonzero eigenforms! The conjecture that all  are modular (of level 1) thus predicts that there are no representations of the type contemplated if ` is 2, 3, 5 or 7. In support of the conjecture, the non-existence statement was proved for ` = 2 by J. Tate in a July, 1973 letter to Serre [113]. Soon after, Serre treated the case ` = 3 by methods similar to those of Tate. (See [113, p. 155] for a discussion and a reference to a note in Serre's uvres.) Quite recently, Sharon Brueggeman considered the case ` = 5; she proved that the conjectured result follows from the Generalized Riemann Hypothesis (see [8]). In another direction, Hyunsuk Moon generalized Tate's result and proved that there are only nitely many isomorphism classes of continuous semisimple Galois representations  : GQ ! GL4 (F2 ) unrami ed outside 2 such that eld K=Q corresponding to the kernel of  is totally real (see [76]). Similar work in this direction has been carried out by Joshi [58], under additional local hypothesis. Serre discussed his conjecture with Deligne, who pointed out a number of surprising consequences. In particular, suppose that one takes a  coming from an eigenform f 0 of some weight and of level N > 1. On general grounds,  has the right to be rami ed at primes p dividing N as well as at the prime `. Suppose that, by accident as it were,  turned out to be unrami ed at all primes p j N . Then the conjecture would predict the existence of a level-1 form f 0 (presumably of the same weight as f ) whose mod ` representation was isomorphic to . For example, if N = ` is a power of `, then the conjecture predicts that  arises from a form f 0 of level 1. How could one manufacture the f 0 ? The passage f f 0 comes under the rubric of \level optimization". When you take a representation  that comes from high level N , and it seems as though that representation comes from a lower level N 0 , then to \optimize the level" is to cough up a form of level N 0 that gives . Deligne pointed out also that Serre's conjecture implies that representations  over F` are required to \lift" to -adic representations of Gal(Q=Q). In the recent articles [80] and [81], R. Ramakrishna used purely Galois cohomological techniques to prove results in this direction.

1.2. The weak conjecture of Serre

In the mid 1980s, Gerhard Frey began lecturing on a link between Fermat's Last Theorem and elliptic curves (see [42, 43]). (Earlier, Hellegouarch had also considered links between Fermat's Last Theorem and elliptic curves; see the MathSciNet review and Appendix of [48].) As is now well known, Frey proposed that if a` + b` was a perfect `th power, then the elliptic curve y2 = x(x ; a` )(x + b`) could be proved to be non-modular. Soon after, Serre pointed out that the non-modularity

1.2. The weak conjecture of Serre

153

contemplated by Frey would follow from suitable level-optimization results concerning modular forms [101]. To formulate such optimization results, Serre went back to the tentative conjecture that he had made 15 years earlier and decided to study representations  : Gal(Q=Q) ! GL(2; F` ) that are not necessarily unrami ed at `. The results, of course, were the conjectures of [102]. An important consequence of these conjectures is the so-called \weak conjecture of Serre." As background, we recall that Hecke eigenforms on congruence subgroups of SL(2; Z) give rise to two-dimensional representations of Gal(Q=Q). If we set things up correctly, we get representations over F` . More speci cally, take integers P k  2 and N  1; these are the weight and level, respectively. Let f = an qn be a normalized Hecke eigenform in the space Sk (;1 (N )) of complex weight-k cusp forms on the subgroup ;1 (N ) of SL(2; Z). Thus f is nonzero and it satis es f jTn = an f for all n  1. Further, there is a character " : (Z=N Z) ! C so that f jhdi = "(d)f for all d 2 (Z=N Z) , where hdi is the diamond-bracket operator. Again, let O be the ring of integers of the eld Q(: : : an : : : ) generated by the an ; this eld is a number eld that is either totally real or a CM eld. Consider a ring homomorphism ' : O ! F` as before. Associated to this set-up is a representation  : Gal(Q=Q) ! GL(2; F` ) with properties that connect it up with f (and '). First, the representation is unrami ed at all p not dividing `N . Next, for all such p, we have tr((Frobp )) = ap ; det((Frobp )) = pk;1 "(p); the numbers ap and pk;1 "(p), literally in O, are mapped tacitly into F` by '. The representation  is determined up to isomorphism by the trace and determinant identities that are displayed, plus the supplemental requirement that it be semisimple. We are interested mainly in the (generic) case in which  is irreducible; in that case, it is of course semisimple. The construction of  from f , k and ' was described in [24]. In this article, Deligne sketches a method that manufactures for each non-archimedean prime  of E a representation ~ : Gal(Q=Q) ! GL(2; E ), where E denotes the completion of E at . Given ', we let  = ker ' and nd a model of ~ over the ring of integers O of E . Reducing ~ modulo , we obtain a representation over the nite eld O =O , and ' embeds this eld into F` . In fact, as Shimura has pointed out, the machinery of [24] can be avoided if one seeks only the mod  representation attached to f (as opposed to the full -adic representation ~ ). As the rst author pointed out in [87], one can use congruences among modular forms to nd a form of weight two and level N`2 that gives rise to . Accordingly, one can nd  concretely by looking within the group of `-division points of a suitable abelian variety over Q: the variety J1 (`2 N ), which is de ned in Section 2.3 and in Conrad's Appendix. Which representations of Gal(Q=Q) arise in this way (as k, N , f and ' all vary)? As in the case N = 1 (i.e., that where ;1 (N ) = SL(2; Z)), any  that comes from modular forms is an odd representation: we have det((c)) = ;1 when c 2 Gal(Q=Q) is a complex conjugation. To see this, we begin with the fact that "(;1) = (;1)k , which generalizes; (1.4); this follows from the functional equation az+b ) to f (z ) when a b
is an element of ;0 (N ) (see Exercise 7). On that relates f ( cz cd +d the other hand, using the Cebotarev density theorem, we nd that det  = k;1 ", where  is again the mod ` cyclotomic character and where " is regarded now as a map Gal(Q=Q) ! F` in the obvious way, namely by composing " : (Z=N Z) ! F`

154

RIBET AND STEIN, SERRE'S CONJECTURES

with the mod N cyclotomic character. The value on c of the latter incarnation of " is the number "(;1) = (;1)k . Since (c) = ;1, we deduce that (det )(c) = ;1, as was claimed. Serre's weak conjecture states that, conversely, every irreducible odd representation  : Gal(Q=Q) ! GL(2; F` ) is modular in the sense that it arises from some f and '. A concrete consequence of the conjecture is that all odd irreducible 2-dimensional Galois representations  come from abelian varieties over Q. Given , one should be able to nd a totally real or CM number eld E , an abelian variety A over Q of dimension [E : Q] that comes equipped with an action of the ring of integers O of E , and a ring homomorphism ' : O ! F` with the following property: Let  = ker '. Then the representation A[] O= F` is isomorphic to . (In comparing A[] and , we use ' : O= ,! F` to promote the 2-dimensional A[] into a representation over F` .) Much of the evidence for the weak conjecture concerns representations taking values in GL(2; Fq ) where the nite eld Fq has small cardinality. In his original article [102, x5], Serre's discusses a large number of examples of such representations. Serre uses theorems of Langlands [68] and Tunnell [115] to establish his weak conjecture for odd irreducible representations with values in GL(2; F2 ) and GL(2; F3 ). Further, he reports on numerical computations of J.-F. Mestre that pertain to representations over F4 (and trivial determinant). Additionally, Serre remarks [102, p. 219] that the weak conjecture is true for those representations with values in GL(2; Fp ) that are dihedral in the sense that their projective images (in PGL(2; Fp )) are dihedral groups. (See also [29, x5] for a related argument.) This section of Serre's paper concludes with examples over F9 and F7 . More recently, representations over the elds F4 and F5 were treated, under somewhat mild hypotheses, by Shepherd-Barron and Taylor [105]. For example, Shepherd-Barron and Taylor show that  : Gal(Q=Q) ! GL(2; F5 ) is isomorphic to the 5-torsion representaton of an elliptic curve over Q provided that det  is the mod 5 cyclotomic character. Because elliptic curves over Q are modular, it follows that  is modular.

1.3. The strong conjecture

Fix an odd irreducible Galois representation

 : GQ ! GL(2; F` ): As discussed above, the weak conjecture asserts that  is modular, in the sense that there exists integers N and k such that  comes from some f 2 Sk (;1 (N )). The strong conjecture goes further and gives a recipe for integers N () and k(), then asserts that  comes from Sk() (;1 (N ())). In any particular instance, the strong conjecture is, a priori, easier to verify or disprove than the weak conjecture because Sk() (;1 (N ())) is a nite-dimensional vector space that can be computed (using, e.g., the algorithm in [73]). The relation between the weak and strong conjectures is analogous to the relation between the assertion that an elliptic curve is modular of some level and the assertion that an elliptic curve A is modular of a speci c level, the conductor of A.

1.3. The strong conjecture

155

For each prime p, let Ip  GQ denote an inertia group at p. The optimal level is a product Y N () = pn(p) ; p6=`

where n(p) depends only on jIp . The optimal weight k() depends only on jI` . The integer n(p) is a conductor in additive notation. In particular, n(p) = 0 if and only if  is unrami ed at p. View  as a homomorphism GQ ! Aut(V ), where V is a two-dimensional vector space over F` . It is natural to consider the subspace of inertia invariants: V Ip := fv 2 V : ()v = v; all  2 Ip g: For example, V Ip = V if and only if  is unrami ed at p. De ne n(p) := dim(V=V Ip ) + Swan(V ); where the wild term Swan(V ) is the Swan conductor 1 X Swan(V ) := [G 1: G ] dim(V=V Gi )  0: i=1 0 i Here G0 = Ip and the Gi  G0 are the higher rami cation groups. Suppose that  arises from a newform f 2 Sk (;1 (N )). A theorem of Carayol [12], which was proved independently by Livne [70, Prop. 0.1], implies that N () j N . It is productive to study the quotient N=N (). Let O be the ring of integers of the eld generated by the Fourier coecients of f and let ' : O ! F` be the map such that '(ap ) = tr((Frobp )): Let  be a prime of O lying over ` and E be the completion of Frac(O) at . Deligne [24] attached to the pair f;  a representation  : GQ ! GL(2; E ) = Aut(V~ ) where V~ is a two-dimensional vector space over E . The representation  can be conjugated so that its images lies inside GL(2; O ); the reduction of  modulo  is then . The following diagram summarizes the set up: /  GL(2;6 E ) = Aut(V~ )

GQ /

❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ( ❧ ❧ ❧ ❧

GL(2; O )❘ 

❘ ❘ ❘ ❘ ❘ ❘' ❘ ❘ ❘ ❘ ❘ ❘ ( / GL(2; ` ) = Aut(V )

F

Let m(p) be the power of p dividing the conductor of  . In [12], Carayol proves that m(p) = ordp N . As above, m(p) = dim(V~ =V~ Ip ) + (wild term), and the wild term is the same as for . The power of p dividing N=N () is dim(V~ =V~ Ip ) ; dim(V=V Ip ) = dim V Ip ; dim V~ Ip . Though V~ and V are vector spaces over dierent elds, we can compare the dimensions of their inertia invariant subspaces. The formula (1.1) ordp (N ) = n(p) + (dim V Ip ; dim V~ Ip ) indicates how this dierence is the deviation of N from the optimal level locally at p. This is the description of n(p) that is used in proving that if  is modular at

156

RIBET AND STEIN, SERRE'S CONJECTURES

all, then it is possible to re ne N and k to eventually discover that  arises from a newform in Sk() (;1 (N ())). After much work (see [26, 87]) it has been shown that for ` > 2 the weak and strong conjectures are equivalent. See [9] for equivalence in many cases when ` = 2. Rearranging (1.1) into n(p) = ordp (N ) ; (dim V Ip ; dim V~ Ip ) provides us with a way to read o N () from ordp (N ), dim V Ip , and dim V~ Ip . If f 2 Sk (;1 (N )) gives rise to  and ` N , then k()  k. In contrast, if we allow powers of ` in the level then the weight k can always be made equal to 2. -

1.4. Representations arising from an elliptic curve

Equations for elliptic curves can be found in the Antwerp tables [4] and the tables of Cremona [20]. For example, consider the elliptic curve B given by the equation y2 + y = x3 + x2 ; 12x +2: This is the curve labeled 141A1 in [20]; it has conductor N = 3
47 and discriminant
= 37
47. There is a newform f 2 S2 (;0 (141)) attached to B . Because N is square free, the elliptic curve B is semistable, in the sense that B has multiplicative reduction at each prime. The curve B is isolated in its isogeny class; equivalently, for every ` the representation ` : GQ ! Aut(B [`])  GL(2; F` ) is irreducible (see Exercise 4 and Exercise 5). We will frequently consider the representations ` attached to B . The following proposition shows that because B is semistable, each ` is surjective [93]. Proposition 1.1. If A is a semistable elliptic curve over Q and ` is a prime such that ` is irreducible, then ` is surjective. Proof. Serre proved this when ` is odd; see [93, Prop. 21], [103, x3.1]. If 2 isn't surjective, then by [93, Prop. 21(b)] and Theorem 2.10 it's unrami ed outside 2. This contradicts [113]. To give a avor of Serre's invariants, we describe N (` ) and k(` ) for the representations ` attached to B . (Note that we still have not de ned k().) At primes p of bad reduction, after a possible unrami ed quadratic extension of Qp , the elliptic curve B is a Tate curve. This implies that for p 6= `, the representation ` is unrami ed at p if and only if ordp (
)  0 (mod `); for more details, see Section 2.4. The optimal level N (` ) is a divisor of 3
47; it is divisible only by primes for which ` is rami ed, and is not divisible by `. The representation ` is unrami ed at 3 if and only if ` j ord3 (
) = 7, i.e., when ` = 7. Furthermore, ` is always rami ed at 47. First suppose ` 62 f3; 47g. If in addition ` 6= 7 then N (` ) = 3
47, and k(` ) = 2 since ` 3
47. If ` = 7 then N (`) = 47, and again k(` ) = 2. The remaining cases are ` = 3 and ` = 47. If ` = 47 then N (` ) = 3, and because ` ; 1 is the order of the cyclotomic character, k(` )  2 (mod 47 ; 1); Serre's recipe then gives k(` ) = 2 + (47 ; 1) = 48. Similarly, when ` = 3, we have N (`) = 47 and k(` ) = 2 + (3 ; 1) = 4. The following table summarizes the Serre invariants: -

1.5. Background material

157

Table 1.4. The Serre invariants of `

` N (` ) k(` ) 3 47 4 7 47 2 47 3 48 all other ` 141 2 To verify the strong conjecture of Serre for ` = 3; 47, we use a standard tradeo of level and weight, which relates eigenforms in S2 (;0 (141); F`) to eigenforms in S2+`;1 (;0 (141=`); F`) (see Section 3.1). The only exceptional prime is ` = 7, for which the minimal weight k() is 2. The strong conjecture of Serre predicts the existence of an eigenform g 2 S2 (;0 (47)) that gives rise to ` . Our initial instinct is to look for an elliptic curve A of conductor 47 such that A[`]  = B [`], as GQ -modules. In fact, there are no elliptic curves of conductor 47. This is because S2 (;0 (47)) is four having basis the Galois conjugates of a single eigenform g = P dimensional, cn qn . The Fourier coecients cn of g generate the full ring of integers in the eld K obtained from Q by adjoining a root of h = x4 ; x3 ; 5x2 + 5x ; 1: The discriminant 1957 = 19
103 of K equals the discriminant of h, so a root of h generates the full ring of integers. The eigenvalue c2 satis es h(c2 ) = 0. Since h  (x + 2)(x3 + 4x2 + x + 3) (mod 7), there is a prime  lying over 7 such that O= = F7 ; the isomorphism is given by c2 7! ;2 mod 7. As a check, note that #B (F2 ) = 5 so a2 = 3 ; 5 = ;2 = '(c2 ). More generally, for p 7
141, we have '(cp )  ap mod 7. This equality of traces implies that the representation g; is isomorphic to  = A;7 , so A is modular of level 47. -

1.5. Background material

In this section, we review the cyclotomic character, Frobenius elements, modular forms, and Tate curves. We frequently write GQ for Gal(Q=Q). Many of these basics facts are also summarized in [23].

1.5.1. The cyclotomic character

The mod ` cyclotomic character is de ned by considering the group ` of `th roots of unity in Q; the action of the Galois group GQ on the cyclic group ` gives rise to a continuous homomorphism (1.2) ` : GQ ! Aut(` ): Since ` is a cyclic group of order `, its group of automorphisms is canonically the group (Z=`Z) = F` . We emerge with a map GQ ! F` , which is the character in question. Let A be an elliptic curve and ` be a prime number. The Weil pairing e` (see [109, III.8]) sets up an isomorphism of GQ -modules 2 ^



= : A[`] ;;;;! ` The determinant of the representation A;` is the mod ` cyclotomic character ` . Suppose now that c 2 GQ is the automorphism \complex conjugation." Then the determinant of A;`(c) is ` (c). Now c operates on roots of unity by the map

(1.3)

e` :

158

RIBET AND STEIN, SERRE'S CONJECTURES

 7!  ;1 , since roots of unity have absolute value 1. Accordingly, (1.4) det A;` (c) = ;1; ;
one says that A;` is odd. If ` 6= 2, then A;`(c) is conjugate over F` to 10 ;01 (Exercise 7). If ` = 2 then the characteristic polynomial of A;` (c) is (x + 1)2 so A;`(c) is conjugate over F` to either the identity matrix or ( 10 11 ).

1.5.2. Frobenius elements

Let K be a number eld. The Galois group Gal(K=Q) leaves the ring OK of integers of K invariant, so that one obtains an induced action on the ideals of OK . The set of prime ideals p of OK lying over p (i.e., that contain p) is permuted under this action. For each p, the subgroup Dp of Gal(K=Q) xing p is called the decomposition group of p. Meanwhile, Fp := OK =p is a nite extension of Fp . The extension Fp =Fp is necessarily Galois; its Galois group is cyclic, generated by the Frobenius automorphism 'p : x 7! xp of Fp . There is a natural surjective map Dp ! Gal(Fp =Fp ); its injectivity is equivalent to the assertion that p is unrami ed in K=Q. Therefore, whenever this assertion is true, there is a unique p 2 Dp whose image in Gal(Fp =Fp ) is 'p . The automorphism p is then a well-de ned element of Gal(K=Q), the Frobenius automorphism for p. The various p are all conjugate under Gal(K=Q) and that the Frobenius automorphism for the conjugate of p by g is gp g;1. In particular, the various p are all conjugate; this justi es the practice of writing p for any one of them and stating that p is well de ned up to conjugation. We next introduce the concept of Frobenius elements in GQ = Gal(Q=Q). Let p again be a prime and let p now be a prime of the ring of integers of Q lying over p. To p we associate: (1) its residue eld Fp , which is an algebraic closure of Fp , and (2) a decomposition subgroup Dp of GQ . There is again a surjective map Dp ! Gal(Fp =Fp ). The Frobenius automorphism 'p topologically generates the target group. We shall use the symbol Frobp to denote any preimage of 'p in any Dp corresponding to a prime lying over p, and refer to Frobp as a Frobenius element for p in GQ . This element is doubly ill de ned. The ambiguity in Frobp results from the circumstance that p needs to be chosen and from the fact that Dp ! Gal(Fp =Fp ) has a large kernel, the inertia subgroup Ip of Dp . The usefulness of Frobp stems from the fact that the various p are all conjugate, so that likewise the subgroups Dp and Ip are conjugate. Thus if  is a homomorphism mapping GQ to some other group, the kernel of  contains one Ip if and only if it contains all Ip . In this case, one says that  is unrami ed at p; the image of Frobp is then an element of the target that is well de ned up to conjugation. Consider an elliptic curve A over Q and let ` be a prime number. The xed eld of A;` is a nite Galois extension K` =Q whose Galois group G` is a subgroup of GL(2; F` ). A key piece of information about the extension K`=Q is that its discriminant is divisible at most by ` and primes dividing the conductor of A. In other words, if p 6= ` is a prime number at which A has good reduction, then K`=Q is unrami ed at ` (see Exercise 15); one says that the representation A;` is unrami ed at p. Whenever this occurs, the construction described above produces a Frobenius element p in G` that is well de ned up to conjugation. Fix again an elliptic curve A and a prime number `, and let A;` : GQ ! GL(2; F`) be the associated representation. For each prime p not dividing ` at which A has good reduction the Frobenius p = A;` (Frobp ) is well de ned only up

1.5. Background material

159

to conjugation. Nevertheless, the trace and determinant of p are well de ned. The determinant of A;` is the mod ` cyclotomic character , so p = (Frobp ) = p 2 F` . On the other hand, one has the striking congruence tr(A;` (Frobp )) = p + 1 ; #A~(Fp ) (mod `):

1.5.3. Modular forms

We now summarize some background material concerning modular forms. Serre's book [96] is an excellent introduction (it treats only N = 1). One might also read the survey article [27] or consult any of the standard references [65, 66, 75, 108]. The modular group SL(2; Z) is the group of 2  2 invertible integer matrices. For each positive integer N , consider the subgroup 





a b 2 SL(2; Z) : N j c and a  d  1 (mod N ) : c d Let h be the complex upper half plane. A cusp form of integer weight k  1 and level N is a holomorphic function f (z ) on h such that     az + b = (cz + d)k f (z ) for all a b 2 ; (N ); f cz (1.5) 1 c d +d we also require that f (z ) vanishes at the cusps (see [108, x2.1]). We denote by Sk (;1 (N )) the space of weight-k cusp forms of level N . It is a nite dimensional complex vector space. When k  2 a formula for the dimension can be found in ;1 (N ) :=

[108, x2.6]. Modular forms are usually presented as convergent Fourier series

f (z ) =

1 X

n=1

an qn

where q := e2iz . This is possible because the matrices ( 10 1b ) lie in ;1 (N ) so that f (z + b) = f (z ) for all integers b. For the forms that most interest us, the complex numbers an are algebraic integers. The space Sk (;1 (N )) is equipped with an action of (Z=N Z) ; this action is given by



f (z ) 7! f jhdi(z ) := (cz + d);k f az + b



cz + d where ac db 2 SL(2; Z) is any matrix such that d  d (mod N ). The operator hdi = hdi is referred to as a \diamond-bracket" operator. For each integer n  1, the nth Hecke operator on Sk (;1 (N )) is an endomorphism Tn of Sk (;1 (N )). The action is generally written on the right: f 7! f jTn. The various Tn commute with each other and are interrelated by identities that express a given Tn in terms of the Hecke operators indexed by the prime factors of n. If p N is a prime de ne the operator Tp on Sk (;1 (N )) by ;




-

f jTp (z ) =

1 X

n=1

anp qn + pk;1

1 X

n=1

an (f jhpi)qnp :

160

RIBET AND STEIN, SERRE'S CONJECTURES

For p j N prime, de ne Tp by

f jTp (z ) =

1 X n=1

anp qn :

The Hecke algebra associated to cusp forms of weight k on ;1 (N ) is the subring

T := Z[: : : Tn : : : hdi : : : ]  End(Sk (;1(N ))) generated by all of the Tn and hdi. It is nite as a module over Z (see Exercise 20). The diamond-bracket operators are really Hecke operators, in the sense that they lie in the ring generated by the Tn; thus T = Z[: : : Tn : : : ]: An eigenform is a nonzero element f 2 Sk (;1 (N )) that is P a simultaneous eigenvector for every element of the Hecke algebra T. Writing f = an qn we nd

that an = a1 cn where cn is the eigenvalue of Tn on f . Since f is nonzero, a1 is also nonzero, so it is possible to multiply f by P a;1 1 . The resulting normalized eigenform wears its eigenvalues on its sleeve: f = cn qn . Because f is an eigenform, the action of the diamond bracket operators de nes a character " : (Z=N Z) ! C ; we call " the character of f . Associated to an eigenform f 2 Sk (;1 (N )) we have a system (: : : ap : : : ), p N , of eigenvalues. We say that f is a newform if this system of eigenvalues is not the system of eigenvalues associated to an eigenform g 2 Sk (;1 (M )) for some level M j N with M 6= N . Newforms have been extensively studied (see [2, 13, 69, 75]); the idea is to understand where systems of eigenvalues rst arise, and then reconstruct the full space Sk (;1 (N )) from newforms of various levels. -

1.5.4. Tate curves

The Tate curve is a p-adic analogue of the exponentiation of the representation C= of the group of an elliptic curve over C. In this section we recall a few facts about Tate curves; for further details, see [110, V.3]. Let K be a nite extension of Qp ; consider an elliptic curve E=K with split multiplicative reduction, and let j denote the j -invariant of E . By formally inverting the well-known relation j (q(z )) = q(1z ) + 744 + 196884q(z ) +


between the complex functions q(z ) = e2iz and j (z ), we nd an element q 2 K  with j = j (q) and jqj < 1. There is a Gal(Qp =K )-equivariant isomorphism E (Qp )  = Qp =qZ . The Tate curve, which we suggestively denote by Gm =qZ , is a scheme whose Qp points equal Qp =qZ . As a consequence, the group of n-torsion points on the Tate curve is identi ed with the Gal(Qp =K )-module fna (q1=n )b : 0  a; b  n ; 1g; here n is a primitive nth root of unity and q1=n is a xed nth root of q in Qp . In particular, the subgroup generated by n is invariant under Gal(Qp =K ), so the local Galois representation on E [n] is reducible. It is also known that the group of connected component of the reduction of the Neron model of E over Fp is a cyclic group whose order is ordp (q). The situation is summarized by the following table (taken from [88]):

1.5. Background material Complex case C=

161

p-adic case no p-adic analogue

exponential map e2iz

no exponential available C=qZ K  =qZ . Remark 1.2. When E has non-split multiplicative reduction over K , there is an unrami ed extension L over which E aquires split multiplicative reduction. 

1.5.5. Mod ` modular forms

There are several excellent papers to consult when learning about mod ` modular forms. The papers of Serre [95] and Swinnerton-Dyer [112] approach the subject from the point of view of Galois representations. Katz's paper [59] is very geometric. Edixhoven's paper [32] contains a clear description of the basic facts. See also Jochnowitz's paper [56].

CHAPTER 2

Optimizing the weight In [102, x2] Serre associated to an odd irreducible Galois representation  : GQ ! GL(2; F` ) two integers N () and k(), which are meant to be the minimal level and weight of a form giving rise to . Conjecture 2.1 (Strong conjecture of Serre). Let  : GQ ! GL(2; F`) be an odd irreducible Galois representation arising from a modular form. Then  arises from a modular form of level N () and weight k(). In this chapter, we are concerned with k(). We consider a mod ` representation  that arises from an eigenform of level N not divisible by `. Using results of Fontaine and Deligne, we motivate Serre's recipe for k(). In [32], Edixhoven also de nes an \optimal" weight, which sometimes diers from Serre's k(). Our de nition is an \average" of the two; for example, we introduce a tiny modi cation of k() when ` = 2. We appologize for any confusion this may cause the reader. Using various arguments involving the Eichler-Shimura correspondence and Tate's -cycles, Edixhoven showed in [31] that there must exist another form of weight at most k(), also of level N , which gives rise to . Some of Edixhoven's result rely on unchecked compatibilities that are assumed in [46]; however, when ` 6= 2 these results were obtained unconditionally by Coleman and Voloch in [17]. We sketch some of Edixhoven's arguments to convey the avor of the subject. Remark 2.2 (Notation). We pause to describe a notational shorthand which we will employ extensively in this chapter. If  : G ! Aut(V ) is a two-dimensional representation over a eld F, we will frequently write 

   



to mean that there is a basis for V with respect to which    ( x ) ( x ) (x) = (x) (x) 2 GL2 (F) for all x 2 G. If we do not wish to specify one of the entries we will simply write . Thus \  ( 0 1 )" means that  possesses a one-dimensional invariant subspace, and the action on the quotient is trivial.

2.1. Representations arising from forms of low weight

We rst consider irreducible Galois representations associated to newforms of low P weight. Fix a prime ` and suppose f = an qn is a newform of weight k and 163

164

RIBET AND STEIN, SERRE'S CONJECTURES

level N , such that ` N and 2  k  ` + 1. Let " : (Z=N Z) ! C denote the character of f . Fix a homomorphism ' from the ring of integers O of Q(: : : an : : : ) to F` . To abbreviate, we often write an for '(an ); thereby thinking of an as an element F` . Let  = f;' : GQ ! GL(2; F` ) be the two-dimensional semisimple odd Galois representation attached to f and ', and assume that  is irreducible. The recipe for N () depends on the local behavior of  at primes p other than `; the recipe for k() depends on the restriction jI` of  to the inertia group at `. Motivated by questions of Serre, Fontaine and Deligne described jI` in many situations. We distinguish two cases: the ordinary case and the non-ordinary case, which we call the \supersingular case." -

2.1.1. The ordinary case

Deligne (see [46, Prop. 12.1]) considered the ordinary case, in which  arises from a weight-k newform f with a` (f ) 6= 02 F` .He showed that  has a one-dimensional unrami ed quotient , so jD`  0  with (I` ) = 1 and  = k;1 ". The mod N character " is also unrami ed at `because` N . Since the mod ` cyclotomic k;1 character  has order ` ; 1 and jI`  0 1 ; the value of k modulo ` ; 1 is determined by jI` . In the case when k is not congruent to 2 modulo ` ; 1, the restriction jI` determines the minimal weight k(). We will discuss the remaining case in Section 2.2. -

2.1.2. The supersingular case and fundamental characters

Fontaine (see [32, x6]) investigated the supersingular case, in which  arises from a newform f with a` (f ) = 0 2 F` . We call such a newform f supersingular. To describe the restriction jI` of  to the inertia group at `, we introduce the fundamental characters of the tame inertia group. Fix an algebraic closure Q` of the eld Q` of `-adic numbers; let Qnr `  Q` denote the maximal unrami ed extension of Q` , and Qtm  Q the maximal tamely rami ed extension of Qnr ` ` `. tm nr The extension Q` is the compositum of all nite extensions of Q` in Q` of degree prime to `. Letting D` denote the decomposition group, I` the inertia group, It the tame inertia group, and Iw the wild inertia group, we have the following diagram:

Q` Iw

Qtm ` D`

It

Qnr` Zb

Q`

I`

2.2. Representations of high weight

p

165

n It is a standard fact (see, e.g., [44, x8]) that the extensions Qnr ` ( `), for all n tm not divisible by `, generate Q` . For n not divisible by `, the nth roots of unity n are contained in Qnr ` . Kummer theory (see [3]) gives, for each n, a canonical isomorphism p pn (pn `) : !  ; nr ) ;; Gal(Qnr ( ` ) = Q  ! 7 n ` ` n

`

Each isomorphism lifts to a map I` ! n that factors through the tame quotient It of I` . The groups n = n (Q` ) lie in the ring of integers Z` of Q` . Composing any of the maps It ! n with reduction modulo the maximal ideal of Z` gives a mod ` character It ! F` , as illustrated: / F It ❈ ❈ ✇ ; ` ❈❈ ❈❈ ❈!

= /

n (Q` )

✇✇ - ✇ ✇ ✇

n (F` ):

Let n = ` ; 1 with  > 0. The map It ! n de nes a character " : I` ! F` . Composing with each of the  eld embeddings F` ! F` gives the  fundamental characters of level  :

It

GG GG GG GG GG #

✇ ✇✇ ✇✇✇✇✇ ✇ ✇✇ ✇

F`  F`

F

; ✇ ✇ ; `: ✇✇ ✇ ✇ ✇ ✇✇ ✇ ; ✇ ✇✇✇✇✇ ✇✇

 maps

For example, the unique fundamental character of level 1 is the mod ` cyclotomic character (see Exercise 16). When  = 2, there are two fundamental characters, denoted and 0 ; these satisfy ` = 0 and ( 0 )` = . Let A be an elliptic curve over Q` with good supersingular reduction. In [93], Serre proved that the representation It ! Aut(A[`])  GL(2; F` ) is the direct sum of the two fundamental characters and 0 . One of the characters is It ! F`  GL(2; F` ) where F` is contained in GL(2; F` ) as a non-split Cartan subgroup of GL(2; F` ). More precisely, F` is embedded in GL(2; F` ) via the action of the multiplicative group of a eld on itself after a choice of basis. More generally, in unpublished joint work, Fontaine and Serre proved in 1979 that if f is a supersingular eigenform of weight k  `, then jI` : I` ! GL(2; F` ) factors through It and is a direct sum of the two character k;1 and ( 0 )k;1 . Note that k is determined by this representation, because it is determined modulo `2 ; 1. 2

2

2

2.2. Representations of high weight

Let D` be a decomposition group at ` and consider a representation  : D` ! GL(2; F` ) that arises from a newform f of possibly large weight k. Let ss denote the semisimpli cation of ; so ss =  if  is irreducible, otherwise ss is a direct

166

RIBET AND STEIN, SERRE'S CONJECTURES

sum of two characters  and . The following lemma of Serre (see [93, Prop. 4]) asserts that ss is tamely rami ed. Lemma 2.3. Any semisimple representation  is tame, in the sense that (Iw ) = 0. Proof. Since the direct sum of tame representations is tame, we may assume that  is simple. The wild inertia group Iw is the pro nite Sylow `-subgroup of I` : it is a Sylow `-subgroup because each nite Galois extension of Qtm ` has degree a power of `, and the order of It is prime to `; it is unique, because it is the kernel of Gal(Q` =Q`) ! Gal(Q` =Qtm ` ), hence normal. Because  is continuous, the image of D` is nite and we view  as a representation on a vector space W over a nite extension of F` . The subspace W Iw = fw 2 W : ( )w = w for all  2 Iw g is invariant under D`. It is nonzero, as can be seen by writing the nite set W as a disjoint union of its orbits under Iw : since Iw is a pro-`-group, each orbit has size either 1 or a positive power of `. The orbit f0g has size 1, and #W is a power of `, so there must be at least ` ; 1 other singleton orbits fwg; for each of these, w 2 W Iw . Since  is simple and W Iw is a nonzero D` -submodule, it follows that W Iw = W , as claimed. The restriction ss jI` is abelian and semisimple, so it is given by a pair of characters ; : I` ! F` : Let n be an integer not divisible by `, and consider the tower of elds p Qnr` ( n `) n

G

Qnr` Zb =hFrob` i

Q` pn nr nr in which G = Gal(Qnr = Gal(Qnr ` ( `)=Q` ), n  ` ( `)=Q` ), and Gal(Q` =Q` ) is topologically generated by a Frobenius element at `. Choose a lift g 2 G of Frob` , pand consider an element h 2 n corresponding to an element  2 n nr Gal(Qnr ` ( `)=Q` ). Then since g acts as the `th powering map on roots of unity, p p p p gg;p1 ( n `) = g(gp; n `) = g(g;ph n `) = g(hp) n ` = h` : n n n n pn

`

1

`

1

`

`

Applying the conjugation formula ghg;1 = h` to ss gives ss (ghg;1) = ss (h` ) =

ss (h)` . The two representations h 7! ss (h)` and h 7! ss (h) of It are thus equivalent via conjugation by ss (g); we have ss (g)ss (h)ss (g;1 ) = ss (ghg;1) = ss (h)` : Consequently, the pair of characters f; g is stable under the `th power map, so as a set f; g = f`; ` g: There are two possibilities: | The ordinary case: ` =  and ` = . | The supersingular case: ` = 6=  and ` =  6= .

2.2. Representations of high weight

167

In the rst case  and take values in F` and in the second case they take values in F` but not in F` . By the results discussed in Section 2.1, this terminology is consistent with the terminology introduced above. We rst consider the supersingular case. Let denote one of the fundamental characters of level 2, and write  = n, = n` , with n an integer modulo `2 ; 1. Next write the smallest non-negative representative for n in base `: n = a + `b with 0  a; b  ` ; 1. Then `n  b + `a (mod `2 ; 1). Switching  and permutes a and b so, relabeling if necessary, we may assume that a  b. If a = b, then  = a( 0 )a = a , so  takes values in F` , which is not the supersingular case; thus we may assume that 0  a < b  ` ; 1: We now factor out by a power of the cyclotomic character:  = n = a ( 0 )b = a ( 0 )a ( 0 )b;a = a ( 0 )b;a = a b;a: Put another way,  b;a  0 ss a   0 ( 0 )b;a :  k;  The untwisted representation is 0 ( 0 0 )k; , where k = 1 + b ; a. Since 2  1+ b ; a  ` ; 1, the weight of the untwisted representation is in the range considered above. Thus we are in good shape to de ne k(). Before giving k() it is necessary to understand how the weight changes upon twisting by a power of the cyclotomic character . This problem is addressed by the theory of mod ` modular forms, rst developed by Serre [95] and Swinnerton-Dyer [112], then generalized by Katz [59]. A brief review of the geometric theory, which gives an excellent de nition of mod ` modular forms, can be found in [32, x2], [35, x1], or [46, x2]. In [61], Katz de ned spaces of mod ` modular forms, and a q-expansion map M  : Mk (;1 (N ); F` ) ! F` [[q]]: 2

1

1

k0

This map is not injective, because both the Hasse invariant of weight ` ; 1 and the constant 1 have the same q-expansion. De nition 2.4. The minimal weight ltration w(f ) 2 Z of an element f of the ring of mod ` modular forms is the smallest integer k such that the q-expansion of f comes from a modular form of weight k; if no such k exists, do not de ne w(f ). De nition 2.5. De ne the operator  = q dqd on q-expansions by (P anqn) = P nan qn . For example, if f is an eigenform of weight k, then there isPa mod ` eigenform f of weight k + ` + 1, still of level N , whose q-expansion is ( an qn ). Theorem 2.6. Let f be a mod ` modular form. Then w(f ) = w(f ) + ` + 1 if and only if ` w(f ). In addition, if ` j w(f ) then w(f ) < w(f ) + ` + 1. -

2.2.1. The supersingular case

We now give Serre's recipe for k() in the supersingular case. The minimal weight before twisting is 1+ b ; a, which is a positive integer that is not divisible by `. Each

168

RIBET AND STEIN, SERRE'S CONJECTURES

twist by  adds ` + 1 to the weight, so in the supersingular case we are motivated to de ne k() := (1 + b ; a) + a(` + 1) = 1 + `a + b: We have to check that at each step the weight is prime to `, so the minimal weight does not drop during any of the a twists by . Since 1 < 1 + b ; a < ` and (1 + b ; a) + a(` + 1)  (` ; 1) + (` ; 2)(` + 1) < `2 ; the weight can only drop if there exists c with 1  c < a such that (1 + b ; a) + c(` + 1)  0 (mod `): If this occurred, then c  a ; b ; 1 (mod `). But 1  c < a  ` ; 2, so either c = a ; b ; 1, which implies c  0 since a < b, or c = ` + a ; b ; 1 = a + ` ; 1 ; b  a, which would be a contradiction. Assume that  : GQ ! GL(2; F` ) arises from an eigenform f such that a` (f ) = 0 2 F` . Now we sketch Edixhoven's proof that  arises from a mod ` eigenform of weight k(). Let ss denote the semisimpli cation of the restriction of  to a decomposition group at `. The restriction of ss to the inertia group at ` is  n  ss jI`  0 ( 0 0 )n ; where and 0 = ` are the two fundamental characters of level 2. If necessary, reorder and 0 so that n = a + b` with 0  a < b  ` ; 1. Then n = a+b` = a ( 0 )b = a ( 0 )a ( 0 )b;a = a ( 0 )b;a ; and  0 b;a  0 ss jI`  a ( 0 ) b;a : Recall that, motivated by Fontaine's theorem on Galois representations arising from supersingular eigenforms, we de ned k() = a(` + 1) + (b ; a + 1) = 1 + `a + b: The rst step in showing that  arises from a form of weight k(), is to recall the well known result that, up to twist, all systems of mod ` eigenvalues occur in weight at most ` + 1. This is the subject of the next section.

2.2.2. Systems of mod ` eigenvalues Theorem 2.7. Suppose  is modular of level N and some weight k, and that ` N . Then some twist  ; is modular of weight  ` + 1 and level N . -

This is a general theorem, applying to both the ordinary and supersingular cases. See Serre [97, Th. 3] when N = 1; signi cant further work was carried out by Jochnowitz [55] and Ash-Stevens [1, Thm. 3.5] when `  5. Two proofs are given in [32, Thm. 3.4 and x7]. The original method of Serre, Tate, and Koike for treating questions like this is to use the Eichler-Selberg trace formula. As Serre has pointed out to us, the weight appears in that formula simply as an exponent; this makes more or less clear that a congruence modulo `2 ; 1 gives information on modular forms mod `.

2.2. Representations of high weight

169

As a digression, we pause to single out some of the tools involved in one possible proof of Theorem 2.7. Note that by twisting we may assume without loss of generality that k  2. The group ;1 (N ) acts by matrix multiplication on the real vector space R2 . The Eichler-Shimura correspondence (see [108, x8.2]) is an isomorphism of real vector spaces  = H 1 (; (N ); Symk;2 (R2 )): Sk (;1 (N )) ;;;;! P 1 The parabolic (or cuspidal) cohomology group HP1 is the intersection, over all cusps  2 P1 (Q), of the kernels of the restriction maps res : H 1 (;1 (N ); Symk;2 (R2 )) ! H 1 (; ; Symk;2 (R2 )); where ; denotes the stabilizer of . For xed z0 in the upper half plane, the Eichler-Shimura isomorphism sends a cusp form f to the class of the cocycle c : ;1 (N ) ! Symk;2 (R2 ) induced by

7!  k;2

where z1

(z0 )

Z

z0

 k;2

Re f (z ) z1  

!

dz ;

 

denotes the image of z1


z1 2 Symk;2 (C2 ), and integration

is coordinate wise. There is an action of the Hecke algebra T on HP1 (;1 (N ); Symk;2 (R2 )); such that the Eichler-Shimura correspondence is an isomorphism of T-modules. The forms whose periods are integral form a lattice HP1 (;1 (N ); Symk;2 (Z2 )) inside HP1 (;1 (N ); Symk;2 (R2 )): Reducing this lattice modulo ` suggests that there is a relationship between mod ` modular forms and the cohomology group HP1 (;~ 1 (N ); Symk;2 (F2` )); where ;~ 1 (N ) is the image of ;1 (N ) in SL(2; F` ). Serre and Hida observed that for k ; 2  ` the ;~ (N ) representations Symk;2 (F2` ) are sums of representations 0 ;2 1 2 k arising in Sym (F` ) for k0 < k. This essential idea is used in proving that all systems of eigenvalues occur in weight at most ` + 1.

2.2.3. The supersingular case revisited

Let  be a supersingular mod ` representation that arises from some modular form. By Theorem 2.7 there is a form f of weight k  ` + 1 such that ;   f . In fact, we may assume that 2  k  `; when k = ` + 1 a theorem of Mazur (see [32, Thm. 2.8]) implies that there is a form of weight 2 giving rise to f , and when k = 1 we multiply f by the weight ` ; 1 Hasse invariant. To show that w( f ) = k() we investigate how application  n of the -operator changes the minimal weight. We have (f  )jI`  0 ( 0 0 )n with n = a + b` and a < b. Fontaine's theory (see Section 2.1) identi es the characters corresponding to f jI` as powers k;1 and ( 0 )k;1 of the fundamental characters. This gives an equality of unordered sets f k;1  ; ( 0 )k;1  g = f n; ( 0 )n g: It is now possible to compute w( f ) by considering two cases, corresponding to the ways in which equality of unordered pairs can occur.

170

RIBET AND STEIN, SERRE'S CONJECTURES

Case 1. Suppose that k;1 = ( 0)n . Since  = `+1, we have

k;1+(`+1) = k;1  = ( 0 )n = ( 0 )a+b` = b+a`: Comparing exponents of gives (2.1) k ; 1 + (` + 1)  b + a` (mod `2 ; 1); which reduces modulo ` + 1 to k ; 1  b ; a (mod ` + 1); because 2  k  `, this implies that k = 1 + b ; a. Reducing (2.1) modulo ` ; 1 and substituting k = 1 + b ; a gives b ; a + 2  b + a (mod ` ; 1); we nd the possible solutions  = a + m(` ; 1)=2 with m an integer. No solution  = a + m(` ; 1)=2, with m odd, satis es (2.1), so  = a as an integer mod ` ; 1. Finally, we apply Theorem 2.6 and argue as in the end of Section 2.2, to show that w(a f ) = w(f ) + a(` + 1) = 1 + b ; a + a` + a = 1 + b + a` = k(): Case 2. Suppose that k;1 = n. Then k;1+(`+1) = k;1  = n = a+b` : Comparing powers of , we obtain (2.2) k ; 1 + (` + 1)  a + b` (mod `2 ; 1); which reduces modulo ` + 1 to k ; 1  a ; b (mod ` + 1); thus k = ` + 2 ; (b ; a): The dierence b ; a must be greater than 1; otherwise k = ` + 1, contrary to our assumption that 2  k  `. Reducing (2.2) modulo ` ; 1 gives k ; 1 + 2  a + b (mod ` ; 1); substituting k = ` + 2 ; (b ; a) we nd that  = b ; 1 + m(` ; 1)=2 with m an integer. If m is odd, then  does not satisfy (2.2), so  = b ; 1 as an integer modulo ` ; 1. It remains to verify the equality w(b;1 f ) = w(). Unfortunately, k = ` + 2 ; (b ; a) is not especially telling. The argument of Case 1 does not apply to compute w( f ); instead we use -cycles. Because f is supersingular, Fermat's Little Theorem implies that `;1 f = f . We use Tate's theory of -cycles (see [32, x7] and [55]) to compute w(b;1 f ). The -cycle associated to f is the sequence of integers w(f ); w(f ); w(2 f ); : : : ; w(`;2 f ); w(f ): The -cycle for any supersingular eigenform must behave as follows (see Theorem 2.6):

go up



✈ ✈ ✈ ✈✈ ✈ ✈ ✈✈ ✈✈ ✈

✈ : ✈ ✈ ✈✈ ✈ ✈ ✈ ✈ ✈ ✈ : : :✈ ✈ ✈ ✈✈

❄❄

❥ 5 ✴ ❥ ❥ ❥ ✴✴ ❥ ❄❄ ❥ ❥ ✴✴ ❄ ❄ drop ❥ ❥ ❥ ❥ ❥ ❄❄ ❥ ❥ go up : : : ✴✴ ❥ ❥ ❄❄ ❥ ❥ ❥ ✴✴  ❥ ❥ ❥ ❥ ❥ ✴ ✴ drop  ✴✴ ✴✴ ✴✴ ✴

go up : : : , drop once, go up : : : , drop to original weight



2.3. Distinguishing between weights 2 and ` + 1

171

Knowing this, we can deduce the exact -cycle. List ` numbers starting and ending with k: k; k + (` + 1); k + 2(` + 1); : : : ; k + (` ; k)(` + 1); ` + 3 ; k; (` + 3 ; k) + (` + 1); : : : ; (` + 3 ; k) + (k ; 3)(` + 1);

k

The rst and second lines contain ` + 1 ; k and k ; 2 numbers, respectively. All told, ` numbers are listed; this must be the -cycle. It is now possible to compute w(b;1 f ). If b ; 1  ` ; k = ` ; (` + 2 ; b + a) = ;2 + b ; a; then a  ;1, a contradiction; thus b ; 1 > ` ; k. It follows that w(b;1 f ) = ` + 3 ; k + (` + 1)(b ; 2 ; (` ; k)) = 1 + b + a` = k(); verifying Serre's conjecture in this case.

2.2.4. The ordinary case

We next turn to the ordinary case, in which 

jI`  0 



with ; : I` ! F` powers of the cyclotomic character. View jI` as the twist of a representation in which the lower right entry is 1: 







    ;1  : 0 0 1

To determine the minimal weight of a form giving rise to jI` , it is necessary to develop an ordinary version of -cycles. In general this so we make  isi complicated,    the simplifying assumption that = 1; then jI`  0 1 with 1  i  ` ; 1. Deligne showed that if f is of weight k and = 1, then the associated representation   is 0k; 1 with 2  k  ` + 1. Motivated by this, our rst reaction is to de ne k() to be i +1. This de nition does not distinguish between the extreme weights 2 and ` +1 because they are congruent modulo ` ; 1. Given a representation  arising from a form of weight either 2 or ` + 1, we cannot, in general, set k() = 2. For example, suppose f =
is the level 1 cusp form of weight 12 and  is the associated mod 11 representation. It would be wrong to set k() = 2, because there is no cusp form of weight 2 and level 1. Warning: When ` = 2 and our k() is 3, Serre replaced k() by 4 because there are no weight-3 modular forms whose character is of degree coprime to ` = 2. 1

2.3. Distinguishing between weights 2 and ` + 1

We continue to motivate the de nition of k(). Consider a representation  : GQ ! GL(2; F` ) that arises from a newform f of the optimal level N = N () and weight k satisfying 2  k  ` +1. Assume that f is ordinary in the sense that a`(f ) 6= 0 2 F` . Then, as discussed in Section 2.1, 



jI`  0k; 1 ; 1

172

RIBET AND STEIN, SERRE'S CONJECTURES

so jI` determines k modulo ` ; 1. This suggests a way to de ne k() purely in terms of the Galois representation , at least when k 62 f2; ` + 1g. The key to de ning k() when k = 2 or k = ` + 1 is good reduction. To understand why this is so, we brie y summarize Shimura's geometric construction of Galois representations associated to newforms of weight 2.

2.3.1. Geometric construction of Galois representations

P

Shimura attached mod ` representations to a weight-2 newform f = an qn of level N . Let E be the totally real or CM eld Q(: : : an : : : ). In [108, Thm. 7.14], Shimura described how to associate to f an abelian variety A = Af over Q of dimension [E : Q] furnished with an embedding E ,! EndQ A (see also Conrad's appendix). The mod ` representations attached to f are then found in the `-torsion of A. Over the complex numbers, the abelian variety A is found as a quotient of the Jacobian of the Riemann surface X1 (N ) := ;1 (N )nh = ;1 (N )nh [ fcuspsg: The Riemann surface X1 (N ) has a structure of algebraic curve over Q; it is called the modular curve of level N . Its Jacobian J1 (N ) is an abelian variety over Q which, by work of Igusa, has good reduction at all primes ` N . The dimension of J1 (N ) equals the genus of X1 (N ); for example, when N = 1, the curve X1 (1) is isomorphic over Q to the projective line and J1 (1) = 0. There are (at least) two functorial actions of the Hecke algebra T on J1 (N ), and (at least) two de nitions of J1 (N ). In the next section we will x choices, and then construct A as the quotient of J1 (N ) by the image of the annihilator in T of f . -

2.3.1.1. Hecke operators on J1 (N ). We pause to formulate a careful de nition of X1 (N ) and of our preferred functorial action of the Hecke operators Tp on J1 (N ). For simplicity, we assume that N > 4 and p N . Following [46, Prop. 2.1] there is a smooth, proper, geometrically connected algebraic curve X1 (N ) over Z[1=N ] that represents the functor assigning to each Z[1=N ]-scheme S the set of isomorphism classes of pairs (E; ), where E is a generalized elliptic curve over S and  : (N )S ,! E sm [N ] an embedding of group schemes over S whose image meets every irreducible component in each geometric ber. Let X1 (N; p) be the ne moduli scheme over Z[1=N ] that represents the functor assigning to each Z[1=N ]-scheme S the set of isomorphism classes of triples (E; ; C ), where E is a generalized elliptic curve over S ,  : (N )S ,! E sm [N ] an embedding of group schemes over S , and C a locally free subgroup scheme of rank p in E sm [p], such that im()  C meets every irreducible component in each geometric ber of E . Let 1 ; 2 : X1 (N; p) ! X1 (N ) over Z[1=N ] be the two standard degeneracy maps, which are de ned on genuine elliptic curves by 1 (E; ; C ) = (E; ) and 2 (E; ; C ) = (E 0 ; 0 = '), where E 0 = E=C and ' : E ! E 0 is the associated p-isogeny. The Hecke operator Tp = (Tp ) acts on divisors D on X1 (N )=Q by Tp (D) = (1 )  2 D: For example, if (E; ) is a non-cuspidal Q-point, then X Tp (E; ) = (E 0 ; '    [p];1 ); -

2.3. Distinguishing between weights 2 and ` + 1

173

The Hecke operatorThe Hecke operator Tp = (Tp ) acts on divisors D on X1 (N )=Q by Tp (D) = (1 )  2 D: For example, if (E; ) is a non-cuspidal Q-point, then X

Tp (E; ) = (E 0 ; '    [p];1 ); where the sum is over all isogenies ' : E ! E 0 of degree p, and Tp = (Tp ) acts on divisors D on X1 (N )=Q by Tp (D) = (1 )  2 D: For example, if (E; ) is a non-cuspidal Q-point, then X Tp (E; ) = (E 0 ; '    [p];1 ); where the sum is over all isogenies ' : E ! E 0 of degree p, and where the sum is over all isogenies ' : E ! E 0 of degree p, and [p];1 is the inverse of pth powering on N . This map on divisors de nes an endomorphism Tp of the Jacobian J1 (N ) associated to X1 (N ) via Picard functoriality. For each prime p there is an involution hpi of X1 (N ) called a diamond bracket operator, de ned functorially by

hpi(E; ) = (E;   [p]):

The diamond bracket operator de nes a correspondence, such that the induced map (hpi) on J1 (N ) is (hpi) (E; ) = (E;   [p;1 ]): If (Tp ) denotes the pth Hecke operator as de ned in [46, x3], then (Tp ) = Tp  (hp;1 i) ; Thus our Tp diers from Gross's (Tp ) . Furthermore, upon embedding X1 (N ) into J1 (N ) and identifying weight-2 cusp forms with dierentials on J1 (N ), Gross's (Tp ) induces, via Albanese functoriality, the usual Hecke action on cusp forms, whereas ours does not. In addition, we could have de ned X1 (N ) by replacing the group scheme N by (Z=N Z). In this connection, see the discussion at the end of Section 5 of [26] and [35, x2.1]. 2.3.1.2. The representations attached to aPnewform. Again let O be the ring of integers of E = Q(: : : an : : : ), where f = an qn is a weight-2 modular forms on ;1 (N ). Recall that A = Af is the quotient of J1 (N ) by the image of the annihilator in T of f . In general, O need not be contained in End A. However, by replacing A by an abelian variety Q-isogenous to A, we may assume that O is contained in End A (see [108, pg. 199]). Let  be a maximal ideal of O and set A[] := fP 2 A(Q) : xP = 0 all x 2 g: By [108, Prop. 7.20, pg 190], dimO= A[] = 2, so A[] aords a 2-dimensional Galois representation, which is well-de ned up to semisimpli cation. Let f; : GQ ! A[]ss be the semisimpli cation of A[].

174

RIBET AND STEIN, SERRE'S CONJECTURES

2.3.1.3. Good reduction. De nition 2.8. A nite group scheme G over Qnr` is said to have good reduction, or to be nite at, if it extends to a nite at group scheme over the ring of integers OQ` of Qnr` . Proposition 2.9. The representation f; is nite at at each prime p N . Proof. The nite at group scheme extending A[] is the scheme theoretic closure of A[] in a good model A=OQ` of A. Such a model exists because A has good reduction at p. Consider again a Galois representation  as in the beginning of Section 2.3 such that jI`  0k; 1 . If k 6 2 (mod ` ; 1) then k() is de ned to equal k. If k  2 (mod ` ; 1), then ( if  is nite at, k() := 2 ` + 1 otherwise. nr

-

nr

1

2.4. Representations arising from elliptic curves

Theorem 2.10. Suppose A=Q is a semistable elliptic curve and that A;` is irreducible. Let
A denote the minimal discriminant of A. The representation A;` is nite at at ` if and only if ` j ord`
A . If p 6= `, then A;` is unrami ed at p if and only if ` j ordp
A : Proof. The rst statement is Proposition 5 of [102]. When A has good reduction at p, the second statement holds (see Exercise 15). Suppose A has multiplicative reduction at p. There is an unrami ed extension K of Qp such that A has split multiplicative reduction at p. Consider the Tate curve Gm=qZ over K associated to A. Thus Qp=qZ = A(Qp ) as Gal(Qp=K )-modules. The `-torsion points A[`] correspond to the points f`a (q1=` )b : 0  a; b < `g in the Tate curve. The extension K (` ; q1=` ) of K is unrami ed because ` 6= p and ordp (q) = ordp (
A ) is divisible by `. Since an unrami ed extension of an unrami ed extension is unrami ed, the extension K (` ; q1=` ) of Qp is unrami ed, which proves the second part of the theorem.

2.4.1. Frey curves

Using Theorem 2.10 we see that the Shimura-Taniyama conjecture together with Serre's conjecture implies Fermat's Last Theorem. Suppose (a; b; c) is a solution to the Fermat equation a` + b` = c` with `  11 and abc 6= 0. Consider the Frey curve A given by the equation y2 = x(x ; a` )(x + b` ); it is an elliptic curve ` with discriminant
A = ((abc2 ) ) : By [93, x4.1, Prop. 6] the representation A[`] is irreducible. Theorem 2.10 implies that A;` is unrami ed, except possibly at 2 and `. Thus N () j 2, and k() = 2 since ` j ord` (
A ). But there are no cusp forms of level 2 and weight 2. The modularity of A (proved in [114, 117]), together with the weak conjecture of Serre (enough of which is proved in [84]), leads to a contradiction. 2

8

2.4.2. Examples

2.5. Companion forms

175

Using Theorem 2.10 we can frequently determine the Serre invariants N () and k() of a representation  attached to an elliptic curve. When N () < N , it is illustrative to verify directly that there is a newform of level N () that also gives rise to . For example, there is a unique weight-2 normalized newform f = q + q2 ; q3 ; q4 ; 2q5 ; q6 + 4q7 ; 3q8 + q9 +


on ;0 (33). One of the elliptic curves associated to f is the curve A given by the equation y2 + xy = x3 + x2 ; 11x: The discriminant of A is
= 36
112 and the conductor is N = 3
11. Because A is semistable and there are no elliptic curves 3-isogenous to A, the associated mod 3 representation  = A;3 : GQ ! Aut(A[3]) is surjective (see Section 1.4). Since 3 j ord3
A , the Serre weight and level are k() = 2 and N () = 11. As predicted by Serre's conjecture, there is a weight-2 newform on ;0 (11) such that if B is one of the three elliptic curves of conductor 11 (it does not matter which), then B [3]  A[3] as representations of GQ . Placing the eigenforms corresponding to A and B next to each other, we observe that their Fourier coecients are congruent modulo 3: fA = q +q2 ;q3 ;q4 ;2q5 ;q6 +4q7 ;3q8 +q9 +


fB = q ;2q2 ;q3 +2q4 +q5 +2q6 ;2q7 ;2q9 +


: Next consider the elliptic curve A cut out by the equation y2 + y = x3 + x2 ; 12x + 2: It has conductor N = 141 = 3
47 and discriminant
= 37
47. Since ord3 (
) is divisible by 7, the mod 7 representation A;7 has Serre invariants k(A;7 ) = 2 and N (A;7) = 47. In con rmation of Serre's conjecture, we nd a form f 2 S2 (;0 (47)) that gives rise to A;7 . The Fourier coecients of f generate a quartic eld. Next consider A;3 , whose Serre invariants are N (A;3 ) = 47 and, since 3 does not divide ord3 (
), k(A;3 ) = ` + 1 = 4. In S4 (;0 (47)) there are two conjugacy classes of eigenforms, which are de ned over elds of degree 3 and 8, respectively. The one that gives rise to A;3 is g = q + aq2 + (;1=2a2 ; 5=2a ; 1)q3 + (a2 ; 8)q4 + (a2 + a ; 10)q5 +


; where a3 + 5a2 ; 2a ; 12 = 0.

2.5. Companion forms

Suppose f is a newform of weight k with 2  k  ` +1. Let ` be an ordinary prime, so a`(f ) is not congruent to 0 modulo a prime  lying over ` and 



k;1  f; jI`  0 1 :

Is this representation split or not? Put another way, can  be taken equal to 0, after an appropriate choice of basis? For how many ` do these representations split? We suspect that the ordinary split primes ` are in the minority, among all primes. How can we quantify the number of split primes? If  = 0, then   1 0 jI`  0 k;1 ;

176 so

RIBET AND STEIN, SERRE'S CONJECTURES 



`;k jI` `;k  0 01 : Assume that 2  1 + ` ; k  ` + 1, so k( `;k ) = 1 + ` ; k. Using the -operator we see that  `;k is modular, of some weight and level. To say that it is modular of Serre's conjectured weight k() is to make a much strong statement. If  `;k is indeed modular of weight 1+ ` ; k, then by de nition there exists an eigenform g of weight 1 + ` ; k with g  f `;k . Such an eigenform g, if it exists, is called a companion of f . The existence of g is far from obvious. We can extend the notion of companion form to the case when k() = `. In this case the companion has weight 1. If  is unrami ed at `, then we expect  to

also arise from a weight-1 eigenform. The existence of a companion form was proved (assuming unchecked compatibilities) in most cases in which k < ` by Gross in [46] and in a few cases when k = `. Using new methods, Coleman and Voloch [17] proved all cases except k = ` = 2. The arguments of Coleman and Voloch do not require veri cation of Gross's unchecked compatibilities.

CHAPTER 3

Optimizing the level Consider an irreducible Galois representation  : GQ ! GL(2; F` ) that arises from a newform of weight k and level N . Serre de ned integers k() and N (), and conjectured that  arises from a newform of weight k() and level N (). In Chapter 2 we sketched Edixhoven's proof that if ` N then  arises from an newform of weight k() and level N . In this chapter, we introduce some of the techniques used in proving that  arises from a newform level N (). For more details, see [84, 87]. In [102, x1.2] Serre de ned the optimal level N () Q as the prime-to-` part of the Artin conductor of . Recall that N () is a product pn(p) over prime numbers p 6= `. The integer n(p) is de ned by restricting  to a decomposition group Dp at p. Consider the sequence of rami cation groups G0  G1 


 Gi 


where G0 is the inertia subgroup Ip of Dp . Let V be a vector space over F` aording the representation , and for each i  0 let Vi be the subspace of V consisting of those v 2 V that are xed by Gi . Then -

n(p) :=

1 X

1 dim V=V : i ( G : Gi ) 0 i=0

3.1. Reduction to weight 2

The optimal level N () is not divisible by `. The rst step in level optimization is to strip the power of ` from N . When ` is odd, this is done explicitly in [87, x2]; for the case ` = 2 see [9, x1]. Many of the arguments and key ideas are due to Serre [94]. This proof that ` can be stripped from the level uses concrete techniques of Serre [95, x3], [98, Thm. 5.4], and Queen [78, x3]; it involves multiplying f by suitable Eisenstein series and taking traces. Katz's theory of `-adic modular forms suggests an alternative method. A classical form of weight 2 and level M`m is an `-adic form of level M ; the mod ` reduction of this form is classical of level M and some weight, and is congruent to f . See the appendices of [60] and the discussions in [49, x1] and [50, x1]. The next step is to replace f by a newform of weight between 2 and ` + 1 that gives rise to a twist of . Twisting  by the mod ` cyclotomic character  preserves N ; this is because   arises from (f ) = q dqd (f ), which also has level N . Theorem 2.7 asserts that some twist  i of  arises from a form g of weight between 2 and ` + 1. If  i arises from a newform of level N , then  also arises from a newform of the same level, so we can replace f by g and k by the weight of g. By results discussed in Chapter 2, we may assume that k = k( i ). For the case ` = 2 see [9, Prop. 1.3(a)]. 177

178

RIBET AND STEIN, SERRE'S CONJECTURES

We have reduced to considering a representation  that arises from a newform f of weight k() and level N not divisible by `. The weight satis es 2  k()  ` + 1, but N need not equal N (). That N is a multiple of N () is a theorem proved by both Carayol [12] and Livne [70, Prop. 0.1]. In order to lower N it is convenient to work systematically with form of weight 2. Paradoxically, even though we have just taken all powers of ` out of N , we are now going to allow one power of ` back into N . This allows us to reduce to weight 2 and realize  as a group of torsion points on an abelian variety. An alternative approach (see [41, 57]) is to avoid this crutch and work directly with representations coming from arbitrary weights between 2 and ` + 1; these are realized in etale cohomology groups. This later approach has the advantage that X0 (N ) has good reduction at `. Reduction to weight 2 is accomplished using a general relationship that originates with ideas of Koike and Shimura. In characteristic `, eigenforms of level N whose weights satisfy 2 < k  ` + 1 correspond to eigenforms of weight 2 and level `N (see [87, Thm. 2.2]): n

2 < k  ` + 1, level N

o

o /o /o /o /o /o /

n

o

k = 2, level `N :

Thus we can and do work with weight 2 and level (

N  := N

if k = 2, N` if k > 2.

3.2. Geometric realization of Galois representations

To understand representations arising from modular forms, it is helpful to realize these representations inside of geometric objects such as J := J1 (N  ). These representations are constructed geometrically with the help of the Hecke algebra T := Z[: : : Tn : : : ]; which was de ned in Section 2.3. Recall that T is a commutative subring of EndQ J that is free as a module over Z, and that its rank is equal to the dimension of J . When N is cube free, T is an order in a product of integer rings of number elds; this is a result of Coleman and Edixhoven (see [16, Thm. 4.1]). In contrast, the Hecke operators Tp , for p3 j N , are usually not semisimple (see Exercise 3). It is fruitful to view a newform f as a homomorphism T ! O = Z[: : : an : : : ]; Tn 7! an: Letting ' : O ! F` be the map sending ap to tr((Frobp )) 2 F` , we obtain an exact sequence 0 ! m ! T ! F` with m a maximal ideal. Let  : GQ ! GL(2; F` ) be an irreducible Galois representation that arises from a weight-2 newform f . The next step, after having attached a maximal ideal m to f and ', is to nd a T=m-vector space aording  inside of the group of `-torsion points of J . Following [71, xII.7], we consider the T=m-vector space J [m] := fP 2 J (Q) : tP = 0 all t 2 mg  J (Q)[`]  (Z=`Z)2g : Since the endomorphisms in T are Q-rational, J [m] comes equipped with a linear action of GQ .

3.3. Multiplicity one

179

That tr((Frobp )) and det((Frobp )) both lie in the sub eld T=m of F` suggests that  has a model over T=m, in the sense that  is equivalent to a representation taking values in GL(2; T=m)  GL(2; F` ). Lemma 3.1. The representation  has a model m over the nite eld T=m. Proof. This is a classical result of I. Schur. Brauer groups of nite elds are trivial (see e.g., [100, X.7, Ex. a]), so the argument of [99, x12.2] proves the lemma. Alternatively, when the residue characteristic ` of T=m is odd, the following more direct proof can be used. Complex conjugation acts through  as a matrix with distinct F` -rational eigenvalues; another well known theorem of Schur [90, IX a] (cf. [116, Lemme I.1]) then implies that  can be conjugated into a representation with values in GL(2; T=m).

3.3. Multiplicity one

Let Vm be a vector space aording m. Under the assumption that m is absolutely irreducible, Boston, Lenstra, and Ribet (see [6]) proved that J [m] is isomorphic as a GQ -module to a sum of copies of Vm :

J [m] 

t M i=1

Vm :

The number of copies of Vm is called the multiplicity of m. When ` is odd, the hypothesis of irreducibility of m is equivalent to absolute irreducibility (see Exercise 3). Proposition 3.2. The multiplicity t is at least 1. Proof. Let T  End(J ) be the Hecke algebra associated to J . Because T Z` is an algebra of nite rank over the local ring Z` , we have a decomposition M T Z` = T ; j`

where  runs through the maximal ideals of T lying over `, and T denotes the completion of T at  (see, e.g., [37, Cor. 7.6]). The Tate module n Tate` J := Hom(Q` =Z` ; [n1 J [`n ])  = lim ; J [` ] is a free Z` -module of rank equal to twice the dimension of J . For each maximal ideal  of T lyingPover `, let e 2 T Z` denote the corresponding idempotent; P thus e2 = e and j` e = 1. The map x 7!  e x gives a decomposition 

= Tate` J ;;;;!

M

j`

e Tate` J:

The ring End(J ) Z` operates faithfully on Tate` J (see, e.g., [74, Lem. 12.2]), so each summand e Tate` J is nonzero. Set Tate J := Hom(Q` =Z` ; [n1 J [n ]): We claim that Tate J is identi ed with e Tate` J under the natural inclusion Tate J  Tate` J . Denote by ~ the maximal ideal in T Z` generated by . Let n be a positive integer, and let I be the ideal in T generated by `n . Because T is

180

RIBET AND STEIN, SERRE'S CONJECTURES

a local ring with maximal ideal ~, there is an integer m such that ~m  I . Since I is principal and generated by `n , and T acts on e J [`n ] through T , we have e J [`n ] = (e J [`n ])[I ]  (e J [`n ])[~m ]  (e J [`n ])[m ]  J [m ]: This shows that e Tate` J  Tate J . Next suppose 0 6=  and let n be a positive integer. Since T acts on J [n ] through T=n = T =~n , we have e0 J [n ] = 0, so X J [ n ] = e0 J [n ] = e J [n ]: all 0

The other inclusion Tate J = e Tate J  e Tate` J , which we need to prove equality, then follows. We apply the above conclusion with  = m. Since Tatem J 6= 0, some J [mr ] is nonzero; let r be the smallest such integer. Following [71, p. 112], observe that for each generating set of elements a1 ; : : : ; at of the T=m-vector space mr;1=mr , the map x 7! a1 x 


 at x is an injection of the module J [mr ]=J [mr;1] into the direct sum of t copies of J [m]. Thus J [m] is nonzero. The special case t = 1, in which the multiplicity is one, plays a central role in the development of the theory. A detailed summary of multiplicity one results can be found in [32, x9], and some supplementary results are contained in [117, Thm. 2.1]. In general, the multiplicity can be greater than one (see [72, x13] and [63]).

3.3.1. Multiplicity one representations

Let  : GQ ! GL(2; F` ) be an irreducible modular Galois representation such that 2  k()  ` + 1: Consider pairs (N; ) where N  1 is an integer with the property that ` N if k() = 2 and ` jj N if k() > 2, together with maps  : TN ! F` , such that (Tp ) = tr((Frobp )) and (phpi) = det((Frobp )) for almost all p. Here TN is the Hecke algebra associated to S2 (;1 (N )). Note that if (N; ) is such a pair and m = ker(), then -

  m T=m F` ;

where  : T=m ,! F` and m is the unique (up to isomorphism) semisimple representation over F` such that tr(m (Frobp )) = (Tp ) det(m (Frobp )) = (phpi) for almost all p. De nition 3.3.  is a multiplicity one representation if J1(N )[ker ] has dimension 2 for all pairs (N; ) as above. Remark 3.4. 1. If J1(N )[ker ] has dimension 2 then m = J1 (N )[ker ] by Eichler-Shimura, see [6]. 2. The de nition extends to arbitrary modular Galois representations  as follows. As explained in Section 2.2, every  has a twist  i by some power of the cyclotomic character such that k( i )  ` + 1. We say that  is a multiplicity one representation if  i is a multiplicity one representation.

3.3. Multiplicity one

181

3.3.2. Multiplicity one theorems

Techniques for proving multiplicity one results were pioneered by Mazur in [71] who considered J0 (p) with p prime. Let f be an eigenform and x a nonzero prime  of the ring generated by the Fourier coecients of f such that f; is absolutely irreducible. View the Hecke algebra T as a subring of End(J0 (p)), and let m be the maximal ideal associated to f and . Let Vm again be a two-dimensional T=m-vector space that aords m : GQ ! GL(2; T=m). Mazur proved (see Prop. 14.2, ibid.) that J [m]  Vm , except perhaps when m is ordinary of residue characteristic ` = 2. The missing ordinary case can be treated under suitable hypothesis. If m restricted to a decomposition group at 2 is not contained in the scalar matrices, then J [m]  Vm (see, e.g., [9, Prop. 2.4]). The results of Mazur are extended in [72] and [84, x5]. Theorem 3.5. An irreducible modular Galois representation  : GQ ! GL2(F`) is a multiplicity one representation, except perhaps when all of the following hypothesis on  are simultaneously satis ed: | k() = `; |  is unrami ed at `; |  is ordinary at `; | jD`  ( 0  ) with  = . Proof. See [32, x9], [117, Thm. 2.1], and [9, Prop. 2.4] for the case ` = 2. In [46, x12] Gross proves multiplicity one when  6= , k()  `, and  is ordinary; he uses this result in his proof of the existence of companion forms. In contrast, Coleman and Voloch [17] prove the existence of companion forms when  = and ` > 2 using a method that avoids the need for multiplicity one. Remark 3.6. L. Kilford of London, England has recently discovered an example at prime level 503 in which multiplicity one fails. Let E1 , E2 , and E3 be the three elliptic curves of conductor 503, and for each i = 1; 2; 3, let mi be the maximal ideal of T  End(J0 (503)) generated by 2 and all Tp ; ap (Ei ), with p prime. Each of the Galois representations Ei [2] is irreducible, and one can check that m1 = m2 = m3 . If multiplicity one holds, then E1 [2] = E2 [2] = E3 [2] inside of J0 (503). However, this is not the case, as a modular symbols computation in the integral homology H1 (X0 (N ); Z) reveals that E1 \ E2 = f0g.

3.3.3. Multiplicity one for mod 2 representations

For future reference, we now wish to consider multiplicity one in the following rather extreme situation. Suppose that ` = 2, and let  be a mod ` representation arising from a form of weight either 2 or 3. If the weight is 3 then  is not nite at 2; this can be used to deduce multiplicity one by adapting the arguments of [72] (see the proof of [9, Prop. 2.4]). When the weight is 2, we have the following proposition. Proposition 3.7. Let  : GQ ! GL2(PF2) be an irreducible Galois representation that arises from a weight-2 form f = an qn on ; = ;1 (N ) \ ;0 (2) with N odd, and let " be the character of f . If a22 6 "(2) 2 F2 , then  is a multiplicity one representation.

182

RIBET AND STEIN, SERRE'S CONJECTURES

Proof. Let m be the maximal ideal associated to f in the Hecke algebra T at-

tached to ;. Because the weight of f is 2, the representation  is nite at 2. If  is supersingular then the inertia group I2 operates through the two fundamental characters of level 2. These both have order `2 ; 1 = 3 6= 1, so  is rami ed and this can be used to deduce multiplicity one. If  is ordinary then jD  ( 0  ) with unrami ed and (Frob2 )  T2 mod m. The determinant  of jD is 
" where  is the mod 2 cyclotomic character and " is unrami ed at 2. Since , ", and are unrami ed,  is also unrami ed. Since (Frob2 ) = 1 and  = ", we have (Frob2 ) = ;1 (Frob2 )"(2) = a;2 1 "(2) (mod m): The further condition, under which we might not know multiplicity one, is jD = jD ; expressed in terms of the image of Frobenius, this becomes a;2 1 "(2)  a2 (mod m), or equivalently, a22  "(2) (mod m). By hypothesis, this latter condition does not hold. 2

2

2

2

3.4. The key case

We have set our problem up so that level optimization pertains to weight-2 forms of appropriate level, and takes place on Jacobians of modular curves. This level optimization problem was described, and partially treated, in a paper of Carayol [12]. In this paper, Carayol reduced the problem to the following key case.

Key case: Let  : GQ ! GL(2; F`) be a Galois representation that arises from a weight-2 newform f of level pM , with p `M , and character " : (Z=pM Z) ! C . Assume that  is unrami ed at p, and that " factors through the natural map (Z=pM Z) ! (Z=M Z) . Show that  arises from a form of level M . -

In the key case, the character " of f is unrami ed at p. Thus f , a priori on ;1 (pM ), is also on the bigger group ;1 (M ) \ ;0 (p); that is, f lies in S2 (;1 (M ) \ ;0 (p)). Example 3.8. Consider the representation  arising from the 7-division points of the modular elliptic curve A of conductor NA = 3
47 and minimal discriminant
A = 37
47. (The curve A is labeled 141A in Cremona's notation [20].) The newform f corresponding to A is on ;0 (3
47). As in Section 1.4, since ord3 (
A ) = 7, the representation  is unrami ed at 3 and N () = 47. To optimize the level means to nd a form g on ;0 (47) that gives rise to . Example 3.9 (Frey curves). The elliptic curves that Frey associated in [42] to hypothetical solutions of the Fermat equation x` + y` = z ` give rise to mod ` Galois representations. According to Wiles's theorem [117], there is a weight-2 form f of level 2L, with L big and square free, that gives rise to . At the same time, N () = 2. Taking p to be any odd prime dividing L, we are put in the key case. If we can optimize the level, then we eventually reach a contradiction and thus deduce Fermat's Last Theorem. The key case divides into two subcases; the more dicult one occurs when the following conditions are both satis ed: | p  1 (mod `); | (Frobp ) is a scalar matrix. The second condition makes sense because p N (); since det((Frobp )) = k;1 ", we know the scalar up to 1. The complementary case is easier; it can be treated -

3.5. Approaches to level optimization in the key case

183

using \Mazur's principle" (see Section 3.9). Though Example 3.8 falls into the easier case because 3 6 1 (mod 7), the proof of Fermat's Last Theorem requires level optimization in both cases. Consider a modular representation  : GQ ! GL(2; F` ) that arises from a newform of level N and weight k = k(), and assume that ` N . The goal of level optimization is to show that there is a newform of Serre's optimal level N () that gives rise to . P As discussed in Section 3.1,  arises from a newform f = an qn on ;1 (N  ) of weight 2 and some character ". Thus there is a homomorphism ' from O = Z[: : : an : : : ] to F` such that '(ap ) = tr((Frobp)) for all p `N . Let T be the Hecke algebra associated to S2 (;1 (N  )). The maximal ideal m of T associated to  is the kernel of the map sending Tn to '(an ). As was discussed in the previous chapter, the representation  is realized geometrically inside the subspace J [m]  J [`] of the `-torsion of the Jacobian J of X1 (N  ). Problem. Fix a divisor p of N =N (). Find a newform whose level is a divisor of N  =p that also gives rise to . -

-

Lemma 3.10. Let  be as above, and suppose p is a prime such that p j N  but p `N (), so  is unrami ed at p. Let "p denote the p part of ". Then either "p = 1 or p  1 (mod `). Proof. The character " is initially de ned as a homomorphism (Z=N  Z) ! O; the reduction " is obtained by composing " with ' : O ! F` . Since  is unrami ed k ;1 -

at p, the determinant det() = ` " = ` " is also unrami ed at p. Because ` is rami ed only at `, the character " is unrami ed at p. Let M = N  =pr where r = ordp (N  ), and write (Z=N  Z)  = (Z=pr Z)  (Z=M Z) . By restricting " to each factor, we write " as a product of two characters: " = "p
"(p) where "p is a character of (Z=pr Z) and "(p) is a character of (Z=M Z) . The character "(p) has conductor dividing M , so it is unrami ed at p. By class eld theory, "p is totally rami ed at p, so the reduction " is unrami ed at p precisely when "p = 1; equivalently, " is unrami ed at p exactly when "p has order a power of `. If "p is non-trivial, then, since the order of "p divides the order pr;1(p ; 1) of a generator of (Z=pr Z) , a power of ` divides pr;1 (p ; 1), so p  1 (mod `) since ` 6= p. In addition to his conjectures about the optimal weight and level, Serre also made a conjecture about the optimal character of a form giving rise to . Let p be a prime not dividing `N (). Serre's optimal character conjecture implies that , which we know to arise from a form on ;1 (M ) \ ;1 (pr ), arises from a form on ;1 (M ) \ ;0 (pr ), and this has been proved in most cases.

3.5. Approaches to level optimization in the key case

As discussed in Section 3.4, results of Carayol and Livne (see [12, 70]) reduce the level optimization problem to the following key case. The weight-2 newform f , a priori on ;1 (N  ), is in fact on the bigger group ;1 (M ) \ ;0 (p), where Mp = N  , p M , and  is unrami ed at p. The goal is to show that  arises from a newform on ;1 (M ). This has been achieved when ` is odd, and in many cases when ` = 2, using several level optimization techniques. -

184

RIBET AND STEIN, SERRE'S CONJECTURES

I. Mazur's principle If either (Frobp ) is not a scalar matrix or p 6 1 (mod `), then an argument of Mazur, explained in Section 3.9, can be used to optimize the level. II. Multiplicity one It is possible to optimize the level if  is a multiplicity one representation, as explained in [84, 9] and Section 3.11. The cases in which multiplicity one is known were reviewed in Section 3.3. In particular, we do not know multiplicity one in some cases when k() = ` and the eigenvalues of Frobp are not distinct. III. Using a pivot Suppose that M can be written as a product M = qK with q a prime not dividing pK , that  arises from a form on ;1 (K ) \ ;0 (pq), and that  is rami ed at q and unrami ed at p. Then q can be used as a \pivot" to remove p from the level. This approach grew out of [83], and was introduced in the short paper [86]. In Section 3.10 we describe the approach and discuss the terminology. IV. Without multiplicity one When ` is odd and " = 1, the level optimization theorem was proved in [87] using an argument that does not require  to have multiplicity one. The hypothesis ` 6= 2 is used in the proof of Proposition 7.8 of [87] to force splitting of a short exact sequence. In [26], Diamond extended the results of [87] to cover the case of arbitrary character, still under the assumption that ` is odd. One encounters seemingly insurmountable diculties in trying to push this argument through when ` = 2.

3.6. Some commutative algebra

In this section we set up some of the commutative algebra that is required in order to lower levels. There are two injective maps 

S2 (;1 (M ))  

/

/ S2 (;1 (M ) \ ;0 (p))

:

One is the inclusion f (q) 7! f (q) and the other is f (q) 7! f (qp ) (see Exercise 18). The p-new subspace S2 (;1 (M ) \ ;0 (p))p-new is the complement, with respect to the Petersson inner product, of the subspace S generated by the two images of S2 (;1 (M )). The p-new subspace can also be de ned algebraically as the kernel of the natural map from S2 (;1 (M ) \ ;0 (p)) to the direct sum of two copies of S2 (;1 (M )). Let T denote the Hecke algebra acting on S2 (;1 (M ) \ ;0 (p)). If p M , then Tp acts on S as a direct sum of two copies of its action on S2 (;1 (M )); otherwise, Tp usually does not act diagonally (see Exercise 19). The image of T in End(S ) is a quotient T called the p-new quotient. A representation  associated to a maximal ideal m of T arises from level M if and only if m arises by pullback from a maximal ideal of T. Because the map T ! T is surjective, m arises from level M if and only if the image of m inT is not the unit ideal (see Exercise 21). -

3.7. Aside: Examples in characteristic two

Sections 3.7 and 3.8 can be safely skipped on a rst reading.

3.7. Aside: Examples in characteristic two

185

To orient the reader, we focus for the moment on mod 2 representations that arise from elliptic curves. We give examples in which one of the level optimizations methods applies but the others do not. We do not consider method IV because it is not applicable to mod 2 representations. The hypothesis of the \multiplicity one" method II when ` = 2 are discussed after the statement of Theorem 3.19 in Section 3.11. We were unable to nd an example in which none of the level optimization theorems applies. We will repeatedly refer to the following theorem, which rst appeared in [85]. Theorem 3.11. Suppose  arises from a newform in S2(;0(N )). Let p `N be a prime satisfying one or both of the identities tr (Frobp ) = (p + 1) (mod `): Then  arises from a newform of level pN . -

3.7.1. III applies but I and II do not

In this section we give a mod 2 representations in which the pivot hypothesis of

III is satis ed, but the hypotheses of I and II are not. Our example is obtained

by applying Theorem 3.11 to the mod 2 representation attached to a well-chosen elliptic curve. We will nd an elliptic curve E of conductor M = qR such that  = E [2] is absolutely irreducible, rami ed at q, unrami ed at 2, and (Frob2 ) = ( 10 01 ). Because of the last condition, [9, Prop. 2.4] does not imply that  is a multiplicity one representation, so II does not apply. (In fact, following Remark 3.6, one sees that  is not a multiplicity one representation.) Likewise, I does not apply because (Frob2 ) is a scalar and the p we will chose will satisfy p  1 (mod 2). Next we choose a prime p 2qR such that E;2 (Frobp ) = ( 10 01 ). Let f be the newform associated to E . By Theorem 3.11 there is a newform g of level pqR such that -

In particular,

g;  E;2 :

g; (Frobp ) = E;2(Frobp ) = ( 10 01 )

is scalar and p  1 (mod 2), so I does not apply. However, method III does apply with q used as a pivot. For example, consider the elliptic curve E de ned by the equation y2 + xy = x3 ; x2 + 19x ; 32: The conductor of E is N = 19
109, and the discriminant of the eld K = Q(E [2]) is ;193
1093. We select q = 19 as our pivot. The prime p = 73 splits completely in K , so   1 0 E;2 (Frobp ) = 0 1 : By Theorem 3.11 there is a form g of level 109
19
73 that is congruent to the newform f attached to E modulo a prime lying over 2. Method III can be used to optimize the level, but neither method I nor II applies.

3.7.2. II applies but I and III do not

We exhibit a mod 2 representation for which method II can be used to optimize

186

RIBET AND STEIN, SERRE'S CONJECTURES

Z[x]=(x + 1) (2; x + 1) (3; x + 1)

(3; x)

(2; x2 + x + 1) 2 Z[x]=(x + x + 3)

(5; x + 2)

2 2 (7; x2 + x + 3) (13; x + x + 3) (19; x + x + 3) (5; x + 4) (17; x2 + x + 3) (23; x + 5) (11; x + 6) (23; x + 19) (5; x + 1) (7; x + 1) (11; x + 1)

(13; x + 1) Figure 1.

(17; x + 1)

(23; x + 1) (19; x + 1)

The spectrum of T  End(S2 (;0 (33))), with x = T3

the level, but neither method I nor III applies. Let K be the GL2 (F2 )-extension of Q obtained by adjoining all cube roots of 2. Then K = Q(E [2]), where E is the elliptic curve X0 (27) given by the equation y2 + y = x3 ; 7. The prime p = 31 splits completely in K , so by Theorem 3.11 there is a newform f of level 31
27 and a maximal ideal  of the appropriate Hecke algebra such that f;  E [2]. Neither method I nor III can be used to optimize the level of f; . Method I doesn't apply because 31 is odd and f; (Frob31 ) = ( 10 01 ); method III doesn't apply because the only odd prime that is rami ed in K is 3, which does not exactly divide 31
27. If D2 is a decomposition group at 2 then D2 has image in GL2 (F2 ) of order 2, so it is not contained in the scalar matrices and II can be used to optimize the level of f; .

3.8. Aside: Sketching the spectrum of the Hecke algebra

It is helpful to understand the Hecke algebra geometrically using the language of schemes (see, e.g., [38]). The topological space underlying the scheme Spec(T) is the set of prime ideals of T endowed with the Zariski topology, in which the closed sets are the set of prime ideals containing a xed ideal. We can draw Spec(T) by sketching a diagram whose irreducible components correspond to the Galois conjugacy classes of eigenforms, and whose intersections correspond to congruences between eigenforms. When the level is not cube free, T can contain nilpotent elements, and then one might wish to include additional P information. If an qn is an eigenform, then the failure of Z[: : : an : : : ] to be integrally closed can be illustrated by drawing singular points on the corresponding irreducible component; however, we do not do this below. Example 3.12. The spectrum of the Hecke algebra associated to ;0(33) is illustrated in Figure 1. The Hecke algebra T  S2 (;0 (33)) has discriminant ;99, as does the characteristic polynomial of T3 , so T = Z[T3]=((T3 + 1)(T32 + T3 + 3)) = Z[x]=((x + 1)(x2 + x + 3)): We sketch a curve corresponding to each of the two irreducible components. Some of the closed points (maximal) ideals are represented as dots. One component corresponds to the unique newform on ;0 (33), and the other corresponds to the two images of the newform on ;0 (11).

3.9. Mazur's principle

187

141A

141E 2

3 43 3

141D

7

47A

141C 2

141B

141F: Z[T2]=(T22 + T2 ; 4) Figure 2.

The spectrum of T  End(S2 (;0 (141)))

Example 3.13. Figure 2 is a diagram of the Hecke algebra associated to S2(;0(3


47)). We have labeled fewer closed points than in Figure 1. The components are labeled by their isogeny class and the level at which they are new (the notation extends that of [20]). The component labeled 141F corresponds to an eigenform whose Fourier coecients generate a quadratic extension of Q. The newform corresponding to the elliptic curve A from Example 3.8 is labeled 141A. Geometrically, the assertion that the level of A;7 can be optimized is represented by the characteristic-7 intersection between the component labeled 141A and the old component 47A coming from the unique Galois conjugacy class of newforms on ;0 (47).

3.9. Mazur's principle

A principle due to Mazur can be used to optimize the level in the key case, provided that a mild hypothesis is satis ed. The principle applies whenever p 6 1 (mod `) and also in the case when p  1 (mod `) but (Frobp ) is not a scalar. This principle rst appeared in [84, x6], then in [26, x4], and most recently when ` = 2 in [9, pg. 7].

188

RIBET AND STEIN, SERRE'S CONJECTURES

Theorem 3.14 (Mazur's Principle). Suppose that  : GQ ! GL(2; F`) arises from

a newform f of weight 2 and level Mp, with p - M , and character " of conductor dividing M . Assume that  is unrami ed at p and that either (Frobp ) is not a scalar matrix or p 6 1 (mod `). Then  arises from a modular of level dividing M . We will require the following basic fact later in the proof. Lemma 3.15 (Li). Let f = P anqn be a newform on ;1(M ) \ ;0(p) of weight k. Then a2p = "(p)pk;2 : Proof. Li's proof is an easy application of her generalization to ;1 of the AtkinLehner theory of newforms [69, Thm. 3(iii)]. The newform f is an eigenvector for the operator Wp which is de ned on Sk (;1 (M ) \ ;0 (p)) by   apz + b ; Wp (f ) = pk=2 f Mpz +p 2 where a and b are integers such that ap ; bMp = p. By [69, Lem. 3], g := Tp (f ) + pk=2;1 Wp (f ) lies in Sk (;1 (M )): For all primes q - Mp, the eigenvalue of Tq on the oldform g is the same as the eigenvalue of Tq on the newform f , so g = 0. By [69, Lem. 2] Wp 2 (f ) = "(p)f , so a2p = "(p)pk;2 . Remark 3.16. The case of Lemma 3.15 that we will need can also be understood in terms of the local representation jGp , which resembles the mod ` representation  attached to a Tate curve, in the sense that jGp  (  0  ). Our hypothesis include the assumption that  is unrami ed at p, so the two characters  and  are unrami ed at p. Thus (Frobp ) makes sense; we have (Frobp ) = ap (f ) and (Frobp ) = ap (f )p. Since det(jGp ) = 2  = ", we see that a2p = "(p): This local analysis of  was vastly generalized by Langlands in [67], which extends the analysis to include many `-adic representations of possibly higher weight. See also [13]. Let T be the Hecke algebra associated to ;1 (M ) \ ;0 (p), and let m be the kernel of the following map T ! F` : Tn 7!an ; hdi7!"(d) F`: 0 ;! m ;! T ;;;;;;;;;;;! As in Lemma 3.1, the determinants and traces of elements in the image of  = m lie in T=m  F`, so there is a vector space V  (T=m)2 that aords m . Next we realize m as a group of division points in a Jacobian. The curve X1 (Mp) corresponding to ;1 (Mp) covers the curve X1 (M; p) corresponding to ;1 (M ) \ ;0 (p). The induced map J = Jac(X1 (M; p)) ! J1 (Mp) = Jac(X1 (Mp)) has a nite kernel on which the Galois action is abelian. Just as in Section 2.3.1.1, the Hecke algebra associated to ;1 (M ) \ ;0 (p), can be constructed as a ring of correspondences on X1 (M; p), then viewed as a subring T  EndQ(J ). Inside of J we nd the nonzero GQ -module J [m]  ti=1V . For the purposes of this discussion, we do not need to know that J [m] is a direct sum of copies of V . The following weaker assertion, known long ago to Mazur [71, x14, pg. 112], will suce: J [m] is a successive extension of copies of V . In particular, V  J [m]. A weaker conclusion, true since ` 2 m, is that V  J [`],

X1 (M )

3.9. Mazur's principle

189

X1 (M ) Figure 3.

The reduction mod p of the Deligne-Rapoport model of X1 (M; p)

Our hypothesis that  is unrami ed at p translates into the inclusion V  J [`]Ip , where Ip is an inertia group at p. By [104, Lem. 2], if A is an abelian variety over Q with good reduction at p, then A[`]Ip  = AFp [`]. However, the modular curve X1 (M; p) has bad reduction at p, so J is likely to have bad reduction at p| in this case it does. We are led to consider the Neron model J of J (see, e.g., [5]), which is a smooth commutative group scheme over Z satisfying the following property: the restriction map HomZ (S ; J ) ;! HomQ (SQ ; J ) is bijective for all smooth schemes S over Z. Passing to the scheme-theoretic closure, we have, inside of J , a two-dimensional T=m-vector space scheme V . In Section 2.3.1.1 we only de ned X1 (M; p) as a scheme over Z[1=Mp]. Deligne and Rapoport [25] extended X1 (M; p) to a scheme over Z[1=M ] and computed the reduction modulo p. The introduction to [62] contains a beautiful historical discussion of the diculties involved in extending modular curves over Z. We know a great deal about the reduction of X1 (M; p) at p, which is frequently illustrated by the squiggly diagram in Figure 3. This reduction is the union of 2 copies of X1 (M )Fp intersecting transversely at the supersingular points. The subspace S2 (;1 (M ))  S2 (;1 (M )) of S2 (;1 (M ) \ ;0 (p)) is stable under the Hecke algebra T, so there is a map T ! End(S2 (;1 (M ))  S2 (;1 (M ))). The p-old quotient of T is the image T. Since the map T ! T is surjective, the image of m in T is an ideal m. To optimize the level in the key case amounts to showing that m is not the unit ideal. As is well known (cf. [71, Appendix, Prop 1.4]), the results of M. Raynaud [82] and Deligne-Rapoport [25] combine to produce an exact sequence (3.1)

0 ;! T ;! JF0p ;! J1 (M )Fp  J1 (M )Fp ;! 0;

where T is a torus, i.e., TFp  Gm 


 Gm , and JF0p is the identity component of JFp . There is a concrete description of T and of the maps in the exact sequence. Each object in the sequence is equipped with a functorial action of the Hecke algebra T, and the sequence is T-invariant. The p-old quotient T can be viewed as coming from the action of T on J1 (M )Fp  J1 (M )Fp . By a generalization of [104, Lem. 2], the reduction map J (Qp )[`]Ip ! JFp (Fp ) is injective. Thus V = VFp (Fp )  JFp (Fp ). The component group  = JFp =JF0p is Eisenstein, in the sense that it does not contain irreducible representations arising from eigenforms. Since V is irreducible, as a Galois module  does not contain an

190

RIBET AND STEIN, SERRE'S CONJECTURES

isomorphic copy of V , so VFp  JF0p and we have the following diagram: 0 /

T

/ _

JFO0p ?

VFp

/ J1 (M )Fp 8 p p p p p p p p p pp p p

 J1 (M )Fp

/

0:

Since m acts as 0 on V , the image m of m acts as 0 on the image of V in J1 (M )Fp  J1 (M )Fp . If m 6= (1) then we can optimize the level, so assume m = (1). Then the image of V in J1 (M )Fp  J1 (M )Fp is 0, so VFp ,! T . Let Xp (J ) := Hom(T ; Gm ) be the character group of T . The action of T on T induces an action of T on Xp (J ). Furthermore, Xp (J ) supports an action of Gal(Fp =Fp ) which, because tori split over a quadratic extension, factors through the Galois group of Fp . View the Galois action as an action of Frobp 2 Dp = Gal(Qp =Qp). With our conventions, the action of Frobenius on the torus is as follows (cf. [26, pg. 31]). Lemma 3.17. The Frobenius Frobp acts as pTp on T (Fp ). Make the identi cation T  = HomZ (Xp (J ); Gm ), so that V  T (Fp )[`] = HomZ (Xp (J ); ` ): By Lemma 3.17, Frobp acts on V  T (Fp ) as pap 2 T=m, i.e., as a scalar. The determinant of  is ", so we have simulatenously ( det((Frobp )) = p"(p)2 and (pap ) : By Lemma 3.15, a2p = "(p), so p2  p (mod `). Since p 6= `, this can only happen if p  1 (mod `), which completes the proof. 2

3.10. Level optimization using a pivot

In this section we discuss an approach to level optimization that does not rely on multiplicity one results. In this approach, we eliminate a prime p from the level by making use of the rational quaternion algebra that is rami ed precisely at p and at a second prime q. The latter prime is, in the simplest case, an appropriate prime number at which  rami es; in more complicated cases, it is an \auxiliary" prime at which  is unrami ed. The central role of q in the argument, and the fact that q stays xed in the level while p is removed, leads us to refer to q as a \pivot." The following theorem rst appeared in [86]. Theorem 3.18. Let  : GQ ! GL(2; F`) be an irreducible continuous representation that arises from an eigenform f on ;1 (K ) \ ;0 (pq) with p and q distinct primes that do not divide `K . Make the key assumption that the representation  is rami ed at q and unrami ed at p. Then  arises from a weight-2 eigenform on ;1 (K ) \ ;0 (q). The case ` = 2 is not excluded from consideration. Before sketching the proof, we describe a famous application. Edixhoven suggested to the rst author that such an approach might be possible in the context of

3.10. Level optimization using a pivot

191

Fermat's Last Theorem. We associate to a (hypothetical) solution a` + b` + c` = 0 of the Fermat equation with ` > 3 a Galois representation E [`] attached to an elliptic curve E . A theorem of Mazur implies that this representation is irreducible; a theorem of Wiles implies that it arises from a modular` form. Using Tate's algorithm, we nds that the discriminant of E is
E = (abc2 ) , which is aQperfect `th power away from 2, and that the conductor of E is NE = rad(abc) = pjabc p. Let q = 2; then E [`] is rami ed at q because ` ord2 (
E ) = ;8 (see Theorem 2.10), but E [`] is unrami ed at all other primes p, again by Theorem 2.10. To complete the proof of Fermat Last Theorem, we use q = 2 as a pivot and inductively remove each odd factor from N . One complication that may arise (the second case of Fermat Last Theorem) is that ` j N . Upon removing ` from the level (using Section 3.1), the weight may initially go up to ` + 1. If this occurs, since k() = 2 we can use [32] to optimize the weight back to 2. As demonstrated by the application to Fermat, in problems of genuine interest the setup of Theorem 3.18 occurs. There are, however, situations in which it does not apply such as the recent applications of level optimization as a key ingredient to a proof of Artin's conjecture for certain icosahedral Galois representations (see [10]). 2

8

-

3.10.1. Shimura curves

We cannot avoid considering Shimura curves. Denote by X (K; pq) the modular curve associated to ;1 (K ) \ ;0 (pq) and let J := Jac(X (K; pq)) be its Jacobian. Likewise, denote by X pq (K ) the Shimura curve associated to the quaternion algebra of discriminant pq. The curve X pq (K ) is constructed as follows. Let B be an inde nite quaternion algebra over Q of discriminant pq. (Up to isomorphism, B is unique.) Let O be an Eichler order (i.e., intersection of two maximal orders) of level K (i.e., reduced discriminant Kpq) in B . Let ;1 be the group of elements of O with (reduced) norm 1. After xing an embedding B ! M (2; R) (an embedding exists because B is inde nite), we obtain in particular an embedding ;1 ,! SL(2; R) and therefore an action of ;1 on the upper half-plane h. Let X pq (K ) be the standard canonical model, over Q, of the compact Riemann surface ;1 nh, and let J 0 = Jac(X pq (K )) denote its Jacobian. The curve X pq (K ) is furnished with Hecke correspondences Tn for n  1. We write Tn for the endomorphism of J induced by the Tn on X pq (K ) via Pic functoriality. Set J 0 := Jac(X pq (K )) and J := Jac(X (K; pq)). Work of Eichler, JacquetLanglands, and Shimura (see [36, 51, 106]) has uncovered a deep correspondence between certain automorphic forms and certain cusp forms. Combining their work with the isogeny theorem of Faltings [40], we nd (noncanonically!) a map J 0 ! J with nite kernel. The pq-new part of J is Jpq-new := ker(J (K; pq) ;! J (K; p)2  J (K; q)2 ) where the map is induced by Albanese functoriality from the four maps // X (K; p) // X (K; q ): X (K; pq) X (K; pq) and The image of J 0 ! J is the pq-new part of J .

3.10.2. Character groups

Amazingly, there seems to be no canonical map J 0 ! J between the Shimura and classical Jacobians described in the previous section. Surprisingly, there is a

192

RIBET AND STEIN, SERRE'S CONJECTURES

canonical relationship between the character groups of J 0 and J . The C erednikDrinfeld theory gives a description of X pq (K ) in characteristic p (see [14, 30]). Using this we nd a canonical T-equivariant exact sequence (3.2)

0 ! Xp (J 0 ) ! Xq (J ) ! Xq (J 00 ) ! 0

where J 00 = Jac(X (K; q))2 . This exact sequence relates a character group \in characteristic p" to two character groups \in characteristic q". We are now prepared to prove the theorem.

3.10.3. Proof Proof of Theorem 3.18. By our key assumption, the representation  is rami ed at q, so m  T is not q-old. We may as well suppose we are in a situation where we can not optimize the level, so we assume that m is not p-old either and hope for a contradiction. Localization is an exact functor, so the localization (3.3)

0 ;! Xp (J 0 )m ;! Xq (J )m ;! Xq (J 00 )m ;! 0

of (3.2) is also exact. The Hecke algebra T acts on Xq (J 00 ) through a quotient T. Since m is not q-old, the image of m in T generates the unit ideal. Therefore Xq (J 00 )m = 0 and we obtain an isomorphism Xp (J 0 )m  Xq (J )m . If R is a Tmodule then R=mR = Rm =mRm so (3.4)

Xq (J )=mXq (J )  Xp (J 0 )=mXp (J 0 ):

Switching p and q and applying the same argument shows that

Xp (J )=mXp (J )  Xq (J 0 )=mXq (J 0 ): Both (3.4) and (3.5) are isomorphisms of T=m-vector spaces. L 0 By [ 6 ] we have an isomorphism J [ m ]  V i=1 , with  > 0 and J [m]  L Our i=1 V . (It follows from [51] that  > 0, but we will not use this here.) L hypothesis that V is unrami ed automatically propagates to all of J [m]  i=1 V . (3.5)

Since V is irreducible and we are assuming that m is not p-old, the same argument as in Section 3.9 shows that J [m]  T [m] where T is the toric part of JFp . This means that dim(Xp (J )=mXp (J ))  2. Using the same argument with J replaced by J 0 gives that dim(Xp (J 0 )=mXp (J 0 ))  2. As an Iq -module V is an extension of two copies of the trivial character. This follows from results of Langlands [67], since  is a mod ` representation of GQ associated to some newform f whose level divides pqK and is divisible by q. (The admissible representation of GL(2; Qq ) which is associated to f is a special representation.) Because V is rami ed at q and there is an unrami ed line, we see that dim(V Iq ) = 1. Thus dim J [m]Iq = ; since q 6= ` and the action of inertia on character groups is trivial, we see that Hom(Xq (J )=mXq (J ); ` )  J [m]Iq ;

3.11. Level optimization with multiplicity one

193

so dim Xq (J )=mXq (J )  . A similar argument bounds dim Xq (J 0 )=mXq (J 0 ). We obtain the following quadruple of inequalities: dim Xq (J )=mXq (J )  ; dim Xq (J 0 )=mXq (J 0 )  ; dim Xp (J )=mXp (J )  2; dim Xp (J 0 )=mXp (J 0 )  2: Combining these with (3.4, 3.5), we nd that 2  dim Xp (J )=mXp (J ) = dim Xq (J 0 )=mXq (J 0 )



and simulatenously that 2  . Together these imply that 4   so  = 0. But Proposition 3.2 implies that the multiplicity of  in J [m] is strictly positive. This contradiction implies that our assumption that m is not p-old is false, hence m is p-old and  arises from an eigenform on ;1 (K ) \ ;0 (q).

3.11. Level optimization with multiplicity one

Theorem 3.19. Suppose  : GQ ! GL2(F`) is an irreducible multiplicity one representation that arises from a weight-2 newform f on ;1 (M ) \ ;0 (p) and that p

is unrami ed. Then there is a newform on ;1 (M ) that also gives rise to . We sketch a proof, under the assumption that ` > 2. Buzzard [9] has given a proof when ` = 2; his result has been combined with the results of [28] to prove a Wiles-like lifting theorem valid for many representations when ` = 2, and hence (thanks to Taylor) to establish new examples of Artin's conjecture (see [10]). The following diagram illustrates the multiplicity one argument: /o /o /o /o pivot /o /o /o /o /o /o /o / Mq Mpq 1. Let ` be the smallest prime divisor of d and choose a point x 2 ker(') of exact order `. If the order-` cyclic subgroup generated by x is Galois stable, then A;` is reducible, which is contrary to our assumption. Thus ker(') contains the full `-torsion subgroup A[`] of A. In particular, ' factors as illustrated below: A❋❋

'

❋❋ ❋ ❋` ❋❋

❋"

/ B: ; xx x xx x xx

A=A[`] Since A=A[`]  = A, there is an isogeny from A to B of degree equal to d=`2 , which contradicts our assumption that d is minimal.

Solution 6. The Weil pairing ( ; ) : A[`]  A[`] ! ` can be viewed as a map 2 ^



=  A[`] ;! ` sending P ^Q to (P; Q). For any  2 Gal(Q =Q), we have (P  ; Q ) = (P; Q) . With V2    the action (P ^ Q) = P ^ Q , the map A[`] ! ` is a map of Galois modules. ;
To compute det(()) observe that if e1 , e2 is a basis for A[`], and () = ac db ,

then

(e1 ^ e2 ) = (ae1 + ce2 ) ^ (be1 + de2 ) = (ad ; bc)e1 ^ e2 = det(())e1 ^ e2 V2 Thus A[`] gives the one-dimensional representation det(). Since isomorphic to ` it follows that det() = .

V2

A[`] is

Solution 7.; The
de nition of hdi is as follows: choose any matrix d 2 ;0(N ) such that d  d0 d;0 (mod N ); then hdif = f jd : Observe that ;1 (N ) is a normal 1

subgroup of ;0 (N ) and the matrices ; d
with (d; N ) = 1 and d < N are a system of coset representatives. Thus any ac db 2 ;0 (N ) can be written in the form d
g for some g 2 ;1 (N ). We have   az + b ; k f = f jd g = (f jd )jg = ("(d)f )jg = "(d)f jg = "(d)(cz + d) f cz + d :

202

RIBET AND STEIN, SERRE'S CONJECTURES

Solution 8.

1. Since c2 = 1, the minimal polynomial f of (c) divides x2 ; 1. Thus f
is ; either x + 1, x ; 1, or x2 ; 1. If f = x + 1 then (c) = ;1 = ;10 ;01 . This implies that det((c)) = (;1)2 = 1, which is a contradiction since det((c)) = ;1 and the characteristic of the base eld is odd. If f = x ; 1, then (c) = 1; again a contradiction. Thus the minimal polynomial of (c) is x2 ; 1 = (x ; 1)(x + 1). Since ;1 6= 1 there is a basis of eigenvectors for ;
(c) such that the matrix of (c) with respect to this basis is ;10 01 . 2. The following example ;
shows that when ` = 2 the matrix of A;` need not be conjugate to 10 ;01 . Let A be the elliptic curve over Q de ned by y2 = x(x2 ; a) with a 2 Q not square. Then p p A[2] = f1; (0; 0); ( a; 0); (; a; 0)g: p The action of cpon the basis (0; 0); (; a; 0) pis represented by the matrix p ( 10 11 ), since c(; a; 0) = ( a; 0) = (0; 0) + (; a; 0):

p Solution 9. TheQextension Q( d; d 2 Q=(Q)2) is an extension of Q with Galois group X  F2 . The index-two open subgroups of X correspond to the quadratic extensions of Q. However, Zorn's lemma implies that X contains many

more index-two subgroups, which can be seen more precisely as follows. 1. ChooseQa sequence p1 ; p2 ; p3 ; : : : of distinct prime numbers. De ne 1 : GQ ! F2 by

p 1 ()i = 0 if  acts trivially on Q( pi ); (

1 otherwise

Thus 1 is just

GQ ! Gal(Q(pp1 ; pp2 ; : : : )=Q) 

Y

Q

F2 :

2. Let F2  F2 be the subgroup of elements having only nitely many Q nonzero coordinates. Then F2 =F2 is a vector Q space over F2 of dimension > 0. By Zorn's lemma, there is a basis B of F2 =  FQ2 . Let b 2 B and let W be the subspace spanned by B ; fbg. Then V = ( F2 =  F2 )=W is an F2 -vector space of dimensional 1. 3. Let  be the composite map Gal(Q=Q)❘ Q

F2

❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ /

V

❘ ❘ ❘ ❘ ❘ ❘ )

/  = f1g

4. Let H = ker()  Gal(Q=Q). If (ppi ) = ;ppi pand (ppj ) = ppj for i 6= j , then  2 H . Thus H does not x any Q( pi ), so the xed eld of H equals Q. The largest nite Galois group quotient through which  factors is then Gal(Q=Q) = f1g. Since  6= 1, we conclude that  does not factor through any nite Galois group quotient, which proves that  is not continuous.

4.2. Solutions

203

Solution 10. We have f = f1 + f2 with f1; f2 2 S2(;0 (23)) and f1 = q ; q 3 ; q 4 +


f2 = q2 ; 2q3 ; q4 + 2q5 +


: Because S2 (;0 (23)) has dimension 2, it is spanned by f1 and f2. Let (q) = Q q n1 (1 ; qn ). Then g = ((q)(q23 ))2 2 S2 (;0 (23)). Expanding we nd that g = q2 ; 2q3 +


, so g = f2 . Next observe that g is a power series in q2 modulo 2: 1 24

Y

g = q2 (1 ; qn )2 (1 ; q23n )2 Y  q2 (1 ; q2n )(1 ; q46n ) (mod 2) Y  q2 (1 + q2n + q46n + q48n ) (mod 2) Thus the coecient in f2 of qp with p 6= 2 prime is even, and the proposition follows.

Solution 11. V 1. Let  2 ` be a primitive `th root of unity. Since 2 A[`]  = ` , there exists P; Q 2 A[`] such that P ^ Q =  . Since ` > 2 there exists  such that   6=  , hence P  ^ Q = 6 P ^ Q. This is impossible if all `-torsion is rational, since then P  = P and Q = Q. 2. Consider the elliptic curve de ned by y2 = (x ; a)(x ; b)(x ; c) where a; b; c are distinct rational numbers.

Solution 12.

1. Let K be the splitting eld of x3 + ax + b. Then  embeds Gal(K=Q) in GL(2; F2 ): Gal(Q=QN )



NN N N NN NNN N &

/

GL(2; F2 )

q 8 q q q q q q q + q q q

Gal(K=Q)

2. The representation  is reducible exactly when the polynomial x3 + ax + b has a rational root. 3. Examples: y2 = x(x2 ; 23), y2 = x3 + x ; 1.

Solution 13. Consider the character  = "=. By assumption,  (Frobp ) = 1 for all unrami ed p. Let K be an extension of Q such that  factors through Gal(K=Q). For any  2 Gal(K=Q), the Cebotarev density theorem implies that there are

in nitely many primes p such that Frobp = . Thus for any ,  () =  (Frobp ) = 1, so  = 1 and hence  = ".

Solution 14.

1. See, e.g., [109, 2.11]. 2. By [109, 5.5],  ; 1 is separable, so #A(Fp ) = deg( ; 1). Since  has degree p, there exists an isogeny  (the dual isogeny, see [109, III.6]), such

204

RIBET AND STEIN, SERRE'S CONJECTURES

that  = p. Letting bars denote the dual isogeny, we have #A(Fp ) = deg( ; 1) = ( ; 1)( ; 1) =  ;  ;  + 1 = p ; tr() + 1 3. Both maps are pth powering on coordinates.

Solution 15. Since ` 6= p and A has good reduction at p, the natural map A[`] ! A~[`] is an isomorphism. We have the following commutative diagram  / Aut(A[`]) Gal(Qp (A[`])=Qp )  

Gal((O=)Fp )



=

Aut(A~[`]) It follows that the rst vertical map must be injective, which is the same as Qp (A[`]) being unrami ed over Qp . /

Solution 16. The fundamental character of level one is the composition p nr  `;  Gal(Qnr ` ( `)=Q` ) ! `;1 (Q` ) ! `;1 (F` ) = F` : Let  be such that `;1 = `. Then () = () (mod ): Let  2 Q` be a primitive 1

`th root of unity. Now so

`Y ;1

( a ; 1) = `;

a=1

( ; 1)`;1

`Y ;1  a ; 1 a=1

;  ;1 =`

and (this is where Wilson's theorem is used), `Y ;1  a ; 1 `  a=1 ; 1

 1 (mod  ; 1):

Since the polynomial x`;1 ; 1 has roots over F` , by Hensel's lemma there is a unit u 2 Q`() such that `Y ;1 a u`;1 =  ;;11 : a=1 We can take  = ( ; 1)u. Then () = ( () ; 1)(u)  ( ; 1)u ();1


+ 1) (u) = ( ; 1)( ( ; + 1)u  (  ) ; 1 = ( +


+ 1)(u)=u  () (mod  ; 1):

4.2. Solutions

205

Solution 17. We write N = N () and k = k() to save space. The essential tool

is Theorem 2.10. 1. ` = 5: N = 6, k = 6, ` > 5, N = 30, k = 2. 2. ` = 5: N = 2
3
7, k = 6; ` = 7: N = 2
3
5, k = 8; ` > 7: N = 2
3
5
7, k = 2. 3. ` = 3: N = 2
5
11, k = 4; ` = 5: N = 2
3
11, k = 6; ` = 7: N = 2
3
5
11, k = 2; ` = 11: N = 2
3
5, k = 12; ` > 11: N = 2
3
5
11, k = 2. 4. ` = 3: N = 7
13, k = 2; ` = 5: N = 7
13, k = 6; ` = 7: N = 5
13, k = 8; ` = 13: N = 5
7, k = 14; ` = 11; ` > 13: N = 5
7
13, k = 2. 5. ` = 3: N = 2
11
19, k = 2; ` = 7: N = 2
11
19, k = 8; ` = 11: N = 2
7
19, k = 12; ` = 19: N = 2
7
11, k = 20; ` other: N = 2
7
11
19, k = 2.

Solution 20. One approach is to view J1 (N ) as a complex torus, and note that

the endomorphism ring is the set of automorphism of a complex vector space that x a lattice. Another approach is to use the deeper niteness theorems that are valid in arbitrary characteristic, see, e.g., [74, Thm. 12.5].

CHAPTER 5

Appendix by Brian Conrad: The Shimura construction in weight 2 The purpose of this appendix is to explain the ideas of Eichler-Shimura for constructing the two-dimensional `-adic representations attached to classical weight-2 Hecke eigenforms. We assume familiarity with the theory of schemes and the theory of newforms, but the essential arithmetic ideas are due to Eichler and Shimura. We warn the reader that a complete proof along the lines indicated below requires the veri cation of a number of compatibilities between algebraic geometry, algebraic topology, and the classical theory of modular forms. As the aim of this appendix is to explain the key arithmetic ideas of the proof, we must pass over in silence the veri cation of many such compatibilities. However, we at least make explicit what compatibilities we need. To prove them all here would require a serious digression from our expository goal; see [18, Ch. 3] for details. It is also worth noting that the form of the arguments we present is exactly the weight-2 version of Deligne's more general proof of related results in weight > 1, up to the canonical isomorphism Q` Z` lim; Pic0X=k [`n](k) = He1t(X; Q`(1)) = He1t;c(Y; Q`(1)) for a proper smooth connected curve X over a separably closed eld k of characteristic prime to `, and Y a dense open in X . Using `-adic Tate modules allows us to bypass the general theory of etale cohomology which arises in the case of higher weight.

5.1. Analytic preparations p

Fix i = ;1 2 C for all time. Fix an integer N  5 and let X1 (N )an denote the classical analytic modular curve, the \canonical" compacti cation of Y1 (N )an = ;1 (N )nh, where h = fz 2 C : Im z > 0g and ;1 (N )  SL2 (Z) acts on the left via linear fractional transformations. The classical theory identi es the C-vector space H 0 (X1 (N )an ; 1X (N ) ) with S2 (;1 (N ); C), the space of weight-2 cusp forms. Note that the classical Riemann surface X1 (N )an has genus 0 if we consider N < 5, while S2 (;1 (N ); C) = 0 if N < 5. Thus, assuming N  5 is harmless for what we will do. The Hodge decomposition for the compact Riemann surface X1 (N )an supplies us with an isomorphism of C-vector spaces 1

an

207

208

CONRAD, THE SHIMURA CONSTRUCTION

S2 (;1 (N ); C)  S2 (;1 (N ); C) =H 0 (X1 (N )an ; 1X (N ) )  H 0 (X1 (N )an ; 1X (N ) )  H 1 (X (N )an ; C) ;! 1 =H 1 (X1 (N )an ; Z) Z C (where A denotes the constant sheaf attached to an abelian group A). This 1

an

1

an

will be called the (weight-2) Shimura isomorphism. We want to de ne \geometric" operations on H 1 (X1 (N )an ; Z) which recover the classical Hecke operators on S2 (;1 (N ); C) via the above isomorphism. The \geometric" (or rather, cohomological) operations we wish to de ne can be described in two ways. First, we can use explicit matrices and explicit \upper-half plane" models of modular curves. This has the advantage of being concrete, but it provides little conceptual insight and encourages messy matrix calculations. The other point of view is to identify the classical modular curves as the base of certain universal analytic families of (generalized) elliptic curves with level structure. A proper discussion of this latter point of view would take us too far a eld, so we will have to settle for only some brief indications along these two lines (though this is how to best verify compatibility with the algebraic theory via schemes). ;
Choose a matrix n 2 SL2 (Z) with n  n0 ; n (mod N ), for n 2 (Z=N Z) . The action of n on h induces an action on Y1 (N )an and even on X1 (N )an . Associating to each z 2 h the data of the elliptic curve C=[1; z ] = C=(Z + Zz ) and the point 1=N of exact order N , we may identify Y1 (N )an as a set with the set of isomorphism classes of pairs (E; P ) consisting of an elliptic curve E over C and a point P 2 E of exact order N . The map Y1 (N )an ! Y1 (N )an induced by n can then described on the underlying set by (E; P ) 7! (E; nP ), so it is \intrinsic", depending only on n 2 (Z=N Z) . We denote by In : X1 (N )an ! X1 (N )an the induced map on X1 (N )an . Once this data (E; P ) is formulated in a relative context over an analytic base, we could de ne the analytic map In conceptually, without using the matrix n . We ignore this point here. ;1 on h induces a map Y1 (N )an ! Y1 (N )an which extends to The map z 7! Nz an wN : X1 (N ) ! X1 (N )an . More conceptually and more generally, if  2 N (C) is a primitive N th root of unity, consider the rule w that sends (E; P ) 2 Y1 (N )an to (E=P; P 0 mod P ), where P 0 2 E has exact order N and hP; P 0 iN =  , with h ; iN the Weil pairing on N -torsion points (following the sign conventions of [62, 77]; opposite the convention of [109]). More speci cally, on C=[1; z ] we have h N1 ; Nz iN = e2i=N . The map w extends to an analytic map X1 (N )an ! X1 (N )an . When  = e2i=N , we have w = wN due to the above sign convention. We have induced pullback maps w ; In : H 1 (X1 (N )an ; Z) ! H 1 (X1 (N )an ; Z): We write hni rather than In . Finally, choose a prime p. De ne ;1 (N; p)  SL2 (Z) to be ;1 (N; p) = ;1 (N ) \ ;0 (p) when p N and ;1 (N; p) = ;1 (N ) \ ;0 (p)t when p j N , where the group ;0 (p)t is the transpose of ;0 (p). De ne Y1 (N; p)an = ;1 (N; p)nh and let X1 (N; p)an be its \canonical" compacti cation. Using the assignment z 7! (C=[1; z ]; 1 ; h 1 i) 1

-

N p

5.1. Analytic preparations

209

when p N and -

z 7! (C=[1; z ]; N1 ; h zp i) when p j N , we may identify the set Y1 (N; p)an with the set of isomorphism classes of triples (E; P; C ) where P 2 E has exact order N and C  E is a cyclic subgroup of order p, meeting hP i trivially (a constraint if p j N ). Here and below, we denote by hP i the (cyclic) subgroup generated by P .

There are unique analytic maps 1(p) ; 2(p) : X1 (N; p)an ! X1 (N )an determined on Y1 (N; p)an by 1(p) (E; P; C ) = (E; P ) and 2(p) (E; P; C ) = (E=C; P mod C ): For example, 1(p) is induced by z 7! z on h, in terms of the above upper half plane uniformization of Y1 (N )an and Y1 (N; p)an . We de ne Tp = (1(p) )  (2(p) ) : H 1 (X1 (N )an ; Z) ! H 1 (X1 (N )an ; Z) where (1(p) ) : H 1 (X1 (N; p)an ; Z) ! H 1 (X1 (N )an ; Z) is the canonical trace map associated to the nite map 1(p) of compact Riemann surfaces. More speci cally, we have a canonical isomorphism H 1 (X1 (N; p)an ; Z)  = H 1 (X1 (N )an ; (1(p) ) Z) since (1(p) ) is exact on abelian sheaves, and there is a unique trace map of sheaves (1(p) ) Z ! Z determined on stalks at x 2 X1 (N )an by Y

(5.1)

1(p) (y)=x

Z!Z

(ay ) 7! y ey ay where ey is the rami cation degree of y over x via 1(p) . A fundamental compatibility, whose proof we omit for reasons of space, is: Theorem 5.1. The weight-2 Shimura isomorphism Sh; (N ) : S2 (;1 (N ); C)  S2 (;1 (N ); C)  = H 1 (X1 (N )an ; Z) Z C from (5:1) identi es hni  hni with hni 1, Tp  T p with Tp 1, and wN  wN with we i=N 1. Let T1 (N )  EndZ (H 1 (X1 (N )an ; Z)) be the subring generated by the Tp's and  hni 's. By Theorem 5.1, this is identi ed via the Shimura isomorphism with the classical (weight-2) Hecke ring at level N . In particular, this ring is commutative (which can be seen directly via cohomological considerations as well). It is clearly a nite at Z-algebra. The natural map (5.2) T1 (N ) Z C ,! EndC(H 1(X1(N )an ; Z) Z C) 1

2

210

CONRAD, THE SHIMURA CONSTRUCTION

induces an injection T1 (N ) C ,! EndC (S2 (;1 (N ); C)), by Theorem 5.1. This is the classical realization of Hecke operators in weight 2. Another compatibility we need is between the cup product on H 1 (X1 (N )an ; Z) and the (non-normalized) Petersson product on S2 (;1 (N ); C). To be precise, we de ne an isomorphism H 2 (X1 (N )an ; Z)  = Z using the i-orientation of the complex manifold X1 (N )an (i.e., the \idz ^ dz" orientation), so we get via cup product a (perfect) pairing ( ; ); (N ) : H 1 (X1 (N )an ; Z) Z H 1 (X1 (N )an ; Z) ! H 2 (X1 (N )an ; Z)  = Z: This induces an analogous pairing after applying Z C. For f; g 2 S2 (;1 (N ); C) we de ne Z hf; gi; (N ) = f (z )g(z )dxdy 1

;1 (N )nh

1

where this integral is absolutely convergent since f and g have exponential decay near the cusps. This is a perfect Hermitian pairing. Theorem 5.2. Under the weight-2 Shimura isomorphism Sh; (N ), ;
Sh; (N ) (f1 + g1 ); Sh; (N ) (f2 + g2 ) ; (N ) = 4
(hf1 ; g2i; (N ) ; hf2 ; g1i; (N ) ): Note that both sides are antilinear in g1 , g2 and alternating with respect to interchanging the pair (f1 ; g1 ) and (f2 ; g2 ). The extra factor of 4 is harmless for our purposes since it does not aect formation of adjoints. What is important is that in the classical theory, conjugation by the involution wN takes each T 2 T1 (N ) to its adjoint with respect to the Petersson product. The most subtle case of this is T = Tp for p j N . For p N the adjoint of Tp is hp;1 i Tp and the adjoint of hni is hn;1 i . These classical facts (especially for Tp with p j N ) yield the following important corollary of Theorem 5.2. Corollary 5.3. With respect to the pairing [x; y]; (N ) = (x; w y); (N ) with  = e2i=N , the action of T1 (N ) on H 1 (X1 (N )an ; Z) is equivariant. That is, [x; Ty]; (N ) = [Tx; y]; (N ) for all T 2 T1 (N ). With respect to ( ; ); (N ) , the adjoint of Tp for p N is hp;1 i Tp and the adjoint of hni is hn;1 i for n 2 (Z=N Z) . Looking back at the \conceptual" de nition of w for an arbitrary primitive N th root of unity  2 N (C), which gives an analytic involution of X1 (N )an , one can check that wj  w = hj i for j 2 (Z=N Z) . Since hj i is a unit in T1 (N ) and T1(N ) is commutative, we conclude that Corollary 5.3 is true with  2 N (C) any primitive N th root of unity (by reduction to the case  = e2i=N ). Our nal step on the analytic side is to reformulate everything above in terms of Jacobians. For any compact Riemann surface X , there is an isomorphism of complex Lie groups Pic0X  (5.3) = H 1 (X; OX )=H 1(X; Z) via the exponential sequence 1

1

1

1

1

1

-

1

1

1

1

-

1




0 ! Z ! OX ;e;;;! OX ! 1 and the identi cation of the underlying group of Pic0X with H 1 (X; OX )  = H 1 (X; OX ) ; 2i( )

5.1. Analytic preparations

211

where the line bundle L with trivializations 'i : OUi  = LjUi corresponds to the class of the C ech 1-cocycle Y f';j 1  'i : OUi \Uj = OUi \Uj g 2 H 0 (Ui \ Uj ; OX ) i<j

for an ordered open cover fUig. Beware that the tangent space isomorphism T0(Pic0X )  = H 1 (X; OX ) coming from (5.3) is ;2i times the \algebraic" isomorphism arising from 0 ! OX ! OX ["] ! OX ! 1; where X ["] = (X; OX ["]="2 ) is the non-reduced space of \dual numbers over X ". This extra factor of ;2i will not cause problems. We will use (5.3) to \compute" with Jacobians. Let f : X ! Y be a nite map between compact Riemann surfaces. Since f is nite at, there is a natural trace map fOX ! OY , and it is not dicult to check that this is compatible with the trace map f Z ! Z as de ned in (5.1). In particular, we have a trace map f : H 1 (X; OX )  = H 1 (Y; f OX ) ! H 1 (Y; OY ): Likewise, we have compatible pullback maps f OY  = OX and f  Z  = Z. Thus, any such f gives rise to commutative diagrams  H 1 (Y;O OY ) f / H 1 (X;O OX )

H 1 (Y; Z)

H 1 (X;O OX ) f / H 1 (Y;O OY )

f / 1 H (X; Z)

H 1 (X; Z) f / H 1 (Y; Z) ;

where the columns are induced by the canonical maps Z ! OY and Z ! OX . Passing to quotients on the columns therefore gives rise to maps f  : Pic0Y ! Pic0X ; f : Pic0X ! Pic0Y of analytic Lie groups. These maps are \computed" by Lemma 5.4. In the above situation, f  = Pic0 (f ) is the map induced by Pic0 functoriality and f = Alb(f ) is the map induced by Albanese functoriality. These are dual with respect to the canonical autodualities of Pic0X , Pic0Y . The signi cance of the theory of Jacobians is that by (5.3) we have a canonical isomorphism T` (Pic0X (N ) )  = H 1 (X1 (N )an ; Z`) (5.4) = H 1 (X1 (N )an ; Z) Z Z` ; connecting the `-adic Tate module of Pic0X (N ) with the Z-module H 1 (X1 (N )an ; Z) that \encodes" S2 (;1 (N ); C) via the Shimura isomorphism. Note that this isomorphism is de ned in terms of the analytic construction (5.3) which depends upon the choice of i. The intrinsic isomorphism (compatible with etale cohomology) has Z above replaced by 2iZ = ;2iZ. De nition 5.5. We de ne endomorphisms of Pic0X (N ) via Tp = Alb(1(p) )  Pic0 (2(p) ); hni = Pic0 (In ); w = Pic0 (w ): 1

an

1

1

an

212

CONRAD, THE SHIMURA CONSTRUCTION

By Lemma 5.4, it follows that the above isomorphism (5.4) carries the operators on T`(Pic0X (N ) ) over to the ones previously de ned on H 1 (X1 (N )an ; Z) (which are, in turn, compatible with the classical operations via the Shimura isomorphism). By the faithfulness of the \Tate module" functor on complex tori, we conclude that T1(N ) acts on Pic0X (N ) in a unique manner compatible with the above de nition, and (5.4) is an isomorphism of T1 (N ) Z Z` -modules. We call this the ( ) -action of T1 (N ) on Pic0X (N ) . We must warn the reader that under the canonical isomorphism of C-vector spaces S2 (;1 (N ); C)  = H 0 (X1 (N )an ; 1X (N ) ) = H 0 (Pic0X (N ) ; 1Pic ) X N = Cot0 (Pic0X (N ) ); the ( ) -action of T 2 T1 (N ) on Pic0X (N ) does not go over to the classical action of T on S2 (;1 (N ); C), but rather the adjoint of T with respect to the Petersson pairing. To clear up this matter, we make the following de nition: 1

an

1

an

an

1

an

1

an

1

)an

1(

an

1

1

0

an

De nition 5.6.

(Tp ) = Alb(2(p) )  Pic0 (1(p) ); hni = Alb(In ); (w ) = Alb(w ): Since In;1 = In; and w;1 = w on X1 (N )an , we have (w ) = w and hni = ; 1 hn i . We claim that the above ( ) operators are the dual morphisms (with respect to the canonical principal polarization of Pic0X (N ) ) of the ( ) operators and induce exactly the classical action of Tp and hni on S2 (;1 (N ); C), so we also have a well-de ned ( ) -action of T1 (N ) on Pic0X (N ) , dual to the ( ) -action. By Theorem 5.2, Corollary 5.3, and Lemma 5.4, this follows from the following general fact about compact Riemann surfaces. The proof is non-trivial. Lemma 5.7. Let X be a compact Riemann surface, and use the i-orientation to n 2 i=` de ne H 2 (X; Z)  Z . Use 1 ! 7 e to de ne Z=`n = `n (C) for all n. The = diagram / Z` H 1 (X; Z` ) Z` H 1 (X; Z` ) [ 1

an

1

an

1



 =



=

lim ; `n (C) anticommutes (i.e., going around from upper left to lower right in the two possible ways gives results that are negatives of each other), where the bottom row is the `adic Weil pairing (with respect to the canonical principal polarization Pic0X  = Pic0X for the \second" Pic0X in the lower left.) Note that the sign doesn't aect formation of adjoints. It ultimately comes from the sign on the bottom of [77, pg. 237] since our Weil pairing sign convention agrees with [77]. We now summarize our ndings in terms of V` (N ) = Q` Z` T`(Pic0X (N ) ), which has a perfect alternating Weil pairing ( ; )` : V` (N ) V` (N ) ! Q` (1)

T`(Pic0X ) Z` T`(Pic0X )

/

[

1

an

5.1. Analytic preparations

213

and has two Q` T1 (N )-actions, via the ( ) -actions and the ( ) -actions. Since (w ) = w , we simply write w for this operator on V` (N ). Theorem 5.8. Let T1(N ) act on V`(N ) with respect to the ( ) -action or with respect to the ( ) -action. With respect to ( ; )` , the adjoint of Tp for p N is hpi;1 Tp and the adjoint of hni is hni;1 for n 2 (Z=N Z) . With respect to [x; y]` = (x; w (y))` for  2 N (C) a primitive N th root of unity, the action of T1 (N ) on V` (N ) is selfadjoint. In general, adjointness with respect to ( ; )` interchanges the ( ) -action and ( ) -action. It should be noted that when making the translation to etale cohomology, the ( ) -action plays a more prominent role (since this is what makes (5.4) a T1 (N )equivariant map). However, when working directly with Tate modules and arithmetic Frobenius elements, it is the ( ) -action which gives the cleaner formulation of Shimura's results. An important consequence of Theorem 5.8 is Corollary 5.9. The Q` Z T1 (N )-module V` (N ) is free of rank 2 for either action, and HomQ (Q T1 (N ); Q) is free of rank 1 over Q T1 (N ) (hence likewise with Q replaced by any eld of characteristic 0). Remark 5.10. The assertion about HomQ(Q T1(N ); Q) is equivalent to the intrinsic condition that Q T1 (N ) is Gorenstein. Also, this freeness clearly makes the two assertions about V` (N ) for the ( ) - and ( ) -actions equivalent. For the proof, the ( ) -action is what we use. But in what follows, it is the case of the ( ) -action that we need! Proof. Using (5.4) and the choice of ( ) -action on V`(N ), it suces to prove  H 1 (X1 (N )an ; Q) is free of rank 2 over Q T1 (N ),  HomQ (Q T1 (N ); Q) is free of rank 1 over Q T1 (N ). Using [ ; ]; (N ) , we have (5.5) H 1 (X1 (N )an ; Q)  = HomQ (H 1 (X1 (N )an ; Q); Q) as Q T1 (N )-modules, so we may study this Q-dual instead. Since Q T1 (N ) is semilocal, a nite module over this ring is locally free of constant rank if and only if it is free of that rank. But local freeness of constant rank can be checked after faithfully at base change. Applying this with the base change Q ! C, and noting that C T1 (N ) is semilocal, it suces to replace Q by C above. Note that if the right hand side of (5.5) is free of rank 2, so is the left side, so choosing a basis of the left side and feeding it into the right hand side shows that HomQ (Q T1 (N )2 ; Q) is free of rank 2. In particular, the direct summand HomQ (Q T1 (N ); Q) is at over Q T1 (N ) with full support over Spec(Q

T1(N )), so it must be locally free with local rank at least 1 at all points of Spec(Q

T1(N )). Consideration of Q-dimensions then forces HomQ(Q T1(N ); Q) to be locally free of rank 1, hence free of rank 1. In other words, it suces to show that HomQ (H 1 (X1 (N )an ; Q); Q) is free of rank 2 over T1 (N ) Q, or equivalently that HomC (H 1 (X1 (N )an ; C); C) is free of rank 2 over T1 (N ) C. Via the Shimura isomorphism (in weight 2), which is compatible with the Hecke actions, we are reduced to showing that Hom(S2 (;1 (N ); C); C) is free of rank 1 over -

1

214

CONRAD, THE SHIMURA CONSTRUCTION

C T1 (N ). For this purpose, we will study the C T1(N )-equivariant C-bilinear

pairing

S2 (;1 (N ); C) C (C T1 (N )) ! C (f; T ) 7! a1 (Tf ) were a1 (
) is the \Fourier coecient of q". This is C T1 (N )-equivariant, since T1(N ) is commutative. It suces to check that there's no nonzero kernel on either side of this pairing. Since

C T1(N ) ! EndC(S2(;1 (N ); C))

is injective (as noted in (5.2)) and a1 (TTnf ) = an (Tf ) for T 2 T1 (N ), the kernel on the right is trivial. Since a1 (Tn f ) = an (f ), the kernel on the left is also trivial.

5.2. Algebraic preliminaries

Let S be a scheme. An elliptic curve E ! S is a proper smooth group scheme with geometrically connected bers of dimension 1 (necessarily of genus 1). It follows from [62, Ch.2] that the group structure is commutative and uniquely determined by the identity section. Fix N  1 and assume N 2 H 0 (S; OS ) (i.e., S is a Z[ N1 ]scheme). Thus, the map N : E ! E is nite etale of degree N 2 as can be checked on geometric bers. A point of exact order N on E is a section P : S ! E which is killed by N (i.e., factors through the nite etale group scheme E [N ]) and induces a point of exact order N on geometric bers. It follows from the stack-theoretic methods in [25] or the more explicit descent arguments in [62] that for N  5 there is a proper smooth Z[ N1 ]-scheme X1 (N ) equipped with a nite at map to P1Z[ N ] , such that the open subscheme Y1 (N ) lying over P1Z[ N ] ; f1g = A1Z[ N ] is the base of a universal object (E1 (N ); P ) ! Y1 (N ) for elliptic curves with a point of exact order N over variable Z[ N1 ]-schemes. Moreover, the bers of X1 (N ) ! Spec Z[ N1 ] are geometrically connected, as this can be checked on a single geometric ber and by choosing the complex ber we may appeal to the fact (whose proof requires some care) that there is an isomorphism (X1 (N ) Z[ N ] C)an  = X1 (N )an identifying the \algebraic" data (C=[1; z ]; N1 ) in Y1 (N )(C)  X1 (N )(C) with the class of z 2 h in ;1 (N )nh = Y1 (N )an  X1 (N )an (and X1 (N )an is connected, as h is). These kinds of compatibilities are somewhat painful to check unless one develops a full-blown relative theory of elliptic curves in the analytic world (in which case the veri cations become quite mechanical and natural). Again xing N  5, but now also a prime p, we want an algebraic analogue of 1 ]. Let (E; P ) ! S be an elliptic curve with a point of exact X1 (N; p)an over Z[ Np 1 order N over a Z[ Np ]-scheme S . We're interested in studying triples (E; P; C ) ! S where C  E is an order-p nite locally free S -subgroup-scheme which is not contained in the subgroup generated by P on geometric bers (if p j N ). Methods in [25] and [62] ensure the existence of a universal such object (E1 (N; p); P; C ) ! 1 ]-scheme which naturally sits as the complement Y1 (N; p) for a smooth ane Z[ Np 1 ]-scheme X1 (N; p) which is of a relative Cartier divisor in a proper smooth Z[ Np nite at over P1Z[ Np ] (with Y1 (N; p) the preimage of A1Z[ Np ] ). Base change to C 1 ] has and analyti cation recovers X1 (N; p)an as before, so X1 (N; p) ! Spec Z[ Np geometrically connected bers. 1

1

1

1

1

1

5.2. Algebraic preliminaries

215

1 ]-schemes (respectively, Z[ 1 ]-schemes) There are maps of Z[ Np N

In Y (N ) Y1 (N ) ;! 1

Y1 (N; p❑ )

❑ ❑ ❑ (p) ❑ ❑ ❑ 2 ❑❑❑ %

1(p) s s s s s

ss ys s s

Y1 (N )[ p1 ]

Y1 (N )[ p1 ]

  determined by (E; P; C ) ;;! (E; P ) and (E; P; C ) ;;! (E=C; P ) (which makes sense in Y1 (N ) if p j N by the \disjointness" condition on C and P ) and In (E; P ) = (E; nP ). Although 2(p) is not a map over A1Z[ Np ] , it can be shown that these all uniquely extend to (necessarily nite at) maps, again denoted 1(p) , 2(p) , In between X1 (N; p), X1 (N )[ p1 ], X1 (N ). A proof of this fact requires the theory of minimal regular proper models of curves over a Dedekind base; the analogous fact over Q is an immediate consequence of basic facts about proper smooth curves over a eld, but in order to most easily do some later calculations in characteristic p N it is convenient to know that we have the map Ip de ned on X1 (N ) over Z[1=N ] (though this could be bypassed by using liftings to characteristic 0 in a manner similar to our later calculations of Tp in characteristic p). Likewise, over Z[ N1 ; N ] we can de ne, for any primitive N th root of unity  = Ni (i 2 (Z=N Z) ), an operator w : Y1 (N )=Z[ N ;N ] ! Y1 (N )=Z[ N ;N ] via w (E; P ) = (E=hP i; P 0 ) where hP i is the order-N etale subgroup-scheme generated by P and P 0 2 (E [N ]=hP i)(S ) is uniquely determined by the relative Weil pairing condition hP; P 0 iN =  (with P 0 2 E [N ](S ) here). This really does extend to X1 (N )=Z[ N ;N ] , and one checks that w j w = Ij for j 2 (Z=N Z) . In particular, w2 = 1. Since X1 (N ) ! Spec Z[ N1 ] is a proper smooth scheme with geometrically connected bers of dimension 1, Pic0X (N )=Z is an abelian scheme over Z[ N1 ] and N hence is the Neron model of its generic ber. We have scheme-theoretic Albanese and Pic0 functoriality for nite ( at) maps between proper smooth curves (with geometrically connected bers) over any base at all, and analyti cation of such a situation over C recovers the classical theory of Pic0 as used in Section 5.1. For example, we have endomorphisms hni = Pic0 (In ); hni = Alb(In ) (p) 1

(p) 2

1

-

1

1

1

1

[ 1 ]

on Pic0X (N )=Z , 1

[ 1 ]

on Pic0X (N )=Z

N

1

1 [N

w = Pic0 (w ) = Alb(w ) = (w ) ;N ]

, and

Tp = Alb(1(p) )  Pic0 (2(p) ) (Tp ) = Alb(2(p) )  Pic0 (1(p) )

on Pic0X (N )=Z . A key point is that by the Neronian property, Tp and (Tp ) Np uniquely extend to endomorphisms of Pic0X (N )=Z , even though the i(p) do not 1

[ 1 ]

1

[ 1 ] N

216

CONRAD, THE SHIMURA CONSTRUCTION

make sense over Z[ N1 ] from what has gone before. In particular, it makes sense to study Tp and (Tp ) on the abelian variety Pic0X (N )=Fp over Fp for p N . This will be rather crucial later, but note it requires the Neronian property in the de nition. Passing to the analyti cations, the above constructions recover the operators de ned on Pic0X (N ) in Section 5.1. The resulting subring of End(Pic0X (N )=Z )  End(Pic0X (N ) ) N generated by Tp, hni (respectively, by (Tp ) , hni ) is identi ed with T1 (N ) via its ( ) -action (respectively, via its ( ) -action) and using 0 0 n (5.6) lim ; PicX (N )=Z [` ](Q) = T` (PicX (N ) ) -

1

1

an

[ 1 ]

1

1

an

1

1 ] [N

an

1

(using Q  C) endows our \analytic" V` (N ) with a canonical continuous action of GQ = Gal(Q=Q) unrami ed at all p N` (via Neron-Ogg-Shafarevich) and commuting with the action of T1 (N ) (via either the ( ) -action or the ( ) -action). We also have an endomorphism w = w = (w ) on Pic0X (N )=Z ; and it is easy N N to see that (g;1 ) wg( ) g = w on Q-points, where g 2 Gal(Q=Q) and g denotes the natural action of g on Qpoints (corresponding to base change of degree 0 line bundles on X1 (N )=Q ). Since w = w ; (as (E; P )  = (E; ;P ) via ;1), we see that w is de ned over the real sub eld Q(N )+ . By etale descent, the operator w is de ned over Z[ N1 ; N ]+ . In any case, w acts on V` (N ), recovering the operator in Section 5.1, and so this conjugates the ( ) -action to the ( ) -action, taking each T 2 T1 (N ) (for either action on V` (N )) to its Weil pairing adjoint, via the canonical principal polarization of the abelian scheme Pic0X (N )=Z . Using Corollary 5.3 and (5.6) we obtain -

1

[ 1

]

1

1

[ 1 ]

N

Lemma 5.11. Let T1(N ) act on V`(N ) through either the ( ) -action or the ( )  GL(2; Q` T1 (N )) is a continuous action. Then N;` : GQ ! Aut(V` (N )) =

representation, unrami ed at p - N`. The main result we are after is Theorem 5.12. Let T1 (N ) act on Pic0X (N )=Z N via the ( )-action. For any p N`, the characteristic polynomial of N;`(Frobp ) is X 2 ; (Tp ) X + phpi relative to the Q` T1 (N )-module structure on V` (N ), where Frobp denotes an arithmetic Frobenius element at p. The proof of Theorem 5.12 will make essential use of the w operator. For the remainder of this section, we admit Theorem 5.12 and deduce its consequences. Let f 2 S2 (;1 (N ); C) be a newform of levelPN . Let Kf  C be the number eld generated by ap (f ) for all p - N , where f = an (f )qn , so by weak multiplicity one an (f ) 2 Kf for all n  1 and the Nebentypus character f has values in Kf . Let pf  T1 (N ) be the minimal prime corresponding to f (i.e., the kernel of the map T1(N ) ! Kf sending each T 2 T1(N ) to its eigenvalue on f ). We now require T1 (N ) to act on Pic0X (N )=Z via its ( ) -action. [ 1 ]

1

1 ] [N

1

De nition 5.13. Af is the quotient of Pic0X (N )=Q by pf  T1 (N ). 1

5.3. Proof of Theorem 5.12

217

By construction, Af has good reduction over Z[ N1 ] and the action of T1 (N ) on Pic0X (N )=Q induces an action of T1 (N )=p on Af , hence an action of Kf  = (T1 (N )=p) Z Q on Af in the \up-to-isogeny" category. Theorem 5.14 (Shimura). We have dim Af = [Kf : Q] and V`(Af ) is free of rank 2 over Q` Q Kf , with Frobp having characteristic polynomial X 2 ; (1 ap (f ))X + 1 pf (p) for all p N`. Proof. By Lemma 5.11 and Theorem 5.12, we just have to check that the Q`

T1(N )-linear map V` (Pic0X (N )=Q ) ! V` (Af ) identi es the right hand side with the quotient of the left hand side by pf . More generally, for any exact sequence B0 ! B ! A ! 0 of abelian varieties over a eld of characteristic prime to `, we claim V` (B 0 ) ! V` (B ) ! V` (A) ! 0 is exact. We may assume the base eld is algebraically closed, and then may appeal to Poincare reducibility (see [77, pg. 173]). Choosing a place  of Kf over ` and using the natural realization of Kf; as a factor of Q` Kf , we deduce from Theorem 5.14: Corollary 5.15. Let f 2 S2(;1 (N ); C) be a newform and  a place of Kf over `. There exists a continuous representation f; : GQ ! GL(2; Kf;) unrami ed at all p N`, with Frobp having characteristic polynomial X 2 ; ap (f )X + pf (p) 2 Kf; [X ]: 1

-

1

-

5.3. Proof of Theorem 5.12 Fix p N and let -

Jp = Pic0X (N )=Fp  = Pic0X (N )=Z 1

1

[ 1 ] N

Z[ N ] Fp 1

with T1 (N ) acting through the ( ) -action. Fix a choice of Frobp , or more specifically x a choice of place in Q over p. Note that this determines a preferred algebraic closure Fp as a quotient of the ring of algebraic integers, and in particular a map Z[1=N; N ] ! Fp . Thus, we may view w as inducing an endomorphism of the abelian variety Jp Fp Fp over Fp (whereas the elements in T1 (N ) induce endomorphisms of Jp over Fp ). The canonical isomorphism V` (Pic0X (N )=Q )  = V` (Pic0X (N )=Z )  = V` (Jp ) 1

1

[ 1 ]

N

identi es the Frobp -action on Q-points on the left hand side with the (arithmetic) Frobenius action on Fp -points on the right hand side. Obviously V` (Jp ) is a module

218

CONRAD, THE SHIMURA CONSTRUCTION

over the ring Q` T1 (N ) and is free of rank 2 as such. For any Fp -schemes Z , Z 0 and any Fp -map f : Z ! Z 0 the diagram (5.7)

f / 0 Z

Z FZ

FZ 0 f / 0 Z



Z

commutes, where columns are absolute Frobenius. Taking Z = Spec Fp , Z 0 = Jp , we see that the Frobp action of V` (Jp ) through Fp -points is identical to the action induced by the intrinsic absolute Frobenius morphism F : Jp ! Jp over Fp . Here is the essential input, to be proven later. Theorem 5.16 (Eichler-Shimura). In EndFp (Jp ),

(Tp ) = F + hpi F _ ; w;1 Fw = hpi; 1 F where F _ denotes the dual morphism. The extra relation involving w is crucial. The interested reader should compare this with [108, Cor. 7.10]. Let us admit Theorem 5.16 and use it to prove Theorem 5.12. We will then prove Theorem 5.16. Using an Fp -rational base point P (e.g., the cusp 0), we get a commutative diagram / Jp X1 (N )=Fp   FX1 (N ) 

X1 (N )=Fp





F

/ Jp

where FX (N ) denotes the absolute Frobenius morphism of X1 (N )=Fp , so by Albanese functoriality F = Alb(FX (N ) ). Thus FF _ = Alb(FX (N ) )  Pic0 (FX (N ) ) = deg(FX (N ) ) = p as X1 (N )=Fp is a smooth curve. We conclude from (Tp ) = F + hpi F _ that F 2 ; (Tp ) F + phpi = 0 on Jp , hence in V` (Jp ). Thus, N;`(Frobp ) satis es the expected quadratic polynomial X 2 ; (Tp ) X + phpi = 0: Let X 2 ; aX + b be the true characteristic polynomial, which N;`(Frobp ) must also satisfy, by Cayley-Hamilton. We must prove that a = (Tp ) , and then b = phpi is forced. It is this matter which requires the second relation. We want trQ` T (N ) (N;`(Frobp )) = (Tp ) or equivalently trQ` T (N ) (V` (F )) = (Tp ) : Using the modi ed Weil pairing [x; y]` = (x; w y)` 1

1

1

1

1

1

1

5.3. Proof of Theorem 5.12

219

and using the fact that V` (Jp )  = V` (Pic0X (N )=Q ) respects Weil pairings (by invoking the relativization of this concept, here over Z[ N1 ]) we may identify (via Theorem 5.8 and a choice Q` (1)  = Q` as Q`-vector spaces) V` (Jp )  = HomQ` (V` (Jp ); Q` ) := V` (Jp ) as Q` T1 (N )-modules, but taking the F -action over to the hpi F _ -action, since adjoints with respect to Weil pairings are dual morphisms and w;1 F _ w is dual to w;1 Fw = hpi; 1 F = F hpi; 1 (absolute Frobenius commutes with all morphisms of Fp-schemes!) Since V` (Jp ) is free of rank 2 over Q` T1 (N ) and HomQ` (Q` T1 (N ); Q` ) is free of rank 1 over Q` T1 (N ), by Corollary 5.9, we conclude trQ` T (N )(F jV` (Jp )) = trQ` T (N ) (hpi F _ jV` (Jp ) ): We wish to invoke the following applied to the Q` -algebra Q` T1 (N ) and the Q` T1(N )-module V` (Jp): Lemma 5.17. Let O be a commutative ring, A a nite locally free O-algebra with HomO (A; O) a locally free A-module (necessarily of rank 1). Let M be a nite locally free A-module, M  = HomO (M; O), so M  is nite and locally free over A with the same rank as M . For any A-linear map f : M ! M with O-dual f  : M  ! M  , automatically A-linear, char(f ) = char(f  ) in A[T ] (these are the characteristic polynomials). Proof. Without loss of generality O is local, so A is semilocal. Making faithfully

at base change to the henselization of O (or the completion if O is noetherian or if we rst reduce to the noetherian case), we may assume that A is a product of local rings. Without loss of generality, A is then local, so M = Aei if free, and HomO (A; O) is free of rank 1 over A. Choose an isomorphism h:A = HomO (A; O) as A-modules, so the projections i : M ! Aei  =A    satisfy ei = h(i)  i in M . These ei are an A-basis of M  and we compute matrices over A: Matfei g (f ) = Matfei g (f  )t : 1

1

1

We conclude that trQ` T (N ) (F jV` (Jp )) = trQ` T (N ) (hpi f _jV` (Jp )): By Theorem 5.16, we have 2(Tp ) = tr((Tp ) jV` (Jp )) = tr(F + hpi F _jV` (Jp )) = 2 tr(F jV` (Jp )): This proves that tr(F jV` (Jp )) = (Tp ) , so indeed X 2 ; (Tp ) X + phpi is the characteristic polynomial. Finally, there remains 1

1

220

CONRAD, THE SHIMURA CONSTRUCTION

Proof of Theorem 5.16. It suces to check the maps coincide on a Zariski dense subset of Jp (Fp ) = Pic0 (X1 (N )=Fp ). If g is the genus of X1 (N )=Z[ N ] and we x an Fp-rational base point, we get an induced surjective map X1 (N )g=Fp ! Jp =Fp ; so for any dense open U  X1 (N )Fp , U g ! (Jp )=Fp hits a Zariski dense subset of Fp-points. Taking U to be the ordinary locus of Y1(N )=Fp , it suces to study what happens to a dierence (x) ; (x0 ) for x; x0 2 Y1 (N )(Fp ) corresponding to (E; P ), (E 0 ; P 0 ) over Fp with E and E 0 ordinary elliptic curves. 1

By the commutative diagram (5.7), the map Jp (Fp ) ! Jp (Fp ) induced by F is the same as the map induced by the pth power map in Fp . By de nition of Pic0 functoriality, this corresponds to base change of an invertible sheaf on X1 (N )=Fp by the absolute Frobenius on Fp . By de nition of Y1 (N )=Fp as a universal object, such base change induces on Y1 (N )(Fp ) exactly \base change by absolute Frobenius" on elliptic curves with a point of exact order N over Fp . We conclude F ((x) ; (x0 )) = (E (p) ; P (p) ) ; ((E 0 )(p) ; P (p) ) where ( )(p) denotes base change by absolute Frobenius on Fp . Since p = FF _ = F _ F and F is bijective on Fp -points, we have F _ ((x) ; (x0 )) = pF ;1 ((x) ; (x0 )) ; ; ; ; = p((E (p ) ; P (p ) ) ; ((E 0 )(p ) ; (P 0 )(p ) )): Thus, hpi F _((x) ; (x0 )) = p(E (p; ) ; pP (p; ) ) ; p((E 0 )(p; ) ; p(P 0 )(p; ) ) so ; ; (F + hpi F _)((x) ; (x0 )) = (E (p) ; P (p) ) + p(E (p ) ; pP (p ) ) ; ((E 0 )(p) ; (P 0 )(p) ) + p((E 0 )(p; ) ; p(P 0 )(p; ) ): 1

1

1

1

1

1

1

1

1

1

1

1

Computing (Tp ) on Jp = Pic0X (N )=Z Z[ N ] Fp is more subtle because (Tp ) N 1 ] (or over Q) as (2 )   and was extended over Z[ 1 ] by was de ned over Z[ Np 1 N the Neronian property. That is, we do not have a direct de nition of (Tp ) in characteristic p, so we will need to lift to characteristic 0 to compute. It is here that the ordinariness assumption is crucial, for we shall see that, in some sense, (Tp ) ((x) ; (x0 )) = (F + hpi F _ )((x) ; (x0 )) as divisors for ordinary points x, x0 . This is, of course, much stronger than the mere linear equivalence that we need to prove. Before we dive into the somewhat subtle calculation of (Tp ) ((x) ; (x0 )), let's quickly take care of the relation w;1 Fw = hpi; 1 F , or equivalently, 1

[ 1 ]

1

Fw = w hp;1 i F:

5.3. Proof of Theorem 5.12

221

All maps here are induced by maps on X1 (N )=Fp , with F = Alb(FX (N ) ), w = Alb(w jX N ), hp;1 i = Alb(Ip; ). Thus, it suces to show FX (N )  w = w Ip; FX (N ) on X1 (N )=Fp , and we can check by studying x = (E; P ) 2 Y1 (N )(Fp ): FX (N )w (x) = FX (N ) (E=P; P 0 ) = (E (p) =P (p) ; (P 0 )(p) ) where hP; P 0 iN =  , so hP (p) ; (P 0 )(p) iN =  p by compatibility of the (relative) Weil pairing with respect to base change. Meanwhile, w Ip; FX (N ) (x) = w (E (p) ; p;1 P (p) ) = (E (p) =(p;1P (p) ); Q) where hp;1 P (p) ; QiN =  , or equivalently hP (p) ; Qi =  p . Since Q = (P 0 )(p) is such a point, this second relation is established. Now we turn to the problem of computing (Tp ) ((x) ; (x0 )) for \ordinary points" x = (E; P ), x0 = (E 0 ; P 0 ) as above. Let R = Zun p , W (Fp ), or more generally any henselian (e.g., complete) discrete valuation ring with residue eld Fp and fraction eld K of characteristic 0. Since p N , R is a Z[ N1 ]-algebra. Since Y1 (N ) is smooth over Z[ N1 ], we conclude from the (strict) henselian property that Y1 (N )(R) ! Y1 (N )(Fp ) is surjective. Of course, this can be seen \by hand": if (E; P ) is given over Fp , choose a Weierstrass model E ,! P2R lifting E (this is canonically an elliptic curve, by [62, Ch 2]). The nite etale group scheme E [N ] is constant since R is strictly henselian. Thus there exists a unique closed immersion of group schemes Z=N Z ,! E [N ] lifting P : Z=N Z ,! E [N ]. Let (E ; P ), (E 0 ; P 0 ) over R lift x, x0 respectively. We view these sections to X1 (N )=R ! Spec R as relative eective Cartier divisors of degree 1. Using the reduction map Pic0X (N )=Z (R) ! Jp (Fp ) N and the de nition of (Tp ) , we see that (Tp ) ((x);(x0 )) is the image of (Tp ) ((E ; P ); 1 ]-algebra but K is, and we have an injection (even (E 0 ; P 0 )). Now R is NOT a Z[ Np bijection) Pic0X (N )=Z (R) ,! Pic0X (N )=Z (K ); N N as Pic0X (N )=Z ! Spec Z[ N1 ] is separated (even proper). N Thus, we will rst compute (Tp ) ((x) ; (x0 )) by working with K -points, where K is an algebraic closure of K . Since p N , we have X (2 ) 1 ((E ; P )=K ) = (EK =C; PK mod C ) 1

1(

1

)

1

1

1

1

1

1

1

-

1

1

1

[ 1 ]

[ 1 ]

1

[ 1 ]

[ 1 ]

-

C

where C runs through all p + 1 order-p subgroups of E=K . Since E ! Spec R has ordinary reduction, and R is strictly henselian, the connected-etale sequence of E [p] is the short exact sequence of nite at R-group schemes 0 ! p ! E [p] ! Z=pZ ! 0: Enlarging R to a nite extension does not change the residue eld Fp , so we may assume that E [p]=K = Z=pZ  Z=pZ:

222

CONRAD, THE SHIMURA CONSTRUCTION

Taking the scheme-theoretic closure in E [p] of the p +1 distinct subgroups of E [p]=K gives p + 1 distinct nite at subgroup schemes C  E realizing the p + 1 distinct C 's over K . Exactly one of these C 's is killed by E [p] ! Z=pZ over R, as this can be checked on the generic ber, so it must be p ,! E [p]. For the remaining C 's, the map C ! Z=pZ is an isomorphism on the generic ber. We claim these maps

C ! Z=pZ over R are isomorphisms. Indeed, if C is etale this is clear, yet C ,! E [p] is a nite

at closed subgroup-scheme of order p, so a consideration of the closed ber shows that if C is not etale then it is multiplicative. But E [p] has a unique multiplicative subgroup-scheme since E [p]_ = E [p] by Cartier-Nishi duality and E [p] has a unique order-p etale quotient (as any such quotient must kill the p we have inside E [p].) Thus, X X (2 ) 1 ((E ; P )=K ) = (E =C ; P mod C ) ; (E 0 =C 0; P 0 mod C 0 ) C

C0

2 Pic0

X1 (N )=Z[ N1 ] (R)

coincides with (Tp ) ((E ; P ) ; (E 0 ; P 0 )) as both induce the same K -point. Passing to closed bers, (Tp ) ((x) ; (x0 )) = (E=p ; P mod p ) + p(E=Z=pZ0 ; P mod Z=pZ) ; (E 0 =p ; P 0 mod p ) + p(E 0 =Z=pZ; P 0 mod Z=pZ) where E [p]  = p  Z=pZ and E 0 [p]  = p  Z=pZ are the canonical splittings of the connected-etale sequence over the perfect eld Fp . Now consider the relative Frobenius morphism FE=Fp : E ! E (p) ; which sends O to O (and P to P (p) ) and so is a map of elliptic curves over Fp . Recall that in characteristic p, for any map of schemes X ! S we de ne the relative Frobenius map FX=S : X ! X (p) to be the unique S -map tting into the diagram

X ❉ ❉ FX=S

FX / X (p)

❉❉ ❉❉ ❉❉ ❉! 

(/

X 

S FS / S

where FS , FX are the absolute Frobenius maps. Since E ! Spec Fp is smooth of pure relative dimension 1, FE=Fp is nite at of degree p1 = p. It is bijective on points, so ker(FE=Fp ) must be connected of order p. The only such subgroup-scheme of E is p ,! E [p] by the ordinariness. Thus E=p  = E (p) is easily seen to take P mod p to P (p) .

5.3. Proof of Theorem 5.12 Similarly, we have

E

223

p (/ / (p) _F E FE=Fp E FE= p

_ is etale of degree p and base extension by Frob;1 : Fp ! Fp gives so FE= Fp p

E (p;1 )

*

E (p; ) ✤ ✤ / p
P (p; ) : P (p; ) / P As the second map in this composite is etale of degree p, we conclude ; ; (E=Z=pZ ; P mod Z=pZ)  = (E (p ) ; pP (p ) ): /

E

/

1

1

1

1

1

Thus, in Pic0X (N ) (Fp ), ; ; (Tp ) ((x) ; (x0 )) = (E (p) ; P (p) ) + p
(E (p ) ; p
P (p ) ) ; ((E 0 )(p) ; (P 0 )(p) ) ; p
((E 0 )(p; ) ; p
(P 0 )(p; ) ) which we have seen is equal to (F + hpi F _ )((x) ; (x0 )). 1

1

1

1

1

CHAPTER 6

Appendix by Kevin Buzzard: A mod ` multiplicity one result In this appendix, we explain how the ideas of [46] can be used to prove the following mild strengthening of the multiplicity one results in x9 of [32]. The setup is as follows. Let f be a normalised cuspidal eigenform of level N , and weight k, de ned over F` , with ` N and 2  k  ` + 1. Let N  denote N if k = 2, and N` if k > 2. Let JQ be the Jacobian of the curve X1 (N  )Q , and let H denote the Hecke algebra in End(JQ ) generated over Z by Tp for all primes p, and all the Diamond operators of level N  . It is well-known (for example by Proposition 9.3 of [46]) that there is a characteristic 0 normalised eigenform F in S2 (;1 (N  )) lifting f . Let m denote the maximal ideal of H associated to F (note that m depends only on f and not on the choice of F ), and let F = H=m, which embeds naturally into F` . Suppose the representation f : GQ ! GL2 (F` ) associated to f is absolutely irreducible, and furthermore assume that if k = ` + 1 then f is not isomorphic to a representation coming from a form of weight 2 and level N . Theorem 6.1. If f is rami ed at `, or if f is unrami ed at ` but f (Frob`) is not a scalar matrix, then JQ (Q)[m] has H=m-dimension two, and hence is a model for (precisely one copy of) f . The motivation for putting ourselves in the setup above is that every absolutely irreducible modular mod ` representation has a twist coming from a modular form of level prime to ` and weight at most ` + 1. In particular, every modular mod ` representation has a twist coming from a form satisfying the conditions of our setup. Furthermore, if f is as in our setup, then Theorems 2.5 and 2.6 of [32] tell us the precise structure of the restriction of f to D` , a decomposition group at `. These results are explained in Section 2.2. Using them, it is easy to deduce Corollary 6.2. Let  be an absolutely irreducible modular mod ` representation, such that f (D` ) is not contained within the scalars. Then some twist of  comes from a modular form satisfying the conditions of the theorem, and hence  is a multiplicity one representation in the sense of Remark 3.4.2. The theorem, commonly referred to as a \multiplicity one theorem", is a mild extension of results of Mazur, Ribet, Gross and Edixhoven. It was announced for ` = 2 as Proposition 2.4 of [9] but the proof given there is not quite complete|in fact, the last line of the proof there is a little optimistic. I would hence like to thank Ribet and Stein for the opportunity to correct this oversight in [9]. 225 -

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Proof of Theorem. Firstly we observe that the only case not dealt with by Theorem 9.2 of [32] is the case when k = ` and f is unrami ed at `, with f (Frob` ) a non-scalar matrix whose eigenvalues are equal. Moreover, using Theorems 2.5 and 2.6 of [32] we see that in this case f must be ordinary at `. We are hence in a position to use the detailed construction of f given in xx11{12 of [46]. We will follow the conventions set up in the present paper for normalisations of Hecke operators, and so in particular the formulae below dier from the ones in [46] by a twist. The maximal ideal m of H associated to f gives rise as in (12.5) of [46] to an idempotent e 2 H` := H Z Zl , such that the completion Hm of H at m is just eH`. Let G denote e(JQ` [`1 ]), the part of the `-divisible group of J which is associated to m. Then Hm acts on G, and it is proved in Propositions 12.8 and 12.9 of [46] that there is an exact sequence of `-divisible groups

0 ! G0 ! G ! Ge ! 0 over Q` , which is Hm -stable. Let 0 ! T0 ! T ! Te ! 0 be the exact sequence of Tate modules of these groups. We now explain explicitly, following [46], how the group D` acts on these Tate modules. If k > 2 then there is a Hecke operator U` in Hm , and we de ne u = U` . If k = 2 then there is a Hecke operator T` in Hm and because we are in the ordinary case we know that T` is a unit in Hm. We de ne u to be the unique root of the polynomial X 2 ; T` X + `h`i in Hm which is a unit (u exists by an appropriate analogue of Hensel's lemma). The calculations of Propositions 12.8 and 12.9 of [46] show that, under our conventions, the absolute Galois group D` of Q` acts on T e as (u), where (x) denotes the unrami ed character taking Frob` to x. Moreover, these propositions also tell us that D` acts on T 0 via the character `(u;1 h`iN )`;2 , where ` is the cyclotomic character and  is the Teichmuller character. The key point is that this character takes values in H  . The next key observation is that a standard argument on dierentials, again contained in the proof of Propositions 12.8 and 12.9 of [46], shows that Ge [m] = m;1`T e =`T e has Hm=m-dimension 1 and that G0 [m] has dimension d0  1. (Note that the fact that Ge [m] has dimension 1 implies, via some simple linear algebra, that the sequence 0 ! G0 [m] ! G[m] ! Ge [m] ! 0 is exact, as asserted by Gross.) Furthermore, because we can identify G0 [m] with m;1 `T 0=`T 0, we see that the action of D` on G0 [m] is via a character which takes values in (H=m) . In particular, D` acts as scalars on G0 [m]. Let us now assume that f is unrami ed at `, and that f (Frob` ) is a nondiagonalisable matrix with eigenvalue  2 H=m. Choose a model  for f de ned over GL2 (H=m). By the theorem of Boston, Lenstra and Ribet, we know that G[m] is isomorphic to a direct sum of d copies of , or more precisely, d copies of the restriction of  to D` . Here d is an integer satisfying 2d = d0 + de . Hence, if G[m] denotes the subspace of G[m] where Frob` acts as , then the H=m-dimension of G[m] is at most d. On the other hand, Frob` acts on G[m]0 as a scalar, and hence this scalar must be , and so we see G[m]0  G[m] . Hence d0  d = (d0 + 1)=2. We deduce that d0  1 and hence d0 = d = 1 and the theorem is proved.

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227

We remark that L. Kilford has found examples of mod 2 forms f of weight 2, such that f is unrami ed at 2 and f (Frob2 ) is the identity, and where JQ (Q)[m] has H=m-dimension 4, and so one cannot hope to extend the theorem to this case. See Remark 3.6 for more details, or [64]. A detailed analysis of what is happening in this case, at least in the analogous setting of forms of weight 2 on J0 (p), with p prime, has been undertaken by Emerton in [39]. In particular, Emerton proves that multiplicity one fails if and only if the analogue of the exact sequence 0 ! T 0 ! T ! T e ! 0 fails to split as a sequence of Hm -modules.

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