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⊂ p m−1 Acris and Sˆ PD ⊂ Acris .
1.2. Classification of p-adic representations 1.2.1. Crystalline representations and de Rham representations (see [F1], [F5]). Let L be a finite totally ramified extension of K. Denote by RepQp (GL ) the category ¯ For a p-adic representation V , define of p-adic representations of GL = Gal(K/L). G DdR (V ) = V ⊗Qp BdR L . Then DdR (V ) is an L-vector space with a decreasing filtration given by Fili DdR (V ) = (V ⊗Qp Fili BdR )GL . The representation V is said to be de Rham if dimL DdR (V ) = dimQp (V ). (In general, one has dimL DdR (V ) dimQp (V ).) For a de Rham representation we have the Hodge-Tate decomposition V ⊗Qp C ⊕ Gr −i DdR (V ) ⊗L C(i), where Gr k (DdR (V )) = Filk DdR (V )/ Filk+1 DdR (V ). Analogously one defines G Dcris (V ) = V ⊗Qp Bcris L . Then Dcris (V ) is a K-vector space equipped with a Frobenius operator and a filtration, induced by the natural inclusion Dcris (V )⊗K L ⊂ DdR (V ). One has dimK Dcris (V ) dimL DdR (V ) dimQp (V ), and V is said to be crystalline if the equality holds here.
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
219
For the resulting categories we have Repcris (GL ) ⊂ RepdR (GL ) ⊂ RepQp (GL ). The functor DdR (resp., Dcris ) restricted on RepdR (GL ) (resp., Repcris (GL )) commutes with tensor products, direct sums, and duals. Moreover, Dcris is an equivalence between Repcris (GL ) and some subcategory of the category of filtered K-modules. 1.2.2. Classification of Zp -adic representations in terms of -modules (see [F2]). Fontaine [F2] found a classification of all p-adic representations in terms of étale modules over a 2-dimensional local ring endowed with semilinear actions of Frobenius and Γ commuting to each other. In particular, his theory allows us to compute the Galois cohomology of p-adic representations using complexes of such modules (see §1.3 below). We review these results only for cyclotomic ground fields. Let A be a commutative Noetherian ring with a flat map ϕ : A → A. An A-module ᏹ is said to be a ϕ-module if it is equipped with a ϕ- semilinear map ϕᏹ : ᏹ → ᏹ. (We often write ϕ instead ϕᏹ .) Let ᏹϕ = Aϕ ⊗A ᏹ be the module obtained from ᏹ by extending scalars ϕ : A → A, and let ᏹ : ᏹϕ → ᏹϕ be the induced linear map; that is, ᏹ (a ⊗ m) = a ⊗ ϕᏹ (m). ᏹ is called an étale module over A if it is finitely generated and if ᏹ is a bijection. Fix n ∈ N and consider the ring ᏻKn , with Frobenius operator ϕ and continuous action of Γn . An étale -module over ᏻKn is an étale ᏻKn -module equipped with a semilinear action of Γn , commuting with ϕ. The category Mét ᏻKn of étale modules over ᏻKn is a ⊗-category. Let RepZp (GKn ) be the category of Zp -representations, that is, the category of finite Zp -modules, equipped with a continuous linear action of GKn . Theorem (J.-M. Fontaine). (i) The functor DᏻKn : RepZp GKn −→ Mét ᏻK , n
GK∞ DᏻKn (T ) = T ⊗Zp ᏻˆ nr Kn is an equivalence of categories. (ii) The functor VᏻKn : Mét ᏻK −→ RepZp GKn , n
VᏻKn (ᏹ) = ᏹ ⊗ᏻKn ᏻˆ nr Kn
ϕ=1
is quasi inverse to DᏻKn . Remarks. (1) Let T be a representation of GKn , and let m n. Then DᏻKm (T ) = ᏻKm ⊗ᏻKn DᏻKn (T ).
220
DENIS BENOIS
(2) Denote by Reptor(GKn ) the category of p-torsion Galois representations, and denote by Mét ᏻKn ,tor the category of p-torsion -modules. Then DᏻKn and VᏻKn induce equivalences of these categories. Passing to inductive limits one obtains an equivalence between the category MGKn ,tor of p-torsion GKn -modules and the cate− ét gory Mind ᏻK ,tor . n
1.2.3. Classification of crystalline representations in terms of -modules (see [F2], [W]). In this section we review the results of Fontaine and Wach. ˜ ˜ Let S˜n = ᏻˆ nr Kn ∩ W (R), and let, for simplicity, S = S0 . For a free Zp -module T of finite rank endowed with a continuous action of GKn , define G DSn (T ) = T ⊗Zp S˜n K∞ . Then DSn (T ) is a free Sn -module with natural actions of ϕ and Γn . One has rangSn DSn (T ) rangᏻKn DᏻKn (T ) = rangZp T . T is called a representation of finite height if rangSn DSn (T ) = rangZp T . Theorem (N. Wach). Let T be a representation of finite height. Then the following two conditions are equivalent: (i) V = T ⊗Zp Qp is crystalline. (ii) There exists a free Sn -submodule N ⊂ DSn (T ) of rank d such that Γn acts trivially on (N/πn N)(−h) for some h ∈ Z. One can also set
G DS (T ) = T ⊗Zp S˜ K∞ .
It is easy to see that DS (T ) = ϕ n DSn (T ) and that (ii) is equivalent to the existence of a free submodule NS ⊂ DS (T ) of rank d such that Γn acts trivially on (NS /πNS )(−h). Recall briefly the implication (ii) ⇒ (i). Taking T (−h) instead of T we can suppose that h = 0, that is, that Γn acts trivially on NS /πNS . Let π π2 πm PD S(n) = S n , ,..., ,... . p 2!p 2n m!p mn i
PD generated by the elements π m /p mn m!, m i, and Denote by I(n) the ideal of S(n) define i PD PD Sˆ(n) = lim S(n) I(n) . ← − i
PD = Sˆ PD . The Frobenius operator on S can be extended to the Note that Sˆ(0) PD → Sˆ PD , and we have, therefore, a homomorphism ϕ n : Sˆ PD → A ϕ : Sˆ(n) cris . (n−1) (n)
map
By successive approximations it is not difficult to show that there exists a unique PD )Γn of the natural projection N ⊗ PD ˆ Sˆ(n) section NS /π NS (NS ⊗ S ˆ Sˆ(n) → NS /πNS .
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
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Consider a sequence of maps n n PD Γn ϕ ⊗ϕ ˆ S Sˆ(n) ˆ S Sˆ PD Γn ⊂ T ⊗ Acris GKn . NS /π NS NS ⊗ −−−−→ NS ⊗
Since rangZp NS /π NS = d, it implies that (T ⊗Acris )GKn contains a W -lattice of rank d. Thus dimQp Dcris (V ) = d, and V is crystalline. ˆ S Sˆ PD )Γn ⊂ (T ⊗ Acris )GKn , the W ˆ S Sˆ PD )Γn ⊂ (DS (T )⊗ Note that since (NS ⊗ module ˆ S Sˆ PD Γn M = DS (T )⊗ is a lattice of Dcris (V ) stable under the Frobenius operator ϕ. 1.3. Computation of Galois cohomology 1.3.1. Construction of complexes (see [H1]). Denote by MGKn ,pm the category of pm -torsion GKn -modules. The functor DᏻKn gives an equivalence between this − ét category and the category Mind ᏻKn ,p m whose objects are inductive limits of étale m -modules of p -torsion. Note that these categories have enough injective objects. Put ᏻKn ,pm = ᏻKn /p m ᏻKn . For any T ∈ MGKn ,pm there is a natural isomorphism HomGKn Z/p m Z, T Hom ᏻKn ,pm , DᏻKn (T ) , and therefore we obtain isomorphisms of derived functors: Ext iGK Z/p m Z, T Ext i ᏻKn ,pm , DᏻKn (T ) . n
The uniqueness of a derived functor implies that Ext iGK (Z/p m Z, T ) is isomorphic to n
H i (GKn , T ) (see [CE, Chap. 6, Prop. 4.1.3]). Fix a generator γn ∈ Γn , and consider the complex β
α
P . : 0 ←− ᏻKn ,pm [ϕ, γn ] ←− ᏻKn ,pm [ϕ, γn ]⊕ ᏻKn ,pm [ϕ, γn ] ←− ᏻKn ,pm [ϕ, γn ] ←− 0, where α(y, z) = (ϕ − 1)y + (γn − 1)z and β(x) = ((γn − 1)x, (1 − ϕ)x). Then P . is a − ét projective resolution of ᏻKn ,pm . To any ᏹ ∈ Mind ᏻK ,tor we associate the complex n
f
g
C . (ᏹ) : 0 −→ ᏹ −→ ᏹ ⊕ ᏹ −→ ᏹ −→ 0, where f (m1 ) = ((ϕ − 1)m1 , (γn − 1)m1 ) and g(m2 , m3 ) = (γn − 1)m2 + (1 − ϕ)m3 . An easy computation shows that Hom (P . , ᏹ) is isomorphic to C . (ᏹ). Thus, if ᏹ = DᏻKn (T ), then H i (GKn , T ) is isomorphic to H i (C . (ᏹ)). Passing to direct limits we obtain a system of natural isomorphisms hi : H i (C . (ᏹ)) H i GKn , T for all T ∈ MGKn ,tor .
222
DENIS BENOIS
In this paper it is more convenient to use continuous Galois cohomology. Let T be a Zp -representation of GKn , and let ᏹ = DᏻKn (T ). Then, repeating the argument of Tate [T, Prop. 2.2], we obtain an isomorphism between H i (GKn , T ) and H i (C . (ᏹ)) which also is denoted by hi . In particular, h0 : ᏹϕ=1,γn =1 H 0 (GKn , T ). The isomorphism h1 also has a rather simple description in terms of cocycles. Tensoring the exact sequence 1−ϕ
ˆ nr 0 −→ Zp −→ ᏻˆ nr Kn −−−→ ᏻKn −→ 0 with T , one obtains an exact sequence 1−ϕ
ˆ nr 0 −→ T −→ ᏹ ⊗ᏻKn ᏻˆ nr Kn −−−→ ᏹ ⊗ᏻKn ᏻKn −→ 0. Proposition 1.3.2. Let m1 , m2 ∈ ᏹ, and assume that (γn − 1)m1 = (ϕ − 1)m2 . 1 Let u ∈ ᏹ ⊗ᏻKn ᏻˆ nr Kn be a solution of the equation (1 − ϕ)u = m1 . Then h sends cl(m1 , m2 ) to the class of the cocycle k(g)−1 g −→ ug − u + 1 + γn + · · · + γn m2 , k(g)
where γn
= g|K∞ .
Proof. Let ᏺm1 ,m2 = ᏹ ⊕ ᏻKn e, where the action of ϕ and γn is given by ϕ(e) = e + m1 and γn (e) = e + m2 . Then the long exact cohomology sequence associated to the short exact sequence (1)
0 −→ ᏹ −→ ᏺm1 ,m2 −→ ᏻKn −→ 0
1 : H 0 (C . (ᏻ )) → H 1 (C . (ᏹ)). An easy gives the connecting homomorphism δ Kn 1 (1). Applying V diagram search shows that cl(m1 , m2 ) = δ ᏻKn to the sequence (1) one obtains an exact sequence α
0 −→ T −→ Tm1 ,m2 −→ Zp −→ 0 1 : Z → H 1 (G , T ). Let (1 − ϕ)u = m . and the connecting homomorphism δGal p Kn 1 nr and (1 − ϕ)(u + e) = 0; that is, u + e ∈ T Then u + e ∈ ᏺm1 ,m2 ⊗ᏻKn Oˆ K m ,m 1 2. n 1 (1) can be represented by the cocycle g −→ (u + e)g − (u + e) = ug − u + Thus δGal k(g)−1 (1 + γn + · · · + γn )m2 . The proposition follows now from commutativity of the diagram
H 0 (C . (ᏹ))
1 δ
h1
h0
Zp
/ H 1 (C . (ᏹ))
1 δGal
/ H 1 GK , T . n
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
223
The following proposition, which can be easily proved by the same method, plays an important role in this paper. Proposition 1.3.3. Let T and U be Zp -representations of GKn . Put ᏹ = DᏻKn (T ) and ᏺ = DᏻKn (U ). Then the cup products ∪ H i GKn , T × H j (GKn , U ) −→ H i+j (GKn , T ⊗ U ) have the following explicit description in terms of complexes C . (ᏹ) and C . (ᏺ): (i) h0 (cl(m)) ∪ h0 (cl(n)) = h0 (cl(m ⊗ n)), (ii) h0 (cl(m)) ∪ h1 (cl(n1 , n2 )) = h1 (cl(m ⊗ n1 , m ⊗ n2 )), (iii) h1 (cl(m1 , m2 )) ∪ h1 (cl(n1 , n2 )) = h2 (cl(m2 ⊗ γn (n1 ) − m1 ⊗ ϕ(n2 ))). Proof. See [H2, Prop. 4.3]. Proposition 1.3.4. (i) DᏻKn (Zp (1)) = ᏻKn (1). (ii) The map TRn : H 2 C . ᏻKn (1) −→ Zp given by the formula TRn (α ⊗ ε) = −
pn αdπn Tr K/Qp ◦ resπn log χ(γn ) 1 + πn
is an isomorphism. Proof. See [H2, Th. 4.4]. Remark. In §2 we show that TRn coincides with the canonical isomorphism H 2 (GKn , Zp (1)) Zp . §2. The explicit reciprocity law of Coleman 2.1. The Kummer map. In this section, K is a finite unramified extension of Qp (p ! = 2) with the ring of integers W = W (k), σ the absolute Frobenius of K, = W [[X]] the ring of formal power series equipped with Frobenius endomorphisms σ and ϕ, and D = (1 + X)d/dX. There is a natural isomorphism between and Sn = W [[πn ]] which sends X to πn . Lemma 2.1.1. (i) For any f (X) ∈ and γ ∈ Γn , one has γf (πn ) ≡ f (πn ) +
χ(γ ) − 1 Df (πn )π pn
In particular, γf (πn ) ≡ f (πn ) (mod π). (ii) We have ϕ(π ) ≡ 0 (mod (pπ, π p )). (iii) Dϕ = pϕD.
mod π 2 .
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DENIS BENOIS
Proof. To simplify notation, put κn (γ ) =
(χ(γ ) − 1) . pn
Taking into account that χ (γ ) ≡ 1 (mod p n ), one can write γ (πn ) = (1 + πn )(1 + π )κn (γ ) − 1 ≡ πn +
χ(γ ) − 1 (1 + πn )π pn
mod π 2 .
Then, expanding γf (πn ) = f (γ (πn )) in powers of π, we obtain γf (πn ) ≡ f (πn ) +
χ(γ ) − 1 Df (πn )π pn
mod π 2 .
The proofs of (ii) and (iii) are straightforward, and they are omitted here. 2.1.2. The function D. Let E0 : → W/(σ − 1)W be the map given by mod (σ − 1)W . E0 (f (X)) = Df (0) Denote by m = (p, X) the maximal ideal of , and put Ꮽ = 1+ m. In [Cn1], Coleman constructed an exact sequence D
0 −→ (1 + X)Zp −→ Ꮽ −→ E0 =0 −→ 0, where the homomorphism D is given by the formula ϕ log F (X). D(F (X)) = 1 − p Put τ = 1/π + 1/2. Lemma 2.1.3. Let F (X) ∈ Ꮽ and f (X) = D(F (X)). Then for any γ ∈ Γn there exists a unique aF,γ (πn ) ∈ Sn such that (i) (ϕ − 1)(aF,γ (πn ) ⊗ ε) = (γ − 1)(f (πn )τ ⊗ ε), (ii) aF,γ (πn ) ≡ p −n (1 − χ(γ ))D log F (πn ) (mod π). Proof. By Lemma 2.1.1 we can write γf (πn ) = f (πn ) +
χ(γ ) − 1 Df (πn )π pn
mod π 2 .
Similarly one has γ (π ) ≡ χ (γ )π +
χ(γ )(χ(γ ) − 1) 2 π 2
These congruences imply that (γ − 1) f (πn )τ ⊗ ε ≡ −a˜ F,γ (πn ) ⊗ ε
mod π 3 .
(mod π),
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
225
where a˜ F,γ (πn ) = p −n (1−χ (γ ))D log F (πn ). Using the identity D ◦ϕ = pϕ ◦D we can write ϕ −n (ϕ − 1)a˜ F,γ (πn ) = p (χ (g) − 1)D 1 − log F (πn ) = p −n (χ(γ ) − 1)Df (πn ). p Hence (ϕ − 1)a˜ F,γ (πn ) ⊗ ε ≡ (γ − 1)(f (πn )τ ⊗ ε) (mod π). Since ϕ − 1 is invertible on πSn , it implies that there exists a unique aF,γ (πn ) ≡ a˜ F,γ (πn ) (mod π ) such that (ϕ − 1)(aF,γ (πn ) ⊗ ε) = (γ − 1)(f (πn )τ ⊗ ε), and the lemma is proved. 2.1.4.
Fix a generator γn of Γn = Gal(K∞ /Kn ), and consider the complex Cn. =
C . (ᏻKn (1)):
f
g
0 −→ ᏻKn (1) −→ ᏻKn (1) ⊕ ᏻKn (1) −→ ᏻKn (1) −→ 0, where f (α) = ((ϕ − 1)α, (γn − 1)α) and g(α1 , α2 ) = (γn − 1)α1 + (1 − ϕ)α2 . The short exact sequence m
p 0 −→ µpm −→ K¯ ∗ −−→ K¯ ∗ −→ 0
gives rise to the connecting homomorphism δn,m : Kn∗ → H 1 (GKn , µpm ). Passing to the inverse limit over m, we obtain the Kummer map δn : Kn∗ −→ H 1 GKn , Zp (1) . The main result of this section is an explicit description of this homomorphism in terms of C . (ᏻKn (1)). Proposition 2.1.5. Let
ιn : Ꮽ −→ H 1 Cn.
be the homomorphism F (X) → cl(f (πn )τ ⊗ε, aF,γn (πn )⊗ε) with f (X) = D(F (X)), and let ρn (F ) = F (ζpn − 1). Then the diagram Ꮽ
−ιn
ρn
Kn∗
δn
/ H 1 C. n
h1
/ H 1 GK , Zp (1) n
is commutative. 2.1.6. Proof of Proposition 2.1.5. In fact, this proposition is the main lemma of [A] reformulated in cohomological terms. In his paper, V. A. Abrashkin uses another choice of Frobenius, given by ϕ(X) = X p , but all of his arguments work in our case, and we repeat them below with some modifications.
226
DENIS BENOIS
2.1.6.1. It follows from Proposition 1.3.2 that h1 (ιn (F (πn ))) coincides with the class of the cocycle ψF (g) ⊗ ε given by k(g)−1 aF,γn (πn ) ⊗ ε , ψF (g) ⊗ ε = g(u ⊗ ε) − (u ⊗ ε) + 1 + γn + · · · + γn k(g)
where (1 − ϕ)u = f (πn )τ and γn = g|K∞ . Since γn (πn ) ≡ πn (mod π), one has k(g)−1 aF,γn (πn ) ⊗ ε ≡ p −n (1 − χ(g))D log F (πn ) ⊗ ε (mod π). 1 + γn + · · · + γn This congruence implies that ψF (g) ≡ χ (g)ug − u + p −n (1 − χ(g))D log F (πn ) (mod π). We now interpret ψF (g) in terms of Acris . Denote by I the ideal of Acris generated by π 2 and π p−1 /p. Lemma 2.1.6.2. There exists a unique x ∈ Fil1 Acris such that x ≡ u(π − π 2 /2) (mod I ) and ϕ x = f (πn ). 1− p p−1 Proof. Let v = i=0 [ε]i/p . Then u1 = uπ1 is a solution of the equation π (v − ϕ)u1 = f (πn ) 1 + . 2 The reduction of (v − ϕ)X modulo p is vX ¯ − Xp . Since R is integrally closed in Fr(R), it follows, by successive approximation modulo p m , that u1 ∈ W (R). Put x˜ = u(π − π 2 /2) = u1 v(1 − π/2). Then x˜ ∈ W 1 (R) and one has ϕ(x) ˜ π2 (ϕ(π) − π) x˜ − = f (πn ) 1 − + ϕ(u1 ) . ϕ(v) 4 2 Consider this equation in Acris . An easy computation shows that ϕ(x) ˜ ϕ(v) ϕ(π) ϕ(x) ˜ − = ϕ(u1 ) 1 − 1− . ϕ(v) p 2 p Summing these two equations we obtain that ϕ x˜ = f (πn ) + α, 1− p where α = −f (πn )π 2 /4 + ϕ(u1 )β and β = (ϕ(π) − π)/2 + (1 − ϕ(π)/2)(1 − ϕ(v)/p). Using the identity = π −1 ((1 + π)p − 1) it is easy to check that β ∈ I . ϕ(v) ∞ Then α ∈ I and the series m=1 (ϕ/p)m (α) converges to some element of I because ϕ/p is topologically nilpotent on I . Thus, there exists a unique x ∈ Fil1 Acris such that x ≡ x˜ (mod I ) and ϕ x = f (πn ). 1− p The lemma is proved.
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
227
2.1.6.3. From the definition of the ring Acris it is easy to see that the maps log : 1 + Fil1 Acris → Fil1 Acris and exp : Fil1 Acris → 1 + Fil1 Acris , given by usual power series log(1 + X) = X − X 2 /2 + · · · and exp(1 + X) = 1 + X + X 2 /2! + · · · , respectively, are isomorphisms inverse to each other. Since θ(F (πn )g /F (πn )) = 1, one has F (πn )g /F (πn ) ∈ 1 + Fil1 Acris . Hence the element F (πn )g g µF (g) = x − x − log F (πn ) belongs to Fil1 Acris for any g ∈ GKn . Since g2 g F (πn )g1 g2 g2 F (πn )g2 1 + x − x − log µF (g1 g2 ) = x − x − log F (πn ) F (πn ) = µF (g1 )g2 + µF (g2 ), the map µF : GKn → Fil1 Acris is a cocycle. We show that µF (g) = ψF (g)t. Indeed (1 − ϕ/p)µF (g) = 0, and hence µF (g) has a form µF (g) = c(g)t with c(g) ∈ Qp . On the other hand, from the congruences F (πn )g ≡ F (πn )+p −n (χ(g)− 1)D log F (πn )π (mod π 2 ) and x g − x ≡ (χ(g)ug − u)π (mod I ), it follows that µF (g) ≡ χ (g)ug − u π + p −n (1 − χ(g))D log F (πn )π ≡ ψF (g)t (mod I ). Hence µF (g) = ψF (g)t. In particular, one has [ε]ψF (g) = exp(µF (g)) =
exp(x)g F (πn ) . exp(x) F (πn )g
2.1.6.4. Let y = exp(x). Then the equation (1 − ϕ/p)x = f (πn ) can be written in the form yp = exp(pf (πn )). yϕ Consider the short exact sequence ν
1 −→ [ε]Zp −→ 1 + W 1 (R) −→ 1 + pW (R) −→ 1, where ν(a) = a p /a ϕ (see [dSh]). It shows that the inclusion W (R) ⊂ Acris gives a one-to-one correspondence between solutions Y of Y p /Y ϕ = exp(pf (πn )) and solutions X = log Y of (1 − ϕ/p)X = f (πn ). Hence, in fact, y ∈ 1 + W 1 (R). Furthermore, by induction on m, it is easy to see that yp ϕ m−1 2 ϕ m−2 m = exp pf (π ) + p f (π ) + · · · + p f (π ) n n n . m yϕ m
On the other hand, from the definition of D it follows that m m m−1 m−2 F (πn )p = F (πn )ϕ exp pf ϕ (πn ) + p 2 f ϕ (πn ) + · · · + pm f (πn ) .
228
DENIS BENOIS m
m
m
m
Hence y p /y ϕ = F (πn )p /F (πn )ϕ . Let z = ϕ −m (yF (πn )−1 ). Applying the map θ : W (R) → OC to both sides of this equation, we obtain that −1 m θ(z)p = F ζpn − 1 . Hence the connecting map δn,m sends F (ζpn − 1) to the class of the cocycle g → θ (z/zg ). On the other hand, one has yF (πn )g z −ψ (g) = θ ◦ ϕ −m [ε]−ψF (g) = ζpm F , θ g = θ ◦ ϕ −m g z y F (πn ) and the proposition is proved. Remark. The function D and its inverse E were first defined for the Frobenius ϕ(X) = X p in connection with explicit reciprocity laws (see [AH], [Sh], [Br], [V], and [Hen]). 2.2. The isomorphism h2 : H 2 (C . (ᏻKn (1))) H 2 (GKn , Zp (1)). In this section we describe the canonical isomorphism H 2 (GKn , Zp (1)) Zp using -modules and the map TRn , defined in Proposition 1.3.4. We start with a purely formal proposition that allows us to rewrite the integral of Coleman [Cn2] in terms of residues. Proposition 2.2.1. For any f (X) ∈ one has dπn = p −n f (ζ − 1). res π −1 f (πn ) 1 + πn ζ ∈µpn
2.2.2. Proof of Proposition 2.2.1
Lemma 2.2.2.1. Let ψ = {f (X) ∈ | ζ ∈µp f (ζ (1 + X) − 1) = 0}. Let m 1 and 0 i < n. Then for any f (X) ∈ ψ one has dπn −m = 0. res πi f (πn ) 1 + πn Proof. Recall that ψ is topologically generated by (1 + X)a , (a, p) = 1 and that D is invertible on ψ . From the identity −(m+1) D πi−m D −1 f (πn ) = πi−m f (πn ) − mp n−i πi (1 + πi )D −1 f (πn ), it follows that res
πi−m f (πn )
dπn 1 + πn
=p
n−i
res
−(m+1) πi f1 (πn )
dπn , 1 + πn
where f1 (πn ) = m(1 + πi )D −1 f (πn ). If i < n, then f1 (X) ∈ ψ . Hence we can continue this process and obtain the congruence dπn dπn −(m+k) −m k(n−i) res πi fk (πn ) =p ≡ 0 mod p k(n−i) res πi f (πn ) 1 + πn 1 + πn for any k ∈ N. The residue is therefore equal to zero and the lemma is proved.
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
229
2.2.2.2. Now we can prove the proposition. Since the ring W [[X]] is topologically i generated by the series (1 + X)m , m ∈ Z, we may assume that f (X) = (1 + X)p a with (a, p) = 1. First suppose i < n. Taking into account that ϕ ◦ res = res ◦ϕ we obtain ϕ i −1 −1 pi a dπn −i a dπn = p res πi (1 + πn ) =0 res π (1 + πn ) 1 + πn 1 + πn by the lemma just proved. On the other hand, in this case, i f (ζ − 1) = ζ p a = 0. ζ ∈µpn
ζ ∈µpn
Suppose now that i n. Then −1 pi a dπn −1 dπn = res π =1 res π (1 + πn ) 1 + πn 1 + πn and ζ ∈µpn f (ζ − 1) = p n . The proposition is proved. 2.2.3. Denote by invn the canonical isomorphism H 2 (GKn , Zp (1)) Zp , and consider the cup product H 1 GKn , Zp (1) × H 1 GKn , Zp −→ H 2 GKn , Zp (1) . Let κn : Γn → Zp be the additive character given by κn (γ ) = p−n log χ(γ ). Then for any α ∈ U1 (Kn ) one has 1 invn δn (α) ∪ κn = − n Tr Kn /Qp log α, p where Tr Kn /Qp denotes the trace map Kn → Qp (see, e.g., [Se, Chap. 14, Prop. 3]). Proposition 2.2.4. Let f (X) ∈ .Then the composite map invn ◦h2 : H 2 Cn. −→ Zp sends cl(π −1 f (πn ) ⊗ ε) to
− log−1 χ(γn ) Tr K/Qp
f (ζ − 1) .
ζ ∈µpn
2.2.5. Proof of Proposition 2.2.4 Lemma 2.2.5.1. Let F (X) ∈ Ꮽ. Put f (X) = D(F (X)) and α = F (ζpn − 1). Then f (ζ − 1) . Tr Kn /Qp log α = Tr K/Qp ζ ∈µpn
230
DENIS BENOIS
Proof. Note that log α is well defined because α ∈ U1 (Kn ). By multiplicativity of both sides of the formula it is sufficient to consider the two following cases. (a) F (X) = a ∈ 1 + pW (k). Then f (ζ − 1) = (1 − ϕ/p) log a and one has 1 Tr K/Qp Tr K/Qp log α = Tr Kn /Qp log α. f (ζ − 1) = p n 1 − p ζ ∈µpn
(b) F (X) ≡ 1 (mod X). Denote by Pm the set of p m -primitive roots of unity. Taking m into account that f ϕ (X)|X=ζpn −1 = 0 for m > n, we obtain Tr Kn /Qp log α = Tr Kn /Qp = Tr Kn /Qp
m=0 n
= Tr K/Qp = Tr K/Qp = Tr K/Qp
∞ m ϕ
f (X)|X=ζpn −1
pm p
−m σ m
f
m=0 n
p −m
m=0
ϕm
ζ ∈Pn
n
ζ
pm
−1
f (ζ − 1)
m=0 ζ ∈Pn−m
f
pm ζpn − 1
f (ζ − 1) .
ζ ∈µpn
The lemma is proved. 2.2.5.2.
We pass to the proof of the proposition. At first, consider the case f (X) ∈
E0 =0 . Then there exists F (X) ∈ Ꮽ such that f (X) = D(F (X)). Put α = F (ζpn −1).
By Proposition 2.1.5, h1 sends the class of the pair −(f (πn )τ ⊗ ε, aF,γn (πn ) ⊗ ε) to δn (α). Similarly, by Proposition 1.3.2, κn corresponds to the class cl(0, κn (γn )). From Proposition 1.3.3 it follows that the cup product of these classes is equal to cl(π −1 f (πn )κn (γn ) ⊗ ε). On the other hand, by Lemma 2.2.5.1, one has 1 f (ζ − 1) . invn δn (α) ∪ κn = − n Tr K/Qp p ζ ∈µpn
Hence invn ◦h2 sends cl(π −1 f (πn )κn ⊗ ε) to −p−n Tr K/Qp ( f (X) ∈ E0 =0 .
ζ ∈µpn
f (ζ − 1)), and
The general case can be reduced to this the proposition is proved for case in the following way. For f (X) = am X m ∈ , put f1 (X) = f (X)−a1 (1+X).
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
231
Then f1 (X) ∈ E0 =0 and one has dπn dπn −1 −1 − resπn π f1 (πn ) = a1 resπn π −1 dπn = 0. resπn π f (πn ) 1 + πn 1 + πn It now follows from Proposition 1.3.4 that cl(π −1 f (πn ) ⊗ ε) = cl(π −1 f1 (πn ) ⊗ ε). On the other hand, f1 (ζ − 1) = f (ζ − 1) − a1 ζ= f (ζ − 1). ζ ∈µpn
ζ ∈µpn
ζ ∈µpn
ζ ∈µpn
This completes the proof of the proposition. Now we can state the main result of this section. Theorem 2.2.6. The isomorphism invn ◦h2 : H 2 Cn. −→ Zp coincides with the map TRn ; that is, invn ◦h2 cl(α ⊗ ε) = −
αdπn pn Tr K/Qp res . log χ(γn ) 1 + πn
Proof. Since K/Qp is unramified, there exists a ∈ W such that Tr K/Qp a = 1. Put z = a/π ⊗ ε. From Propositions 2.2.1 and 2.2.4 it follows that invn ◦h2 (cl(z)) = TRn (cl(z)) = −
pn . log χ(γn )
Since pn / log χ (γn ) is a p-adic unit, the class cl(z) generates H 2 (Cn. ). We have shown that invn ◦h2 coincides with TRn on cl(z). Hence, they coincide everywhere. 2.3. The explicit reciprocity law for Qp (1). In [Ka1], Kato showed that explicit reciprocity laws can be viewed as an explicit computation of cup products in syntomic cohomology theory (see also [Ku]). In this section we use the complex Cn. to give a new proof of the explicit reciprocity law of Coleman [Cn2]. Suppose that n 1, and consider the cup product inv × id ∪ 2 n H 1 GKn , µpn × H 1 GKn , µpn −→ H 2 GKn , µ⊗ pn −−−−−→ µpn , where, to simplify notation, we write invn for invn (mod p n ). Let δn,n : Kn∗ → H 1 (GKn , µpn ) be the connecting map associated to the Kummer exact sequence. The Hilbert symbol ( , )n : Kn∗ × Kn∗ −→ µpn is a bilinear pairing given by (α, β)n = (invn × id)(δn,n (α), δn,n (β)).
232
DENIS BENOIS
Proposition 2.3.1. Let α, β ∈ U (Kn ), and let F (X), G(X) ∈ Ꮽ be such that F (ζpn − 1) = α, G(ζpn − 1) = β. Then n (α, β)n = ζp[F,G] , n
where [F, G]n = Tr K/Qp res
1 1 D(G(πn )) d log F (πn ) − D(F (πn )) d log G(πn )ϕ . π p
Proof. Let f (X) = D(F (X)) and g(X) = D(G(X)). Consider the commutative diagram H 1 Cn. /p n Cn. × H 1 Cn. /p n Cn.
∪
h1 ×h1
H GKn , µpn × H 1 GKn , µpn 1
∪
/ H 2 C . /p n C . n n h2
/ H GK , µ p n ⊗ µ p n n
2
invn × id
µp n . By Proposition 2.1.5 we have δn,n (α) = −h1 cl f (πn )τ ⊗ ε, aF,γn (πn ) ⊗ ε and
δn,n (β) = −h1 cl g(πn )τ ⊗ ε, aG,γn (πn ) ⊗ ε .
Then from Proposition 1.3.3 it follows that δn,n (α) ∪ δn,n (β) = h2 (cl(Hα,β ⊗ ε2 )), where Hα,β = aF,γn (πn )χ(γn )γn (g(πn )τ ) − f (πn )ϕ aG,γn (πn ) τ. Using the congruence 2.1.3(ii) and the congruence γn (g(πn )τ ) ≡ χ −1 (γn )g(πn )τ (mod Sn ), we can write Hα,β ≡
ϕ 1 − χ (γn ) D log F (πn )D(G(πn )) − D log G(πn ) D(F (πn )) n p π
(mod Sn ).
Note that (D log G(πn ))ϕ = (1/p)D(log G(πn )ϕ ). Applying Theorem 2.2.6 and taking into account that p−n log χ(γn ) ≡ p −n (χ(γn ) − 1) (mod p n ), we obtain that invn ◦h2 (cl(Hα,β ⊗ ε)) is congruent modulo pn to 1 1 ϕ D(G(πn )) d log F (πn ) − D(F (πn ))d log G(πn ) . Tr K/Qp res π p The proposition is proved.
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
233
Remark. In this proposition we considered only the case α, β ∈ U (Kn ). The general case is not much more difficult but demands additional computations and is omitted here. Corollary 2.3.2. Using Proposition 2.2.1 we can write the explicit reciprocity law in the form of Coleman: [F, G]n =
1 Tr D log F D(G) − (D log G)ϕ D(F ) X=ζ −1 . K/Q p n p ζ ∈µpn
Corollary 2.3.3. Suppose that α and β are universal norms of K∞ /Kn . Then one can take F and G such that f = D(F ), g = D(G) ∈ ψ , and 1 [F, G]n = n Tr K/Qp Df (ζ − 1)g(ζ − 1) . p ζ ∈µpn
Proof. It follows from the interpolation theorem of Coleman [Cn1] that one can choose F such that F ζ (1 + X) − 1 = F ϕ (X). ζ ∈µp
An easy computation shows that D(F ) ∈ ψ . Analogously we can choose G such that D(G) ∈ ψ . Furthermore, for any u(X) ∈ ψ and v(X) ∈ one has ζ ∈µpn v ϕ (ζ − 1)u(ζ − 1) = 0. Using this formula we obtain that
(D log F )ϕ (ζ − 1)g(ζ − 1) = 0
ζ ∈µpn
and
(D log G)ϕ (ζ − 1)f (ζ − 1) = 0.
ζ ∈µpn
Hence ζ ∈µpn
D log F D(G) − (D log G)ϕ D(F ) X=ζ −1 = Df (ζ − 1)g(ζ − 1), ζ ∈µpn
and the corollary is proved. §3. The functions Ek,n 3.1. Construction of Ek,n . In this section we define a family of functions Ek,n , k ∈ Zp , which play an important role in our construction of cohomology classes.
234
DENIS BENOIS
Lemma 3.1.1. Let m 1, and let t −m = (i) for all i −m one has
∞
i=−m am,i π
i.
Then
i +m vp (am,i ) − ; p−1
+ (ii) pm−1 t −m ∈ π −m S + π Bcris .
Proof. (i) Since t = log(1 + π) = t
−1
=π
−1
∞
j =1 (−1)
j −1 π j /j ,
one has
2 π π2 π π2 1+ − +··· + − +··· +··· . 2 3 2 3
The series t −m consists, therefore, of terms of the form ±
π j1 +···+js −m , (j1 + 1) × · · · × (js + 1)
jk , s 1.
Put i = j1 + · · · + js − m. If we write jk in the form jk = p lk uk − 1,
lk 0, (uk , p) = 1,
then, taking into account the estimate x , p−1
logp (1 + x)
x p − 1,
we obtain that lk [jk /(p − 1)]. Hence
i +m vp (j1 + 1) × · · · × (js + 1) = l1 + · · · + ls , p−1
and the first assertion is proved. (ii) From (i) it immediately follows that if i 0, then m vp p m−1 am,i m − 1 − 0. p−1 Hence to prove (ii) it is sufficient to show that the sequence am,i+1 π i converges to + . We check that zero in Bcris am,i+1 π i ∈ p l W (R)PD , where (2)
l
i − Am , p2
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
and Am does not depend on i. One has π = vπ1 , where v = W 1 (R) and π1 = [ε]1/p − 1. Since p−1 p−1 vp θ π1 = v p ζp − 1 = 1, p−1
one can write π1
p−1
k=0 [ε]
k/p
235 generates
in the form p−1
π1
= pα + vβ,
α, β ∈ W (R).
Let i = (p − 1)k + r, where 0 r p − 1. Then πi = πr
k
p s cs v kp−s ,
cs ∈ W (R).
s=0
Since v kp−s /(kp − s)! ∈ W (R)PD , the inequality (2) follows from an easy estimate kp − s kp − s r +m+1 + + vp am,i+1 p s (kp − s)! s − k − p−1 p p2 k(p − 1) m+1 −1+ − p−1 p2 i − Am , p2 where Am = [(m + 1)/(p − 1)] + 2. The lemma is proved.
i 3.1.2. The ring An . Let K[[πn−1 , πn ]] the set of all power series ∞ i=−∞ ai πn , ai ∈ −1 K. The rings ᏻKn = W {{πn }} and K[[πn ]] can be viewed as subsets of K[[πn , πn ]]. For k, l 0 define p k t l−k . fk,l (π) = l! (i) The series fk,l (π ) has a form i l−k ak,l π i with ai ∈ Qp . If k > l, then Lemma 3.1.1 implies p−2 i (i) k− . vp ak,l p−1 p−1 From this estimate it follows that for any set of gk,l (πn ) ∈ Sn the series gk,l (πn )fk,l (π)
k,l 0
converges to some g(πn ) = bm πnm ∈ K[[πn−1 , πn ]]. In addition, vp (bm ) 1 for m 0 and limm→−∞ bm = 0. Denote by An the subset of K[[πn−1 , πn ]] consisting of all such series. Then An ⊂ p ᏻKn + πn K[[πn ]]. Since l 1 + l2 fk1 +k2 ,l1 +l2 (π), fk1 ,l1 (π )fk2 ,l2 (π) = l1
236
DENIS BENOIS
An has a natural ring structure. Sn and Sˆ PD can be viewed as subrings of An . The action of Γ on these rings extends to An . Let k ∈ Zp and n ∈ N. For f (X) ∈ ψ define Ek,n : ψ −→
1 An p
as follows: Ek,n (f ) =
∞ (1 − k)(2 − k) × · · · × (i − k − 1)
ti
i=1
p n(i−1) D −i (f (πn )).
Since D −i (f (πn )γ ) = χ (γ )−i (D −i f (πn ))γ , the map Ek,n is a W [[Γ ]]-homomorphism. The main properties of Ek,n are collected in the following proposition. Proposition 3.1.3. (i) For any k, l ∈ Zp one has Ek,n (f ) ≡ El,n (f ) mod πn , p n−1 . (ii) Let r ∈ Zp and j 2. Then j k=0
j (−1) Er+k,n (f ) ≡ 0 k k
mod πn , p j (n−1)+1 .
(iii) For any k ∈ Zp one has tEk,n (Df ) = f (πn ) + p n (1 − k)Ek−1,n (f ). (iv) Let γ ∈ Γn and k ! = 0. Then γ Ek,n (f ) ≡ χ (γ )−k Ek,n (f ) +
1 − χ(γ )−k f (πn ) (mod πn ). kp n
For k = 0 this congruence takes the form γ E0,n (f ) ≡ E0,n (f ) +
log χ(γ ) f (πn ) (mod πn ). pn
(v) Let n 1, and let γn be a generator of Γn . Then for any f ∈ ψ one has p−1 j ϕ ◦ γn Ek,n+1 (f ) ⊗ ε k ≡ Ek,n (f σ ) + A(γn )f σ (πn ) ⊗ ε k (mod πn ), j =0
where
1 χ(γn )pk − 1 −1 . A(γn ) = n kp p χ(γn )k − 1
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
237
For k = 0 this congruence takes the form p−1 j ϕ ◦ γn E0,n+1 (f ) ⊗ ε k ≡ E0,n (f σ ) ⊗ ε k j =0
+
p − 1 log χ(γn ) σ f (πn ) ⊗ ε k 2 pn
(mod πn ).
3.1.4. Proof of Proposition 3.1.3 3.1.4.1.
Let i 2. By Lemma 3.1.1 one has
pn(i−1) t −i = p (n−1)(i−1) p i−1 t −i ≡ 0
mod p (n−1)(i−1) , π ,
and (i) immediately follows from this congruence. 3.1.4.2. We repeat the arguments of Perrin-Riou [PR2, Prop. 2.4.4]. Suppose at first that r ∈ Z. Differentiating i times the identity X r−1 (1 − X)j =
j j (−1)k X r+k−1 k k=0
and putting X = 1, one obtains j i j (−1)k (r + k − m) = 0 k k=0
for
0 i j − 1.
m=1
By continuity, this formula holds for all r ∈ Zp . One has j ∞ i in D −i−1 (f ) j j p Er+k,n (f ) = (−1)k (−1)k (m − r − k) k k t i+1 k=0 k=0 i=0 m=1 j −1 −i−1 j i D (f ) in j ·p (−1)k (m − r − k) = k t i+1
j
i=0
k=0
m=1
j ∞ i j D −i−1 (f ) in + ·p (−1)k (m − r − k) . k t i+1 i=j
k=0
m=1
The first term in this sum is equal to zero. From §3.1.4.1 it follows that p i t −(i+1) ∈ p ᏻKn + K[[πn ]] if i 2. Hence pin t −(i+1) ≡ 0 (mod (p j (n−1)+1 , πn )) for i j , and (ii) is proved.
238 3.1.4.3.
DENIS BENOIS
The third property follows from an easy computation
t · Ek,n (Df ) = t ·
∞ (1 − k)(2 − k) × · · · × (i − k − 1)
ti
i=1
= f (πn ) +
p (i−1)n D 1−i f (πn )
∞ (1 − k)(2 − k) × · · · × (i − k − 1)
t i−1
i=2 n
p (i−1)n D 1−i f (πn )
= f (πn ) + p (1 − k)Ek−1,n (f ). 3.1.4.4. Proof of (iv). To simplify notation put κn (γ ) = (χ(γ )−1)/p n . Let f (x) ∈ ψ . At first we show that ∞
γ Ek,n (f ) =
(3)
ck,i D i f (πn )t i ,
i=−∞
where
ck,i =
p n(m−1)
j −m=i j 0, m1
(1 − k)(2 − k) × · · · × (m − k − 1)κn (γ )j . j !χ(γ )m
Since both sides of (3) are W -linear and the series (1 + X)a , (a, p) = 1 generate ψ topologically, it is sufficient to check this formula for f (X) = (1 + X)a . Taking into j account that 1 + π = exp(t) = ∞ j =0 t /j !, one has, in this case, ∞ (1 − k) × · · · × (m − k − 1) n(m−1) γ Ek,n (f ) = (1 + πn ) p (1 + π)aκn (γ ) a m χ(γ )m t m a
= (1 + πn )a = (1 + πn )a
m=1 ∞
∞
(1 − k) × · · · × (m − k − 1) n(m−1) a j κn (γ )j j t p · a m χ(γ )m t m j!
i=1 ∞
j =0
ck,i a i t i =
i=−∞
∞
ck,i D i f (πn )t i ,
i=−∞
and (3) is checked. For i 1 one has ck,−i =
∞
p
n(i+j −1) κn (γ )
j =0
=p
n(i−1)
χ (γ )
−i
i−1 m=1
j χ(γ )−i−j i+j −1
j!
(m − k) ·
(m − k)
m=1
j 0
j k −i χ(γ )−1 − 1 j
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
= pn(i−1) χ (γ )−i
i−1
239
k−i (m − k) · 1 + (χ(γ )−1 − 1)
m=1
= p n(i−1) χ (γ )−k
i−1
(m − k).
m=1
Hence γ Ek,n (f ) ≡ χ (γ )−k Ek,n (f ) + ck,0 (mod πn ). Similarly it is not difficult to compute ck,0 . For k ! = 0 we have 1 1−k n χ (γ )−1 κn (γ ) + p χ(γ )−2 κn (γ )2 + · · · 1! 2! −k i 1 1 − χ(γ )−k = n 1− . 1 − χ(γ )−1 = kp kp n i
ck,0 =
i 0
Now let k = 0. Then c0,0 = p
−n
1 − χ(γ )−1 i−1 = p −n log χ(γ ). i i 1
3.1.4.5. We now prove the congruence (v). Let Dπi = (1+πi )d/dπi . Recall that κn (γ ) = (χ (γ ) − 1)/p n . Put Fa (π1 ) =
p−1
j
(1 + π1 )aκn (γn ) .
j =0
We need the following auxiliary result. Lemma 3.1.4.6. Let fa = (1 + X)a , where (a, p) = 1. Then p−1 j =0
j γn Ek,n+1 (fa ) ⊗ ε k m ∞ (−1)m p nm m Fa (π1 ) −m−1 k = D (i − k)Dπ1 fa (πn+1 ) ⊗ ε . m! t m=0
i=1
Proof. Since p−1 j =0
j
κn (γn )j s (1 + π1 )aκn (γn ) = a −s Dπs 1 Fa (π1 ),
240
DENIS BENOIS
one has p−1
j χ (γn )j k γn
j =0
D −i fa (πn+1 ) ti
= fa (πn+1 )
p−1 j =0
= fa (πn−1 )
s
j k−i j 1 + p n κn γ n (1 + π1 )aκn (γn ) i i at
p ns
k − i Dπs 1 Fa (π1 ) . a s+i t i s
Let m = s + i − 1. Since
(1 − k)(2 − k) × · · · × (i − k − 1) k − i (i − 1)! s m! (1 − k)(2 − k) × · · · × (m − k) = (−1)s , m! s!(i − 1)!
the summing over i 1 gives p−1
j χ (γn )j k γn Ek,n+1 (fa )
j =0
= fa
∞ (−1)m p nm (1 − k) × · · · × (m − k) a m+1 m! m=0 m (−1)i−1 (i − 1)!p i−1 × Dπs 1 Fa (π1 ) ti s i+s=m+1
=
∞ m=0
(−1)m p nm (1 − k) × · · · × (m − k) m Fa (π1 ) Dπ1 D −m−1 fa . m! t
The lemma is proved. 3.1.4.7. We pass to the proof of (v). From the congruence 1/t ≡ 1/π(mod K[[π]]) it follows that (m − 1)! m−1 1 Dπ ≡ (−1)m−1 mod K[[π]] . π tm It is easy to see that Fa (π1 ) ≡
p−1 j =0
(1 + π1 )aj =
(1 + π)a − 1 (1 + π1 )a − 1
(mod π).
It follows that Fa (π1 )/t ≡ 1/π1 (mod K[[π1 ]]), and one has Fa (π1 ) (m − 1)! ϕ Dπm−1 ≡ (−1)m−1 mod K[[π]] . 1 m t t
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
241
Hence (4)
ϕ ◦
p−1 j =0
j γn Ek,n+1 (fa ) ⊗ ε k ≡ Ek,n faσ ⊗ ε k
mod K[[π]] .
Since fa generates ψ , it holds for all f ∈ ψ . Let k ! = 0. Applying the operator p−1 j ϕ◦ j =0 γn to both sides of tEk+1,n+1 (Df ) ⊗ ε k = f (πn+1 ) ⊗ ε k − kp n+1 Ek,n+1 (f ) ⊗ ε k and using (4) for Ek+1,n+1 , we can write that p−1
j k ptEk+1,n Df σ ⊗ ε k ≡ χ γn f σ (πn ) ⊗ ε k j =0
− kp n+1 ϕ ◦
p−1
j γn Ek,n+1 (f ) ⊗ ε k
j =0
modulo πn . Taking into account that tEk+1,n (Df σ ) = f σ (πn ) − kp n Ek,n (f σ ), we obtain (v) for k ! = 0. But by continuity it holds also for k = 0 since p(p − 1) 1 χ (γn )pk − 1 − p = log χ(γn ). k→0 k χ (γn )k − 1 2 lim
The proposition is proved. Remarks 3.1.5. (i) From the proof of property (iv) it follows that ∞ ck,i D i f (πn )t i χ (γ )k γ − 1 En,k (f ) = i=0
with ck,i = p
−(i+1)n
i+1 ∞ (1 − k) × · · · × (j − k) j χ (γ ) − 1 1 − χ(γ )−1 . 1−k (i + j + 1)! χ (γ ) j =0
This formula is used in §3.2.3 below. (ii) Estimating the coefficients of Ek,n more accurately, it is not difficult to prove the following refined version of congruence (i): mod p n , πn if p 5. Ek,n (f ) ≡ El,n (f )
242
DENIS BENOIS
3.2. The residue formula for Ek,n . The main result of this section is the following formula, which plays an important role in our proof of the explicit reciprocity law (see §5). Proposition 3.2.1. Let γ ∈ Γn and f, g ∈ ψ . Then for any k ∈ Z one has dπn −k res Ek+1,n (f ) · χ (γ ) γ − 1 E−k,n (g) 1 + πn −k−1 D f (πn )D k g(πn ) dπn k log χ(γ ) res = (−1) . pn π 1 + πn Proof. The rest of the section is devoted to the proof of this proposition. Lemma 3.2.2. Let f ∈ ψ , g ∈ , and k 0. If 0 m k, then dπn m res t Ek+1,n (f )g(πn ) 1 + πn −k−1 nm D f (πn )D k−m g(πn ) dπn k p k! res . = (−1) (k − m)! π 1 + πn If m > k, the residue is equal to zero. Proof. By definition one has 1 kp n k!p kn Ek+1,n (f ) = D −1 f (πn ) − 2 D −2 f (πn ) + · · · + (−1)k k+1 D −k−1 f (πn ). t t t Suppose that m > k. Then t m Ek+1,n (f ) ∈ K[[πn ]] and hence dπn = 0. res t m Ek+1,n (f )g(πn ) 1 + πn On the other hand, t k Ek+1,n (f ) ≡ (−1)k k!p kn D −k−1 f (πn )/π (mod K[[πn ]]). Hence dπn k res t Ek+1,n (f )g(πn ) 1 + πn −k−1 D f (πn )D k−m g(πn ) dπn k kn . = (−1) p k! res π 1 + πn Thus, the lemma is proved for all m k. The case m < k can be reduced to m = k in the following way. It is easy to see that D(t k+1 Ek+1,n (f )) = t k f (πn ). Using this formula we obtain D t m Ek+1,n (f )g(πn ) = D t m−k−1 t k+1 Ek+1,n (f ) g(πn ) = p n (m − k − 1)t m−1 Ek+1,n (f )g(πn ) + t m−1 f (πn )g(πn ) + t m Ek+1,n (f )Dg(πn ),
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and hence
n
p (k + 1 − m) res t
m−1
dπn Ek+1,n (f ) · g(πn ) 1 + πn dπn m = res t Ek+1,n (f ) · Dg(πn ) . 1 + πn
Successively applying this formula to m = k, k − 1, . . . , 1 we obtain the lemma. 3.2.3. In this subsection we prove Proposition 3.2.1 for k 0. By Remarks 3.1.5 we can write ∞ χ (γ )−k γ − 1 En,−k (g) = c−k,m D m (g)t m , m=0
where
m+1 ∞ (k + 1)(k + 2) × · · · × (k + i) i χ (γ ) − 1 1 − χ(γ )−1 . c−k,m = (m+1)n k+1 (m + i + 1)! p χ (γ ) i=0
Then by Lemma 3.2.2 one has
res Ek+1,n (f ) · χ (γ )
−k
dπn γ − 1 E−k,n (g) 1 + πn −k−1 D f (πn )D k g(πn ) dπn k = (−1) Ak res π 1 + πn
with Ak =
k m=0
k! p mn c−k,m . (k − m)!
p −n log χ(γ ).
It remains to show that Ak = We prove it by induction on k. If k = 0, then A0 = c0,0 = p −n log χ (γ ) by §3.1.4.4. Suppose now that Ak = p −n log χ(γ ) for some k 0. Taking into account the identity k! p mn c−k,m (k − m)!
m+1 (k + 1)! (m+1)n −n (k − m + 1) × · · · × k χ(γ ) − 1 p = c−k−1,m+1 + p , (k − m)! (m + 1)! χ(γ )k+1
we can write Ak+1 = Ak + c−k−1,0 − p −n χ(γ )−k−1
k m+1 (k − m + 1) × · · · × k . χ(γ ) − 1 (m + 1)!
m=0
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DENIS BENOIS
In §3.1.4.4 it was shown that c−k−1,0 =
1 − χ(γ )−k−1 . p n (k + 1)
On the other hand, k m+1 (k − m + 1) × · · · × k χ(γ ) − 1 (m + 1)!
m=0
=
k m+1 1 (k + 1) × · · · × (k + 1 − m) χ(γ ) − 1 k +1 (m + 1)! m=0
k+1 1 1 + χ(γ ) − 1 = −1 k +1 =
χ (γ )k+1 − 1 . k +1
Hence Ak+1 = Ak = p −n log χ(γ ), and the proposition is proved for k 0. Lemma 3.2.4. For any k ∈ Z one has
res E−k,n (g) · χ (γ )
k+1
dπn γ − 1 Ek+1,n (f ) 1 + πn dπn −k = res Ek+1,n (f ) · χ(γ ) γ − 1 E−k,n (g) . 1 + πn
Proof. The proof is straightforward. Using the formulas γ ((1 + πn )−1 dπn ) = χ (γ )(1 + πn )−1 dπn and χ (γ )k+1 γ Ek+1,n (f ) ≡ Ek+1,n (f ) (mod K[[πn ]]), we can write dπn res E−k,n (g) · χ (γ ) γ − 1 Ek+1,n (f ) 1 + πn dπn + res Ek+1,n (f ) · χ(γ )−k γ − 1 E−k,n (g) 1 + πn dπn −k k+1 = res χ (γ ) γ − 1 E−k,n (g) · χ(γ ) γ Ek+1,n (f ) 1 + πn dπn + res χ (γ )k+1 γ Ek+1,n (f ) · E−k,n (g) 1 + πn dπn − res E−k,n (g)Ek+1,n (f ) 1 + πn
k+1
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
245
dπn = res χ (γ )γ E−k,n (g)Ek+1,n (f ) 1 + πn dπn − res E−k,n (g)Ek+1,n (f ) 1 + πn dπn = 0, = res(γ − 1) E−k,n (g)Ek+1,n (f ) 1 + πn and the lemma is proved. 3.2.5. Now we can complete the proof of Proposition 3.2.1. It remains to show that the formula is true for k < 0. In this case −k 1 and one has dπn res Ek+1,n (f ) · χ(γ ) γ − 1 E−k,n (g) 1 + πn dπn = − res E−k,n (g) · χ(γ )k+1 γ − 1 Ek+1,n (f ) 1 + πn k −k−1 D g(πn )D log χ(γ ) f (πn ) dπn = (−1)k res . pn π 1 + πn
−k
Thus, the proposition is proved. Remark 3.2.6. Using Proposition 2.2.1 one can write Proposition 3.2.1 in the form
res Ek+1,n (f ) · χ (γ )
−k
dπn γ − 1 E−k,n (g) 1 + πn log χ(γ ) −k−1 = (−1)k D f (ζ − 1)D k g(ζ − 1). p 2n
ζ ∈µpn
§4. Construction of families of points. For a crystalline representation V , PerrinRiou [PR2] constructed some families of integral elements of H 1 (GKn , V (k)) for k $ 0 using the exponential map of Bloch and Kato. For representations of finite height we extend this construction to all k using complexes C . (D(T (k))). 4.1. Preliminaries 4.1.1. The exponential map of Bloch and Kato [BK]. Let L be a finite extension of Qp , and let L0 denote the maximal unramified subfield of L. For a de Rham 0 (V ). Note (see §1.1.2) that one representation V of GL , put tV (L) = DdR (V )/DdR has the fundamental exact sequence + 0 −→ Qp −→ Bcris −→ Bcris ⊕ BdR /BdR −→ 0.
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DENIS BENOIS
Tensoring this sequence with V , one obtains the long exact cohomology sequence 0 −→ H 0 GL , V −→ Dcris (V ) −→ Dcris (V ) ⊕ tV (L) −→ H 1 GL , V −→ · · · . The last map of this sequence gives rise to the exponential map expV ,L : tV (L) −→ H 1 GL , V with kernel (Dcris (V ))ϕ=1 /H 0 (GL , V ). In particular, let G be a formal group of finite height over the ring of integers OF , T the Tate module of G, and V = T ⊗ Qp . Then V is a de Rham representation, tV (L) is identified with the tangent space of G over L, and expV ,L coincides with the classical exponential map arising from the Kummer exact sequence (see [BK, Example 3.10.1]). 4.1.2. The modules Ᏸ(V ) and Ᏼ(V ) (see [PR1], [PR2]). Let s ∈ N. In the rest of this paper, T is a Zp -representation of finite height of GKs such that V = T ⊗Zp Qp is crystalline. By the theorem of Wach (see §1.2.3), there exists an S-lattice NS ⊂ DS (T ) such that Γs acts trivially on (NS /πNS )(−h) for some h ∈ Z. In all of this section we suppose that h = 0. This assumption is not restrictive because we can replace V by V (−h). In particular, it implies that Fil0 Dcris (V ) = Dcris (V ). The natural embedding ˆ S Sˆ PD )Γs with a W -lattice of Dcris (V ) stable under the Sˆ PD ∈ Bcris identifies (DS (T )⊗ Frobenius ϕ (see §1.2.3). We denote this lattice by M. Note that the ring = W [[X]] is equipped with (i) a W [[Γ ]]-module structure given by γ (X) = (1 + X)χ(γ ) − 1, (ii) a σ -semilinear Frobenius map ϕ such that ϕ(X) = (1 + X)p − 1, (iii) a differential operator D = (1 + X)d/dX. One has D ◦ ϕ = pϕ ◦ D. Following Perrin-Riou, define ψ = f (X) ∈ f ζ (1 + X) − 1 = 0 , ζ ∈µp
Ᏸ(V (k)) = ψ ⊗W Dcris (V (k)),
k ∈ Z.
It may be shown that ψ is a free W [[Γ ]]-submodule of generated by (1 + X) and that D is invertible on ψ . Then Ᏸ(V (k)) can be viewed as a W [[Γ ]] -module equipped with the differential operator D = D ⊗ id. Let ᏴK be the subset of K[[X]] consisting of all power series that converge on the maximal ideal mC of C. As above one can define the action of ϕ on ᏴK . Set
Ᏼ(V (k)) = α ∈ ᏴK ⊗K Dcris (V (k)) | (1 − ϕ)α ∈ Ᏸ(V (k)) . For k = 0 we also introduce the following integral versions of these modules: Ᏸ(T ) = ψ ⊗W M,
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
247
α ζ (1 + X) − 1 = pα ϕ (X) .
Ᏼ(T ) = α ∈ W [[X]] ⊗W M
ζ ∈µp
Let Ej : Ᏸ(V (k)) → Dcris (V (k))/(1 − p j ϕ)Dcris (V (k)) be the map defined by mod 1 − p j ϕ Dcris (V (k)) , Ej (f ) = D j (f )(0) and let E[0,k] = ⊕kj =0 Ej . Then Ᏸ(V (k)) and Ᏼ(V (k)) are related to each other via the following exact sequence (see [PR2, §2.2.7]): −j 0 −→ log(1 + X)j ⊗ Dcris (V (k))ϕ=p −→ Ᏼ(V (k)) j 0 1−ϕ
−−−→ Ᏸ(V (k))E[0,k] =0 −→ 0. Because Fil0 Dcris (V ) = Dcris (V ), one has Dcris (V )ϕ=p for k = 0 this sequence takes the form
−j
= 0 for j > 0. In particular,
1−ϕ
0 −→ Dcris (V )ϕ=1 −→ Ᏼ(V ) −−−→ Ᏸ(V )E0 =0 −→ 0. Lemma 4.1.3. (i) One has Ᏸ(V ) = Ᏸ(T ) ⊗ Qp and Ᏼ(V ) = Ᏼ(T ) ⊗ Qp . (ii) For any k 0 the diagram D k ⊗ek
Ᏼ(V (k))
1−ϕ
1−ϕ
Ᏸ(V (k))E[0,k] =0
/ Ᏼ(V )
D k ⊗ek
/ Ᏸ(V )E0 =0
is commutative. Proof. (i) It is clear that Ᏸ(V ) = Ᏸ(T ) ⊗ Qp . Let α ∈ Ᏼ(T ) and f = (1 − ϕ)α. Then f ζ (1 + X) − 1 = α ζ (1 + X) − 1 − pϕ(α) = 0, ζ ∈µp
ζ ∈µp
and we obtain that f ∈ Ᏸ(T )E0 =0 . Hence Ᏼ(T ) ⊂ Ᏼ(V ). Suppose now that f ∈ Ᏸ(T )E0 =0 . Then we can write f = (1 − ϕ)f0 + f1 , where f0 ∈ M and f1 (0) = 0. ∞ m Since ϕ(M) ⊂ M, the series f + 0 m=0 ϕ (f1 ) converges to some α ∈ ⊗ M and (1−ϕ)α = f . From ζ ∈µp f (ζ (1+X)−1) = 0 it follows that ζ ∈µp α(ζ (1+X)− 1) = pϕ(α). Hence α ∈ Ᏼ(T ), and one has an exact sequence 1−ϕ
0 −→ M ϕ=1 −→ Ᏼ(T ) −−−→ Ᏸ(T )E0 =0 −→ 0. Tensoring this sequence with Qp and taking into account that Ᏸ(T ) ⊗ Qp = Ᏸ(V ), we obtain that Ᏼ(V ) = Ᏼ(T ) ⊗ Qp .
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DENIS BENOIS
(ii) Let α ∈ Ᏼ(V (k)). As ϕ(ek ) = pk ek and ϕ ◦ D k = p−k D k ◦ ϕ, one has the identity ϕ(D k ⊗ ek )α = (D k ⊗ ek )ϕ(α). Hence (1 − ϕ) D k ⊗ ek (α) = D k ⊗ ek (1 − ϕ)α, and the lemma is proved. 4.1.4. In this subsection we review the construction of elements of H 1 (GKn , V ) given by Perrin-Riou. Fix k ∈ N such that Fil−k Dcris (V ) = Dcris (V ). Denote by ᏴK the set of all power series f (X) ∈ K[[X]] that converge on mC . For f ∈ ᏴK put Rk,n (f ) =
k−1 i=0
Then one has
(−1)i
D i f (πn ) i t ∈ Fil0 Bcris . p in i!
f ζpn − 1 ≡ Rk,n (f ) mod Filk BdR .
Tensoring with Dcris (V ) one obtains a map Rk,n : ᏴK ⊗ Dcris (V ) −→ Dcris (V ) ⊗ Fil0 Bcris . Lemma 4.1.5 [PR2, Prop. 2.3.6]. Let α ∈ Ᏼ(V ). (i) There exists A ∈ Fil0 (Bcris ⊗ Dcris (V )) such that (1 − ϕ)A = Rk,n ((1 − ϕ)α). (ii) The element expV ,Kn (α(ζpn − 1)) coincides with the class of the cocycle g g −→ − Ag − A + Rk,n (α) − Rk,n (α). Sketch of proof. The first statement follows from the fundamental exact sequence. To prove (ii) note that mod Dcris (V ) ⊗ Filk BdR . Rk,n (α(πn )) ≡ α ζpn − 1 Since Fil−k Dcris (V ) = Dcris (V ), it implies Rk,n (α(πn )) ≡ α ζpn − 1
+ mod V ⊗ BdR .
Hence
+ mod V ⊗ BdR .
Rk,n (α) − A ≡ α ζpn − 1
On the other hand, one has (1 − ϕ) Rk,n (α) − A = (1 − ϕ)Rk,n (α) − Rk,n ((1 − ϕ)α) = 0. Consider the fundamental exact sequence tensoring with V : f + −→ 0. 0 −→ V −→ V ⊗ Bcris −→ V ⊗ Bcris ⊕ V ⊗ BdR /BdR The last two formulas imply that f (Rk,n (α) − A) = (0, α(ζpn − 1)), and (ii) follows from the definition of the exponential map.
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
249
4.2. The homomorphisms T ,k,n . Using the functions Ek,n studied in the previous section, we construct here a system of maps T ,k,n : Ᏼ(T ) −→ H 1 GKn , T (k) ,
k ∈ Zp .
Let n s. Fix a generator γn of Γn . Recall that for any Γ -module N we denote by T wkε the map T wkε (n) = n ⊗ ε k . Recall that ˆ S Sˆ PD Γs . M = DS (T )⊗ Since Sˆ PD ⊂ An , then for any f ∈ ψ and g ∈ Sˆ PD the product gEk,n (f ) is contained in p −1 An . By linearity we obtain a homomorphism Ᏸ(T ) → p−1 DS (T ) ⊗S An . For k ∈ Zp denote by ET ,k,n the composite map T wkε
ET ,k,n : Ᏸ(T ) −→ p−1 DS (T ) ⊗S An −−−→ p −1 DS (T (k)) ⊗S An . By Proposition 3.1.3(iv) one has (γn − 1)ET ,k,n (f ) ≡
χ(γn )k − 1 f (πn ) ⊗ ε k kpn
mod πn DS (T (k)) .
Let α ∈ Ᏼ(T ). Then there exists (not unique) ᏱT ,k,n (α) ∈ DᏻKn (T (k)) such that ᏱT ,k,n (α) ≡ ET ,k,n ((1 − ϕ)α)
mod πn DSn (T (k)) .
(Since p−1 An ⊂ ᏻKn + πn K[[πn ]], it is sufficient, e.g., to take ET ,k,n ((1 − ϕ)α) truncated modulo πn .) The last congruence implies (γn − 1)ᏱT ,k,n (α) ≡
1 − χ(γn )k (ϕ − 1)α(πn ) ⊗ ε k kp n
mod πn DS (T (k)) .
Since operator 1 − ϕ is invertible on πn DS (T (k)), there exists a unique ᏲT ,k,n (α) ∈ DS (T (k)) ⊗S Sn such that (i) ᏲT ,k,n (α) ≡
1 − χ(γn )k α(πn ) ⊗ ε k kp n
mod πn DSn (T (k)) ,
(ii) (ϕ − 1)ᏲT ,k,n (α) = (γn − 1)ᏱT ,k,n (α). Let ᏹn = DᏻKn (T ). Consider the complex C . (ᏹn (k)): 0 −→ ᏹn (k) −→ ᏹn (k) ⊕ ᏹn (k) −→ ᏹn (k) −→ 0. Then the pair (ᏱT ,k,n (α), ᏲT ,k,n (α)) defines an element of H 1 (C . (ᏹn (k))) which
250
DENIS BENOIS
does not depend on the choice of ᏱT ,k,n (α). Thus we constructed a map Ᏼ(T ) → H 1 (C . (ᏹn (k))). Composing this map with h1 : H 1 (C . (ᏹn (k))) H 1 (GKn , T (k)), one obtains a homomorphism T ,k,n : Ᏼ(T ) −→ H 1 GKn , T (k) , which sends α to h1 ◦ cl(ᏱT ,k,n (α), ᏲT ,k,n (α)). We also consider the homomorphism V ,k,n : Ᏼ(V ) −→ H 1 GKn , V (k) obtained from T ,k,n by ⊗Qp . Theorem 4.3. The homomorphisms T ,k,n satisfy the following properties. (m) (i) For any m ∈ N denote by T ,k,n : Ᏼ(T ) → H 1 (GKn , (T /p m T )(k)) the homom morphism T ,k,n modulo p . Then for any k, l ∈ Zp one has (n−1)
(n−1)
ε T ,k,n = T wk−l ◦ T ,l,n .
(ii) Let j 2 and m = j (n − 1) + 1. Then for any r ∈ Zp one has j (m) k j ε (−1) ◦ resKm /Kn T ,r+k,n = 0. T w−k k k=0
(iii) Let α ∈ Ᏼ(T ). Then
T (−1),k+1,n (D ⊗ e1 )α = −kp n T ,k,n (α).
(iv) For any α ∈ Ᏼ(T ) and n 1 one has
cor Kn+1 /Kn T ,k,n+1 (α) = T ,k,n (σ ⊗ ϕ)α .
(v) Let k 1 and α(X) ∈ Ᏼ(V (k)). Then V ,k,n D k ⊗ ek α = (−1)k (k − 1)!p (k−1)n expV (k),Kn α ζpn − 1 for any n s. 4.4. Proof of Theorem 4.3 4.4.1. Proof of (i). At first we show that, for any f ∈ ψ and g ∈ Sˆ PD , mod πn , p n−1 . gEk,n (f ) ≡ gEl,n Since Sˆ PD is topologically generated by the elements t m /m!, we may assume that g = t m /m!. If m = 0, this congruence follows from Proposition 3.1.3. Let m 1. By the definition of Ek,n it is sufficient to check that, for i 2, gpn(i−1) t −i D −i (f )
i−1 j =1
(j − k) ≡ 0
mod πn , p n−1 ,
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
251
but this is obvious since gp n(i−1) t −i = p (n−1)(i−2)+n
p m−2 p i−m . m! t i−m
Now by definition of ET ,k,n we obtain that ε ◦ ET ,l,n ((1 − ϕ)α) ET ,k,n ((1 − ϕ)α) ≡ T wk−l
mod p n−1 , πn .
Hence we can choose ᏱT ,k,n (α) and ᏱT ,l,n (α) such that ᏱT ,k,n (α) ≡ ᏱT ,l,n (α) (mod p n−1 ) and ᏲT ,k,n (α) ≡ ᏲT ,l,n (α) (mod p n−1 ). Thus (i) is proved. 4.4.2. Proof of (ii). We first prove that (5)
j k=0
j ε (−1) ◦ ᏱT ,r+k,n (α) = 0 T w−k k k
mod p m ,
m = j (n − 1) + 1,
for a suitable choice of ᏱT ,r+k,n (α). For this it is sufficient to show that g
j j (−1)k ET ,r+k,n (f ) = 0 k
mod p m , πn
k=0
for any f ∈ ψ and g ∈ Sˆ PD . As before we may assume that g = t i /i!. If i = 0, this congruence follows from Proposition 3.1.3(ii). Let i 1. In §3.1.4.2 it was shown that j j ∞ l −l−1 (f ) D j j (−1)k · p ln (−1)k (a − r − k) . Er+k,n (f ) = k k t l+1 l=j
k=0
k=0
a=1
On the other hand, from the identity g
i−1 p l−i t i pln p ln l(n−1)+1 p = = p i! t l+1 i! t l−i+1 t l+1
it follows that gp ln /t l+1 ≡ 0 (mod (p j (n−1)+1 , π)) for l j . Thus j ∞ l j D −l−1 (f ) ln ·p (−1)k (a − r − k) ≡ 0 mod p j (n−1)+1 , π , l+1 k t l=j
k=0
a=1
and (5) is proved. The element resKm /Kn T ,r+k,n (α) can be represented by the pair
ᏱT ,r+k,n (α), resKm /Kn ᏲT ,r+k,n (α) ,
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DENIS BENOIS
where resKm /Kn ᏲT ,r+k,n (α) satisfies the following conditions: (i) resKm /Kn ᏲT ,r+k,n (α) ≡
χ r+k (γm ) − 1 α ⊗ ε r+k p n (r + k)
mod DS (T (k)) ⊗ πn K[[πn ]] ,
(ii) (ϕ − 1) resKm /Kn ᏲT ,r+k,n (α) = (γm − 1)ᏱT ,r+k,n (α). From the congruence r+k j j χ (γm ) − 1 ≡0 (−1)k k p n (r + k)
mod p m ,
k=0
we immediately obtain that j k=0
j ε T w−k (−1) ◦ resKm /Kn ᏲT ,r+k,n (α) = 0 k k
mod p m ,
and (ii) is proved. 4.4.3. Proof of (iii). One has M ⊗ e1 = tM. Using the formula tEk+1,n (Df ) = f (πn ) − kp n Ek,n (f ) (see Proposition 3.1.3(iii)), we can write ET (−1),k+1,n (D ⊗ e1 )f = −kp n ET ,k,n (f ) + f (πn ) for any f ∈ Ᏸ(T ). Hence for α ∈ Ᏼ(T ) we can choose ᏱT (−1),k+1,n such that ᏱT (−1),k+1,n (α) ≡ −kp n ᏱT ,k,n (α) + (1 − ϕ)α(πn ) ⊗ ε k
mod DS (T (k)) ⊗ πn K[[πn ]] .
On the other hand, ᏲT (−1),k+1,n (α) ≡ 0 (mod πn DS (T (k))). Take A ∈ πn DS (T (k)) such that A ≡ α(πn ) ⊗ ε k (mod πn ), and put Ᏹ%T (−1),k,n (α) = ᏱT (−1),k,n (α) + (ϕ − 1)A, ᏲT% (−1),k,n (α) = ᏲT (−1),k,n (α) + (γn − 1)A.
Then the pairs (ᏱT (−1),k,n (α), ᏲT (−1),k,n (α)) and (Ᏹ%T (−1),k,n (α), ᏲT% (−1),k,n (α)) are homologous. Moreover, Ᏹ%T (−1),k,n (α) ≡ −kp n ET ,k,n (α) (mod πn ) and (γn − 1)ᏲT% (−1),k,n (α) ≡ χ(γn )k − 1 α ≡ −kp n ᏲT ,k,n (α)
(mod πn ).
Hence cl(Ᏹ%T (−1),k,n (α), ᏲT% (−1),k,n (α)) = −kp n cl(ᏱT ,k,n (α), ᏲT ,k,n (α)), and (iii) is proved.
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ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
4.4.4. Proof of (iv). From an explicit description of the corestriction map it follows that cor Kn+1 /Kn T ,k,n+1 (α) coincides with the class of the cocycle p−1 j γn ᏱT ,k,n+1 (α), ᏲT ,k,n+1 (α) j =0
and, hence, with the class of p−1 j ϕ ◦ γn ᏱT ,k,n+1 (α), ϕ ◦ ᏲT ,k,n+1 (α) j =0
which we denote by (Ᏹ%T ,k,n (α), ᏲT% ,k,n (α)). Take A ∈ DSn (T (k)) such that 1 χ(γn )k − 1 α σ (πn ) ⊗ ε k (mod πn ). A ≡ n 1− kp p χ(γn )k − 1 Then by the congruence (v) of Proposition 3.1.3 we have Ᏹ%T ,k,n (α) ≡ ET ,k,n (σ ⊗ ϕ)α + (ϕ − 1)A (mod πn ). Moreover, from the congruence
ᏲT ,k,n (σ ⊗ ϕ)α ≡
it follows that
1 − χ(γn )k σ α (πn ) ⊗ ε k kp n
mod DS (T (k)) ⊗ p −1 A1n
ᏲT% ,k,n (α) ≡ ᏲT ,k,n (σ ⊗ ϕ)α + (γn − 1)A
mod DS (T (k)) ⊗ πn .
As in the proof of (iii) above, these congruences show that the classes of (Ᏹ%T ,k,n (α), ᏲT% ,k,n (α)) and (ᏱT ,k,n ((σ ⊗ ϕ)α), ᏲT ,k,n ((σ ⊗ ϕ)α)) coincide. 4.4.5. Proof of (v). To prove this property we need some auxiliary results. + GK∞ Lemma 4.4.5.1. (i) Let α ∈ (V ⊗ Bcris )GK∞ and β ∈ (V ⊗ Bcris ) . Assume that 0 (γn − 1)α = (ϕ − 1)β. Then there exists u ∈ Fil (V ⊗ Bcris ) such that (1 − ϕ)u = α, and the map
µα,β : GKn −→ V ,
k(g)−1
µα,β (g) = ug − u + 1 + γn + · · · + γn
β,
k(g)
γn
= g|K∞ ,
is a continuous 1-cocycle. (ii) Let (α1 , β1 ) be an another pair satisfying the same condition and such that + + mod V ⊗ πn Bcris mod V ⊗ πn Bcris , β1 ≡ β . α1 ≡ α Then µα1 ,β1 and µα,β are homologous.
254
DENIS BENOIS
Proof. From the fundamental exact sequence it follows that for any a ∈ Bcris there exists b ∈ Fil0 Bcris such that (1 − ϕ)b = a. Hence there exists u ∈ Fil0 (V ⊗ Bcris ) such that (1 − ϕ)u = α. One has k(g)−1 (1 − ϕ)µα,β (g) = (g − 1)(1 − ϕ)u + 1 + γn + · · · + γn (1 − ϕ)β = (g − 1)α + (1 − g)α = 0. Since µα,β (g) ∈ Fil0 (V ⊗ Bcris ), it implies that µα,β (g) ∈ V . An easy computation shows that µα,β is a cocycle. If u% ∈ Fil0 (V ⊗ Bcris ) is another element such that (1−ϕ)u% = α, then u% = u+v for some v ∈ V , and the cocycle µ% α,β (g) = u% g −u% + k(g)−1
(1 + γn + · · · + γn )β is homologous to µα,β (g). + + since for any a ∈ πn Bcris the We note further that 1 − ϕ is invertible on πn Bcris + i % series i 0 ϕ (a) converges. From the congruence α ≡ α (mod (V ⊗ πn Bcris )) it + )) such that (1 − ϕ)u1 = α1 follows now that there exists u1 ≡ u (mod (V ⊗ πn Bcris + and hence µα1 ,β1 (g) ≡ µα,β (g) (mod (V ⊗ πn Bcris )). Since µα,β (g), µα1 ,β1 (g) ∈ V , it implies that µα,β (g) = µα1 ,β1 (g), and the lemma is proved. Lemma 4.4.5.2. Let α ∈ ᏴK ⊗ Dcris (V (k)). Then for any γ ∈ Γn one has (γ − 1)Rk,n (α) ≡ (−1)k−1
χ(γ )k − 1 k+1 α(πn ) mod Dcris (V (k)) ⊗ Bcris . kn k!p
Proof. To simplify notation put κn (γ ) = (χ(γ ) − 1)/p n . By continuity we may assume that α = g(X) ⊗ d with g(X) = K[X] and d ∈ Dcris (V (k)). Since g(X) can be written in the form a ba (1 + X)a , we may suppose in advance that α(X) = (1 + X)a ⊗ d. Then Rk,n (α) = α(πn )
k−1
(−1)i
i=0
ai t i . i!p ni
Hence Rk,n (α)γ = α(πn )(1 + π)aκn (γ )
k−1
(−1)i
i=0
Writing (1 + π )aκn (γ ) in the form i α(πn ) ∞ i=0 cki t with cki = p
−in i
a
∞
j =0 a
j +m=i 0mk−1
jκ
n (γ )
(−1)
j t j /j !,
m χ(γ )
m
χ(γ )i a i t i . i!pni
one obtains that Rk,n (α)γ =
j χ(γ ) − 1 . m!j !
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
The last formula implies that i i p −in a (−1) i! cki = a k χ(γ )k − 1 k−1 −kn (−1) p k!
255
if 0 i k − 1, if i = k.
Hence (γ − 1)Rk,n (α) ≡ (−1)k−1
χ(γ )k − 1 k+1 α(π ) mod D (V (k)) ⊗ B , n cris cris k!p kn
and the lemma is proved. nr ∩ W (R). Then the following sequence is exact: Lemma 4.4.5.3. Let S˜n = Oˆ K n 1−ϕ
0 −→ Zp −→ π1−k S˜n −−−→ π −k S˜n −→ 0. Proof. By §1.1.1 one has ker(1 − ϕ) = Zp . To prove the surjectivity of 1 − ϕ we use the following method of Fontaine (see [F3]). Let y ∈ π −k y % with y % ∈ S˜n , and p−1 let v = i=0 [ε]i/p . One has π = vπ1 . By successive approximation modulo p m we show that there exists x % ∈ S˜n such that (6)
v k x % − ϕ(x % ) = y % .
% ∈ S˜ such that Suppose that one has xm n % % v k xm − ϕ xm ≡ y% mod p m S˜n ,
m 0.
% % + p m z and a = p −m (y % − v k x % + ϕ(x % )). Then a ∈ S˜ , and the = xm Put xm+1 m m n m m congruence % % v k xm+1 − ϕ xm+1 ≡ y% mod p m+1 S˜n , m 0,
is equivalent to
v k z − ϕ(z) ≡ am
mod p S˜n .
The left side of this congruence is equal to v¯ k X −X p modulo p. Since R is integrally closed in Fr R, there exists z¯ ∈ R such that v¯ k z¯ − z¯ p = a¯ m . Since v¯ k X − X p = y¯ % is a ˆ nr ˆ nr ˆ nr ˜ ˜ separable polynomial, we find that z¯ ∈ R ∩ ᏻˆ nr Kn /p ᏻKn . But R ∩ ᏻKn /p ᏻKn = Sn /p Sn % % by [F2, Prop. 1.8.3], so taking a lifting z ∈ S˜n we find xm+1 . Then x = limm→∞ xm is a solution of (6). Put x = π1−k x % . Then (1 − ϕ)x = π −k (v k − ϕ(x % )) = y, and the lemma is proved.
256
DENIS BENOIS
4.4.5.4. We deduce (v) from the previous lemmas. By linearity we may assume that (D k ⊗ ek )α ∈ Ᏼ(T ). It follows from Lemma 4.1.4 that expV (k),Kn (α(ζpn − 1)) coincides with the class of the cocycle k(g)−1 g → − Ag − A − 1 + γn + · · · + γn B, where (1 − ϕ)A = Rk,n ((1 − ϕ)α), A ∈ Fil0 (V (k) ⊗ Bcris ), and B = (1 − γn )Rk,n (α). On the other hand, V ,k,n ((D k ⊗ ek )α) can be represented by the cocycle k(g)−1 ᏲT ,k,n D k ⊗ ek α g → ug − u + 1 + γn + · · · + γn with (1 − ϕ)u = ᏱT ,k,n ((D k ⊗ ek )α). Tensoring the exact sequence from Lemma 4.4.5.3 with T (k) one obtains that u ∈ π1−k S˜n ⊗T (k) and hence u ∈ Fil0 (V (k)⊗Bcris ). It follows immediately from the definition of ᏱT ,k,n and Rk,n that
ᏱT ,k,n D k ⊗ ek α ≡ (−1)k−1 (k − 1)!p n(k−1) Rk,n ((1 − ϕ)α)
+ ⊗ V (k) . mod πn Bcris
By Lemma 4.4.5.2 we also have the congruence ᏲT ,k,n (α) ≡ (−1)k−1 (k − 1)!p n(k−1) B
+ mod πn Bcris ⊗ V (k) .
Property (v) now follows from Lemma 4.4.5.1, and the theorem is proved. Remark 4.5. The maps T ,k,n depend on the choice of ε. To stress it we often write Tε ,k,n . Using the fact that Ek,n is a Γ -homomorphism, it is easy to verify that τ Tε ,k,n (α) = Tτ ε,k,n (α) = χ(τ )k Tε ,k,n (τ (α)) for any τ ∈ Γ . In particular, let c be the element of Γ defined by c(ε) = ε −1 . Then −1
Tε ,k,n (α) = (−1)k Tε ,k,n (c(α)). These formulas are used in Theorem 5.3.2 below. §5. Explicit reciprocity law 5.1. Cohomological pairing 5.1.1. In this section the main result of the paper is proved. We conserve the notation and conventions of §4. Let T ∗ = HomZp (T , Zp ), and let ᏹ∗ = HomᏻKn (ᏹ, ᏻKn ). Since the category of representations of finite height is stable under the duals, there exist an integer h 0 and an S-lattice NS∗ ∈ ᏹ∗ such that Γs acts trivially on (NS∗ /πNS∗ )(−h). For any
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
257
k ∈ Zp − [1 − h, 0] define V ∗ ,k,n : Ᏼ(V ∗ ) −→ H 1 GKn , V (k) , V ∗ ,k,n (β) =
(−1)h p −hn V ∗ (−h),h+k,n D h ⊗ eh β . k · (k + 1) × · · · × (k + h − 1)
Since H 2 (GKn , Qp (1)) Qp , the cup product gives a pairing ∪ ( , )k,n : H 1 GKn , V (k) × H 1 GKn , V ∗ (1 − k) −→ Qp . On the other hand, the natural duality Dcris (V ) × Dcris (V ∗ ) → Qp can be extended by linearity to the map [ , ] : Ᏸ(V ) × Ᏸ(V ∗ ) −→ Qp ⊗ . Theorem 5.1.2. Let α ∈ Ᏼ(V ), β ∈ Ᏼ(V ∗ ), and let f = (1 − ϕ)α, g = (1 − ϕ)β. Then for any integer k ∈ / [1, h] one has dπn 1 −k V ,k,n (α), V ∗ ,1−k,n (β) k,n = (−1)k Tr K/Qp res D f, D k−1 g . π 1 + πn 5.1.3. Proof of Theorem 5.1.2. It is more convenient to change some notation. Let α ∈ Ᏼ(T ), β ∈ Ᏼ(T (−h)), and let f = (1 − ϕ)α, g = (1 − ϕ)β. Put l = h − k + 1. Then the theorem is equivalent to the following formula T ,k,n (α), T ∗ (−h),l,n (β) k,n h dπn 1 −k −l k nh D f, D ⊗ e−h g = (−1) p (k − m) Tr K/Qp res . π 1 + πn m=1
⊗ε−1
The natural pairing DS (T (k)) × DS (T ∗ (1 − k)) −−−→ S induces a pairing [ , ]D : An ⊗S DS (T (k)) × An ⊗S DS T ∗ (1 − k) −→ An . If m ∈ M and m∗ ∈ M ∗ , then [m ⊗ ε k , m∗ ⊗ eh ⊗ ε l ]D = [m, m∗ ]. By Proposition 1.3.3 and Theorem 2.2.6 one has T ,k,n (α), T ∗ (−h),l,n (β) V ,k,n = − where
pn Tr K/Qp R(α, β), log χ(γn )
R(α, β) = res ᏲT ,k,n (α), γn ᏱT ∗ (−h),l,n (β) D
dπn − ᏱT ,k,n (α), ϕ ᏲT ∗ (−h),l,n (β) D . 1 + πn
258
DENIS BENOIS
Since ᏱT ,k,n (α) ≡ ET ,k,n (f ) (mod A1n ⊗ DS (T (k))), there exists x such that ᏱT ,k,n (α) = ET ,k,n (f ) + (ϕ − 1)x.
Let FT ,k,n (α) = ᏲT ,k,n (α) + (γn − 1)x. Then (1 − ϕ)FT ,k,n (α) = (γn − 1)ET ,k,n (f ). Similarly we can find y and FT (−h),l,n (β) = ᏲT ,k,n (β) + (γn − 1)y such that ᏱT (−h),l,n (β) = ET (−h),l,n (g) + (ϕ − 1)y,
(1 − ϕ)FT (−h),l,n (β) = (γn − 1)ET (−h),l,n (g). Substituting these formulas we obtain R(α, β) = res FT ,k,n (α), γn ET ∗ (−h),l,n (g) D dπn − ET ,k,n (f ), ϕFT ∗ (−h),l,n (β) D 1 + πn dπn − res(γn − 1) A, ET ∗ (−h),l,n (g) D 1 + πn dπn − res(γn − 1) ᏱT ,k,n (α), ϕB D 1 + πn dπn − res(ϕ − 1) A, FT ∗ (−h),l,n (β) D 1 + πn dπn − res(ϕ − 1) ᏲT ,k,n (α), γn B D . 1 + πn One has γn ET ∗ (−h),l,n (g) ≡ ET ∗ (−h),l,n (g) (mod A0n ⊗ DS (T (1 − k))). Taking into account the identity dπn dπn Tr K/Qp res(γn − 1) g(πn ) = Tr K/Qp res(ϕ − 1) g(πn ) =0 1 + πn 1 + πn (see, e.g., [H2, Lemma 3.3]), we obtain Tr K/Qp R(α, β) = Tr K/Qp res FT ,k,n (α), ET ∗ (−h),l,n (g) D − ET ,k,n (f ), ϕFT ∗ (−h),l,n (β) D
dπn . 1 + πn
We further note that [ET ,k,n (f ), ϕFT ∗ (−h),l,n (β)]D has a form m u(πn )v(πn )ϕ /π m , where u(X), v(X) ∈ K ⊗ ψ . Since u(X)v(X)ϕ ∈ K ⊗ ψ by Lemma 2.2.5.1, one has dπn = 0. Tr K/Qp res ET ,k,n (f ), ϕFT ∗ (−h),l,n (β) D 1 + πn
259
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
Analogously we can replace [FT ,k,n (α), ET ∗ (−h),l,n (g)]D by the power series [(1 − ϕ)FT ,k,n (α), ET ∗ (−h),l,n (g)]D . Since (1 − ϕ)FT ,k,n (α) = (γn − 1)ET ,k,n (f ), we obtain dπn Tr K/Qp R(α, β) = Tr K/Qp res (γn − 1)ET ,k,n (f ), ET ∗ (−h),l,n (g) D . 1 + πn The theorem follows now from Proposition 3.2.1. Namely, let f = fi ⊗ mi and " g= gj ⊗ m∗j ⊗ eh , where fi , gi ∈ ψ . Since t h El,n (gj ) ≡ phn hm=1 (k − m) · E1−k,n (D −h gj ) (mod πn ), one has Tr K/Qp R(α, β) dπn = Tr K/Qp mi , m∗j res χ(γn )k γn − 1 Ek,n (fi ) · t h El,n (gj ) 1 + πn i,j
= pnh
h
(k − m) Tr K/Qp
mi , m∗j res χ(γn )k γn − 1 Ek,n (fi )
m=1
i,j
· E1−k,n D
−h
dπn gj 1 + πn
= (−1)k−1 p n(h−1) log χ(γn ) h dπn 1 −k −l × (k − m) Tr K/Qp res D f, D ⊗ e−h g . π 1 + πn m=1
Hence T ,k,n (α), T ∗ (−h),l,n (β) k,n h
= (−1)k p nh
m=1
dπn 1 −k −l D f, D ⊗ e−h g (k − m) Tr K/Qp res . π 1 + πn
The theorem is proved. Corollary 5.1.4. Our result can also be written in the form of Coleman:
V ,k,n (α), V ∗ ,1−k,n (β)
= k,n
(−1)k Tr D −k f, D k−1 g (ζ − 1). K/Q p n p ζ ∈µpn
Corollary 5.1.5. Let k 0. Let α ∈ Ᏼ(V ) and β ∈ Ᏼ(V ∗ (k + 1)). Then V ,−k,n (α), expV ∗ (k+1),Kn β ζpn − 1 −k,n 1 =− Tr Kn /Qp D k α, β ⊗ ek+1 ζpn − 1 . n(k+1) k!p
260
DENIS BENOIS
Proof. Note at first that from Theorem 4.3 it follows that (−1)k+1 V ∗ ,k+1,n D k+1 ⊗ ek+1 β . expV ∗ (k+1),Kn β ζpn − 1 = k!p kn Hence
V ,−k,n (α), expV ∗ (k+1),Kn β ζpn − 1 −k,n =−
D k f, g ⊗ ek+1 (ζ − 1),
1
k!p
Tr K/Qp n(k+1)
ζ ∈µpn
where f = (1 − ϕ)α, g = (1 − ϕ)β. Thus, we must check that D k f, g ⊗ ek+1 (ζ − 1) = Tr Kn /Qp D k α, β ⊗ ek+1 ζpn − 1 . Tr K/Qp ζ ∈µpn
Since ζ ∈µp α(ζ (1 + X) − 1) = pα ϕ (X) and ζ ∈µp β(ζ (1 + X) − 1) = pβ ϕ (X), one has D k α, ϕ(β) ⊗ ek+1 (ζ − 1) = D k ϕ(α), β ⊗ ek+1 (ζ − 1)
ζ ∈µpn
ζ ∈µpn
= p k+1
D k α σ , β σ ⊗ ek+1 (ζ − 1)
ζ ∈µpn−1
ϕ D k α, β ⊗ ek+1 (ζ − 1). = p ζ ∈µpn
Hence
k
D f, g ⊗ ek+1 (ζ − 1) =
ζ ∈µpn
ϕ h (ζ − 1), 1− p
ζ ∈µpn
[D k α, β
where h(X) = ⊗ ek+1 ]. Then the arguments of Lemma 2.2.5.1 show that ζ ∈µpn (1−ϕ/p)h(ζ −1) is equal to Tr Kn /Qp h(ζpn −1), and the corollary is proved. 5.1.6. In the rest of this paper we always take into account the dependence of T ,k,n on ε. Let α ∈ M ϕ=1 . Then ᏱT ,k,n (α) = 0 and hence (1 − ϕ)ᏲT ,k,n (α) = 0. Since DᏻKn (T (k))ϕ=1 = T (k)GK∞ , it implies that ᏲT ,k,n (α) ∈ T (k)GK∞ . Then Tε ,k,n belongs to the image of the inflation map H 1 (Γn , T (k)GK∞ ) → H 1 (GKn , T (k)). Let k∈ / [0, h]. Then zero is not a weight of Hodge of V (k) and therefore V (k)GKn = 0. Hence H 1 (Γn , V (k)GK∞ ) = 0. Then for k ∈ / [0, h] the map Tε ,k,n induces the homomorphism PεV ,k,n : Ᏸ(V )E=0 −→ H 1 GKn , V (k) ,
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
261
defined as follows: PεV ,k,n (f ) = Vε ,k,n (σ ⊗ ϕ)−n (α) ,
where (1 − ϕ)α = f.
Similarly, for k ∈ / [−h, 0] we can define PεV ∗ ,k,n : Ᏸ(V ∗ )E=0 −→ H 1 GKn , V ∗ (k) by PεV ∗ ,k,n (g) = Vε ∗ ,k,n ((σ ⊗ ϕ)−n (β)), where (1 − ϕ)β = g. Then for an integer k∈ / [0, h + 1], Theorem 5.1.2 can be written in the following form. Corollary 5.1.7. Let f ∈ Ᏸ(V )E=0 and g ∈ Ᏸ(V ∗ )E=0 . Then ε (−1)k D −k f, D k−1 g (ζ − 1). PV ,k,n (f ), PεV ∗ ,1−k,n (g) k,n = Tr K/Qp n p ζ ∈µpn
5.2. Interpolation of exponential maps 5.2.1. In this section we review the theory of Perrin-Riou [PR2]. Recall that Λ = Zp [[Γ ]]. Let ι be the Zp -linear map Λ → Λ defined by ι(τ ) = τ −1 . For any s 1 ∞ m denote by Ᏼs the Qp -subspace of Qp [[X]] consisting of all power series m=0 am X ∞ −s such that limm→∞ |am |m = 0. Put Ᏼ∞ = s=1 Ᏼs . Let Ᏼ∞ (Γ1 ) = {f (γ1 − 1) | f (X) ∈ Ᏼ∞ }, where γ1 is a generator of Γ1 , and let Ᏼ∞ (Γ ) = Ᏼ∞ (Γ1 ) ⊗Zp [[Γ1 ]] Λ. The field of fractions of Ᏼ∞ (Γ ) is denoted by ∞ (Γ ). For any Λ-module N the natural homomorphism N → NΓn can be extended to a map Ᏼ∞ (Γ ) ⊗Λ N → Qp ⊗Zp NΓn . We now recall some results on the Iwasawa module associated to T (see [PR1, §2] and [PR2, §3.2.1]). Let Z1∞ (T (k)) = limH 1 GKn , T (k) . ← − cor Then Z1∞ (T (k)) is a Λ-module of rank [K : Qp ] dimQp V with torsion submodule isomorphic to T (k)GK∞ . For any n the natural map Z1∞ (T (k))Γn → H 1 (GKn , T (k)) G is injective, and (T (k))ΓnK∞ is isomorphic to H 1 (Γn , T (k)GK∞ ). So one has a monomorphism 1 H 1 GKn , T (k) Z∞ (T (k)) . πT (k),n : −→ 1 T (k)GK∞ Γn H Γn , T (k)GK∞ Tensoring it with Qp one obtains a map πV (k),n :
Z1∞ (T (k)) T (k)GK∞
H 1 GKn , V (k) , ⊗ Qp −→ 1 H Γn , V (k)GK∞ Γn
262
DENIS BENOIS
which can be extended to a homomorphism
H 1 GKn , V (k) Z1∞ (T (k)) . −→ 1 pr T (k),n : Ᏼ∞ (Γ ) ⊗Λ T (k)GK∞ H Γn , V (k)GK∞
Note that if zero is not a Hodge weight of V (k), then V (k)GKn = 0 and one has H 1 (Γn , V (k)GK∞ ) = 0. The restriction-inflation sequence 0 −→ H 1 Γn , V (k)GK∞ −→ H 1 GKn , V (k) −→ H 1 GK∞ , V (k) gives rise to a monomorphism resK∞ /Kn
H 1 GKn , V (k) −→ H 1 GK∞ , V (k) . : 1 G K ∞ H Γn , V (k)
Proposition 5.2.2. Let s, r ∈ N. Let xn,r+k ∈ im πV (r+k),n (n ∈ N, 0 k s) be a family that satisfies the following properties. (i) One has cor Kn+1 /Kn (xn+1,i ) = xn,i . (ii) For any 0 j s the sequence j (s−j )n k j ε (−1) ◦ resK∞ /Kn (xn,r+k ) T w−k p k k=0
converges to zero in the group H 1 (GK∞ , V (r)). Then there exists a unique element x ∈ Ᏼs (Γ ) ⊗Λ Z1∞ (T (k))/T (k)GK∞ such that pr T (k),n (x) = xn,k for all n and k. Proof. This is [PR3, Prop. 1.8]; see also [PR2, §1.2]. 5.2.3.
Consider the system of maps
PεV ,k,n : Ᏸ(V )E=0 −→ H 1 GKn , V (k) ,
k∈ / [0, h],
defined in §5.1.6 by the formula
PεV ,k,n (f ) = Vε ,k,n (σ ⊗ ϕ)−n (α) ,
(1 − ϕ)α = f.
Theorem 4.3 implies that for k 1 these maps are related to exponential maps by the formula (7) PεV ,k,n (f ) = (−1)k (k − 1)! expV (k),Kn Ξk,n (f ) , where Ξk,n (f ) = p −n (σ ⊗ ϕ)−n Gk (ζpn − 1) and Gk is a solution of the equation (1 − ϕ)Gk = (D −k ⊗ e−k )(f ). In addition one has cor Kn+1 /Kn PεV ,k,n+1 (f ) = PεV ,k,n (f ). Hence we obtain a compatible system of homomorphisms 1 Z∞ (T (k)) E=0 Fn,k : Ᏸ(V )(k)Γn −→ ⊗ Qp , T (k)GK∞ Γn
263
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
ε Fn,k (f ) = PεV ,k,n T w−k (f ) . In [PR2], Perrin-Riou defined the maps PV ,k,n for k > h directly by (7) and showed that they satisfy the interpolation property in Theorem 4.3(ii). Applying to this system Proposition 5.2.2,1 she proved the following fundamental result. Theorem 5.2.4 (Perrin-Riou). There exists a unique Λ-homomorphism (0),ε
PV
: Ᏸ(V )E=0 −→ Ᏼ∞ (Γ ) ⊗ Z1∞ (T )/T GK∞
such that for any k > h the diagram (0),ε
T wkε ◦PV
Ᏸ(V )E=0 Ξk,n
/ Ᏼ∞ (Γ ) ⊗ Z1 (T (k))/T (k)GK∞ ∞
(−1)k (k−1)! expV (k),n
Kn ⊗K Dcris (V (k))
pr T (k),n
/ H GK , V (k) n 1
is commutative. (0),ε
Theorem 4.3 allows us to give an explicit description of pr V (k),n ◦T wkε ◦ PV (f ) for negative k too. Namely, from the uniqueness of p-adic interpolation we immediately obtain the following result. Corollary 5.2.5. For all k ∈ / [0, h] one has (0),ε
PεV ,k,n = pr V (k),n ◦T wkε ◦ PV
(f ).
5.2.6. For γ ∈ Γ put Dm = m − log γ / log χ(γ ). Note that Dm does not depend on the choice of γ , T wkε Dm = Dm−k , and Dm ≡ m (mod (γn − 1)) for any n. The map (0),ε PV gives rise to a system of map (k),ε
PV (k+1),ε
such that PV
: Ᏸ(V )E=0 −→ (Γ ) ⊗Λ Z1∞ (T )
(k),ε
= D k PV
. From Theorem 4.3(iii) it follows that
(1),ε
PV
(0),ε
= −T w1ε ◦ PV (−1) (D ⊗ e1 ).
Iterating this equality one obtains (k+1),ε
PV (1) 1 The
(k),ε
= −T w1ε ◦ PV
(D ⊗ e1 ).
referee pointed out an error in [PR2, Prop. 3.1.4] used in the original proof of this theorem (see [Cz1, remark after Prop. II.1.8] and [PR3]).
264
DENIS BENOIS
5.3. The explicit reciprocity law of Perrin-Riou 5.3.1.
Let V ∗ = HomQp (V , Qp ). The pairing ∪ ( , )T ,n : H 1 GKn , T × H 1 GKn , T ∗ (1) −→ Zp
induces a bilinear form , T : Z1∞ (T ) × Z1∞ T ∗ (1) −→ Λ, which can be described explicitly in the following way. If x = (xn ) ∈ Z1∞ (T ) and y = (yn ) ∈ Z1∞ (T ∗ (1)), then the elements τ −1 xn , yn T ,n τ ∈ Zp Γ /Γn τ ∈Γ /Γn
form a compatible system with respect to the maps Zp [Γ /Γn ] → Zp [Γ /Γn−1 ] and define an element x, yT ∈ Zp [[Γ ]]. In particular, x, yT ≡ τ −1 xn , yn T ,n τ mod (γn − 1) . τ ∈Γ /Γn
It is easy to see that for any λ ∈ Λ one has
# $ λx, yT = λx, yT = x, ι(λ)y T
and
#
$ ε T wkε (x), T w−k (y) T = T w−k x, yT ,
where T w−k : Λ → Λ is the Zp -linear map given by T w−k (τ ) = χ(τ )−k τ for τ ∈ Γ . Hence, , T is a Λ-bilinear form ι , T : Z1∞ (T ) × Z1∞ T ∗ (1) −→ Λ. By linearity this pairing can be extended to ι , V : (Γ ) ⊗Λ Z1∞ (T ) × (Γ ) ⊗Λ Z1∞ T ∗ (1) −→ (Γ ). The formula (1 + X) ∗ (1 + X) = 1 + X defines a unique product structure on ψ compatible with the action of Λ. By linearity it can be extended to a bilinear map Ᏸ(V ) × Ᏸ V ∗ (1) −→ Qp ⊗ , which is denoted by ∗Ᏸ(V ) . Since ψ is a free Λ-module of rank 1, the involution ι acts naturally on ψ by the formula −1
ι(1 + X)a = (1 + X)a . We can now state the explicit reciprocity law of Perrin-Riou.
265
ON IWASAWA THEORY OF CRYSTALLINE REPRESENTATIONS
Theorem 5.3.2. For any f ∈ Ᏸ(V )E=0 and g ∈ Ᏸ(V ∗ (1))E=0 one has & % (0),ε (1),ε−1 PV (f ), PV ∗ (1) (g ι ) (1 + X) = − Tr K/Qp f ∗Ᏸ(V ) g . V
Proof. Since %
(0),ε
PV
Dιm
= −D−m , one can write this formula in the form
& (0),ε−1 ε−1 (f ), T wh+1 ◦ PV ∗ (−h) D h+1 ⊗ eh+1 g ι (1 + X) V
=
h
D−m ◦ Tr K/Qp f ∗Ᏸ(V ) g .
m=1 pn
Put ωn (X) = (1 + X) − 1. From p-adic interpolation theory (see [PR2, no. 1.2]), it follows that it is sufficient to prove the congruences % & (0),ε (0),ε−1 ε−1 ◦ PV ∗ (−h) D h+1 ⊗ eh+1 g ι (1 + X) D k PV (f ), T wh+1 V h (8) ≡ Dk D−m ◦ Tr K/Qp f ∗Ᏸ(V ) g mod ωn (X) m=1 −1
for all n and k $ 0. Taking into account that T wkε = (−1)k T wkε , D k (h(γ − 1)u(X)) = T wk (h(γ − 1))D k u(X), and T wkε D−m ≡ −(m + k) (mod (γn − 1)), we can write the last congruence in the form % & (0),ε (0),ε−1 ε ε−1 ◦ PV (f ), T wk+h+1 ◦ PV ∗ (−h) D h+1 ⊗ eh+1 g ι (1 + X) T w−k V
≡ (−1)k+h
h
(m + k) Tr K/Qp D k f ∗Ᏸ(V ) D k g
mod ωn (X) .
m=1
By definition one has % & (0),ε (0),ε−1 ε ε−1 T w−k ◦ PV (f ), T wk+h+1 ◦ PV ∗ (−h) D h+1 ⊗ eh+1 g ι V h+1 −1 ε ε−1 τ PV ,−k,n (f ), PV ∗ (−h),k+h+1 D ≡ ⊗ eh+1 g ι τ ∈Γ /Γn
mod (γn − 1) .
By the formulas in Remark 4.5 one has −1 τ −1 PεV ,−k,n (f ) = χ(τ )k PεV ,−k,n f τ and
−1 PεV ∗ ,k+1,n (D ⊗ e1 )g ι = (−1)k+1 PεV ∗ ,k+1,n c(D ⊗ e1 )g ι .
−k,n
τ
266
DENIS BENOIS
It is easy to see that cD = −Dc and D −1 ι = ιD. Then, by Corollary 5.1.7 one has −1 τ −1 PεV ,−k,n (f ), PεV ∗ (−h),k+h+1 D h+1 ⊗ eh+1 g ι −k,n
=
h (−1)k+h
pn
(k + m) Tr K/Qp
m=1
( ι −1 D k f ζ χ(τ ) − 1 , D k g ζ −1 − 1 ⊗ e1 .
' ζ ∈µpn
But in [PR2, Prop. 4.3.2], it is shown that ( 1 ' k χ(τ −1 ) k ι −1 D f ζ ζ − 1 , D g − 1 ⊗ e 1 pn ζ ∈µpn
is equal to the coefficient of (1+X)χ(τ ) in the interpolation polynomial of D k f ∗Ᏸ(V ) D k g modulo ωn (X). So the congruence (8) is checked, and the theorem is proved.
References [A] [AV]
[AH] [BK]
[Br]
[CE] [C] [CCz1] [CCz2] [Cn1] [Cn2] [Cn3] [Cz1] [Cz2]
V. A. Abrashkin, The field of norms functor and the Brückner-Vostokov formula, Math. Ann. 308 (1997), 5–19. Y. Amice and J. Vélu, “Distributions p-adiques associées aux séries de Hecke” in Journées Arithmétiques de Bordeaux (Bordeaux, 1974), Astérisque 24–25, Soc. Math. France, Montrouge, 1975, 119–131. E. Artin and H. Hasse, Die beiden Ergänzungssätze zum Reziprozitätsgesetz der l n -ten Einheitzwurzeln, Abh. Math. Sem. Univ. Hamburg 6 (1928), 146–162. S. Bloch and K. Kato, “L-functions and Tamagawa numbers of motives” in The Grothendieck Festschrift, Vol. 1, Progr. Math. 86, Birkhäuser, Boston, 1990, 333– 400. H. Brückner, “Eine explizite Formel zum Reziprozitätsgesetz für Primzahlexponenten p” in Algebraische Zahlentheorie (Oberwolfach, 1964), Bibliographisches Institut, Mannheim, 1967, 31–39. H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, 1956. F. Cherbonnier, Représentations p-adiques surconvergentes, thesis, Orsay, 1996. F. Cherbonnier and P. Colmez, Représentations p-adiques surconvergentes, Invent. Math. 133 (1998), 581–661. , Théorie d’Iwasawa des représentations p-adiques d’un corps local, J. Amer. Math. Soc. 12 (1999), 241–268. R. Coleman, Division values in local fields, Invent. Math. 53 (1979), 91–116. , The dilogarithm and the norm residue symbol, Bull. Soc. Math. France 109 (1981), 373–402. , Local units modulo circular units, Proc. Amer. Math. Soc. 89 (1983), 1–7. P. Colmez, Théorie d’Iwasawa des représentations de de Rham d’un corps local, Ann. of Math. (2) 148 (1998), 485–571. , Représentations cristallines et représentations de hauteur finie, J. Reine Angew. Math. 514 (1999), 119–143.
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J.-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), 529–577. , “Représentations p-adiques des corps locaux, I” in The Grothendieck Festschrift, Vol. 2, Progr. Math. 87, Birkhäuser, Boston, 1990, 249–309. , Le corps des périodes p-adiques, Astérisque 223 (1994), 59–111. , Représentations p-adiques semi-stables, Astérisque 223 (1994), 113–184. , Sur un théorème de Bloch et Kato (lettre à B. Perrin-Riou), Invent. Math. 115 (1994), 151–161. G. Henniart, Sur les lois de réciprocité explicites, I, J. Reine Angew. Math. 329 (1981), 177–203. L. Herr, Sur la cohomologie galoisienne des corps p-adiques, Bull. Soc. Math. France 126 (1998), 563–600. , Une nouvelle approche de la dualité locale de Tate, preprint, Univ. Bordeaux 1, 1999. K. Kato, The explicit reciprocity law and the cohomology of Fontaine-Messing, Bull. Soc. Math. France 119 (1991), 397–441. , “Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR , I” in Arithmetic Algebraic Geometry (Trento, 1991), Notes in Math. 1553, Springer, Berlin, 1993, 50–163. K. Kato, M. Kurihara, and T. Tsuji, Local Iwasawa theory of Perrin-Riou and syntomic complexes, preprint, 1996. M. Kurihara, Computation of the syntomic regulator in the cyclotomic case, appendix to: M. Gros, Régulateurs syntomiques et valeurs de fonctions L p-adiques, I, Invent. Math. 99 (1990), 293–320. B. Perrin-Riou, Théorie d’Iwasawa et hauteurs p-adiques, Invent. Math. 109 (1992), 137– 185. , Théorie d’Iwasawa des représentations p-adiques sur un corps local, Invent. Math. 115 (1994), 81–149. , Théorie d’Iwasawa et loi explicite de réciprocité, Doc. Math. 4 (1999), 219–273, available from http://www.mathematik.uni-bielefeld.de/documenta/. J.-P. Serre, Corps locaux, 2d ed., Publ. Inst. Math. Univ. Nancago 8, Hermann, Paris, 1968. I. R. Shafarevic, A general reciprocity law, Amer. Math. Soc. Transl. 4 (1956), 73–105. E. de Shalit, “The explicit reciprocity law of Bloch-Kato” in Columbia University Number Theory Seminar (New York, 1992), Astérisque 228, Soc. Math. France, Montrouge, 1995, 197–221. J. Tate, Relations between K2 and Galois cohomology, Invent. Math. 36 (1976), 257–274. S. V. Vostokov, Explicit form of the law of reciprocity, Math. USSR-Izv. 13 (1979), 557– 588. N. Wach, Représentations p-adiques potentiellement cristallines, Bull. Soc. Math. France 124 (1996), 375–400. J.-P. Wintenberger, Le corps des normes de certaines extensions infinies de corps locaux; applications, Ann. Sci. École Norm. Sup. (4) 16 (1983), 59–89.
Mathématiques et Informatique, Université Bordeaux I, 351 cours de la Libération, 33405, Talence Cedex, France; [email protected]
Vol. 104, No. 2
DUKE MATHEMATICAL JOURNAL
© 2000
REFLECTIONLESS SCHRÖDINGER OPERATORS, THE DYNAMICS OF ZEROS, AND THE SOLITONIC SATO FORMULA J. F. VAN DIEJEN and H. PUSCHMANN
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 2. Reflectionless Schrödinger operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 2.1. The equations of motion for the zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 2.2. Hamiltonian structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 2.3. Lax representation and integrals of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 279 3. The action-angle transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 3.1. Spectral properties of the Lax matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 3.2. Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 3.3. The integral curve and action-angle diffeomorphism . . . . . . . . . . . . . . . . . . 288 3.4. Symplectic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 4. The wave function and the tau function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.1. A Wilson-type determinantal formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 4.2. The solitonic Sato formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 4.3. The Bethe curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 5. Dynamics and scattering of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 6. The Korteweg–de Vries hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Appendix A. Proof of the no-crossing Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 310 Appendix B. Some Cauchy matrix identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 The zeros of the (Jost) eigenfunction of a 1-dimensional Schrödinger operator with a reflectionless rapidly decreasing potential are related to the spectral data through a nonlinear algebraic system of Bethe-type equations. We show that the behavior of these zeros (with respect to translations) is governed by a rational RuijsenaarsSchneider particle system with harmonic term. The integration of the particle system— via an explicit construction of the action-angle transform—then provides us with detailed information on the solution curve of the Bethe equations. As a result, we find Received 25 May 1999. Revision received 18 November 1999. 2000 Mathematics Subject Classification. Primary 37J35; Secondary 35Q40, 35Q51, 37K10, 70H06, 81Q05. van Diejen supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) through grant number 1980832 and at the Mathematical Sciences Research Institute (MSRI) by National Science Foundation grant number DMS-9701755. 269
270
VAN DIEJEN AND PUSCHMANN
a Wilson-type determinantal formula for the eigenfunction involving RuijsenaarsSchneider (Lax) matrices, and we furthermore recover the solitonic Sato formula (which parametrizes the eigenfunction explicitly in terms of the spectral data). The flows corresponding to the higher integrals of the rational Ruijsenaars-Schneider system with harmonic term give rise to the soliton solutions of the Korteweg–de Vries (KdV) hierarchy. 1. Introduction. In this paper we consider 1-dimensional Schrödinger operators with eigenfunctions that are elementary in the sense that they factorize as a product of a plane wave and a (monic) polynomial in the spectral parameter. It is shown that the dependence of the zeros of such wave functions on the spatial variable is governed by a rational Ruijsenaars-Schneider system with harmonic term (see [RSc], [R2], [Sc], and [BR]). (Here the zeros are thought of as particles, and the spatial variable of the Schrödinger equation corresponds to the time variable of the Ruijsenaars-Schneider system.) The integration of this dynamical system then provides us with very explicit information regarding the structure of the eigenfunctions of our Schrödinger operator. The same class of (reflectionless) Schrödinger operators studied here has been investigated before within the framework of finite-gap theory (where they arise as a (soliton) limiting case) (see [DMN], [DMKM], and [BBEIM]). It should be stressed, however, that—even though in finite-gap theory the dynamics of zeros of Schrödinger wave functions plays a key role—no analysis of the corresponding zero motion in terms of Ruijsenaars-Schneider-type Hamiltonian particle mechanics has been considered so far. (See, however, [K] and [Wi] for a study of the dynamics of poles of the rational Baker-Akhiezer function for the Kadomtsev-Petviashvili (KP) hierarchy in terms of Calogero-Moser particle systems, and see [KWZ] for the pole motion of the elliptic Baker-Akhiezer function for the discretized Kadomtsev-Petviashvili equation in terms of discrete-time Ruijsenaars-Schneider models.) Before presenting a more precise description of our results, let us illustrate the main ideas by means of the simplest nontrivial example: that of an eigenfunction with a polynomial part of degree 1 in the spectral parameter. In this situation the RuijsenaarsSchneider system at issue has only one degree of freedom and its integration is completely elementary. Specifically, we are interested in those Schrödinger equations 2 ∂x + u(x) − z2 ψ(x, z) = 0, −∞ < x < ∞, (1.1) that admit a solution of the form ψ(x, z) = (z − ζ (x)) exp(zx).
(1.2)
(Here z denotes the spectral parameter.) Substitution of ψ(x, z) from equation (1.2) in equation (1.1), and collecting the powers of z, readily entails that the wave function solves the Schrödinger equation if and only if the function ζ (x), characterizing the location of the zero, satisfies the second-order differential equation ζ (x) + 2ζ (x)ζ (x) = 0,
(1.3a)
DYNAMICS OF ZEROS AND THE SATO FORMULA
271
and the potential u(x) is related to ζ (x) via the compatibility condition u(x) = 2ζ (x).
(1.3b)
The differential equation (1.3a) can be brought to Hamiltonian form (here we reduce to the submanifold ζ (x) > 0). To this end we employ the Hamiltonian H = exp(ρ) + ζ 2 ,
(1.4)
with a phase space given by = {(ζ, ρ) ∈ R × R} endowed with the standard symplectic form ω = dρ ∧ dζ . (For relevant background material on classical mechanics/symplectic geometry the reader is referred, e.g., to Arnold’s standard monograph [A].) The Hamilton equations ζ = ∂H /∂ρ, ρ = −∂H /∂ζ associated to H of equation (1.4) read ζ = exp(ρ),
ρ = −2ζ.
(1.5)
The system in equation (1.5) reduces to equation (1.3a) upon differentiation of the first equation and elimination of the ρ-variable. The Hamilton equations are solved by transforming to the action-angle coordinates ˆζ = exp(ρ) + ζ 2 , (1.6a) ζˆ − ζ . (1.6b) ρˆ = log ζˆ + ζ It is not difficult to check that the action-angle mapping (ζ, ρ) − → (ζˆ , ρ) ˆ between ˆ = {(ζˆ , ρ) our phase space and the action-angle phase space ˆ ∈ R × R | ζˆ > 0} (equipped with the symplectic form ωˆ = dρˆ ∧dζˆ ) is in fact a symplectomorphism (i.e., a symplectic diffeomorphism). In the Darboux (i.e., canonical) coordinates ζˆ , ρˆ the Hamiltonian becomes free Hˆ = ζˆ 2 and the corresponding linear Hamilton equations ζˆ = 0, ρˆ = −2ζˆ have as solution
ζˆ (x) = ζˆ0 ,
ρ(x) ˆ = ρˆ0 − 2x ζˆ0 ,
(1.7)
where ζˆ0 = exp(ρ0 ) + ζ02 and ρˆ0 = log(ζˆ0 − ζ0 ) − log(ζˆ0 + ζ0 ) with ζ0 ≡ ζ (0) and ρ0 ≡ ρ(0). We read off from equations (1.6b) and (1.7) that the zero function ζ (x) is recovered as the unique solution of the algebraic equation ζˆ0 − ζ (x) = exp ρˆ0 − 2x ζˆ0 . ζˆ0 + ζ (x) This immediately produces ζ (x) = ζˆ0
1 − exp ρˆ0 − 2x ζˆ0 . 1 + exp ρˆ0 − 2x ζˆ0
(1.8)
(1.9)
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VAN DIEJEN AND PUSCHMANN
The upshot is that all Schrödinger operators on the line, with a (regular) positive potential and an eigenfunction of the factorized form given by equation (1.2), are characterized by the 2-parameter family of potentials (cf. equations (1.3b) and (1.9)) u(x) = 2 ∂x2 log τ (x) with τ (x) = 1 + exp ρˆ0 − 2x ζˆ0 (1.10a) and ρˆ0 ∈ R, ζˆ0 > 0. The corresponding wave function reads (cf. equations (1.2) and (1.9)) 1 + z + ζˆ0 / z − ζˆ0 exp ρˆ0 − 2x ζˆ0 (x, z) = exp(zx), (1.10b) 1 + exp ρˆ0 − 2x ζˆ0 where we have renormalized such that (x, z) → exp(zx) for x → +∞. It is immediate from the explicit formula in equation (1.9) that the zero of our wave function increases monotonically from −ζˆ0 to +ζˆ0 as x runs along the real line in the positive direction. Consequently, the wave function becomes proportional to exp(zx) for x → ±∞, and we are thus dealing with a so-called reflectionless or Bargmann potential (see, e.g., [DT], [AS], and [M4]). (For a general rapidly decreasing potential the Schrödinger wave function has at both sides (i.e., −∞ and +∞) an asymptotics that is composed of linear combinations of the plane waves exp(zx) and exp(−zx). By forming a suitable linear combination of two fundamental solutions to the Schrödinger equation, it is always possible (for generic z) to achieve that the asymptotics of the wave function is of the form ∼ exp(zx) for x → +∞. A potential is said to be reflectionless if one has that at the same time the asymptotics for x → −∞ is also of the form ∼ exp(zx).) By considering the motion of the zero ζ with respect to multiparameter (i.e., multitime) commuting flows associated to the Hamiltonians H m /m, m = 1, 2, . . . (with H taken from equation (1.4)), we recover the 1-soliton solution of the Korteweg–de Vries hierarchy. (See, e.g., [SCM], [AS], [N], and [NMPZ] for relevant preliminaries regarding soliton theory and the KdV equation.) In particular, when the zero function ζ (x, t) is taken to be a solution to the 2-parameter compatible Hamiltonian system ∂H , ∂ρ 1 ∂H 2 ∂t ζ (x, t) = , 2 ∂ρ
∂x ζ (x, t) =
∂H , ∂ζ 1 ∂H 2 ∂t ρ(x, t) = − 2 ∂ζ
∂x ρ(x, t) = −
(1.11a) (1.11b)
(with times x and t), then the above solution method entails for the potential u(x, t) = 2 ∂x ζ (x, t) that u(x, t) = 2 ∂x2 log τ (x, t) with τ (x, t) = 1 + exp ρˆ0 − 2ζˆ0 x − 2ζˆ03 t (1.12) (cf. equation (1.10a)); that is, u(x, t) =
2 ζˆ02 . cosh ζˆ0 x + ζˆ03 t − ρˆ0 /2 2
(1.13)
DYNAMICS OF ZEROS AND THE SATO FORMULA
273
This is the celebrated 1-soliton solution of the KdV equation (see [SCM], [AS], [N], and [NMPZ]) 3 1 ut = uux + uxxx . 2 4
(1.14)
In other words, for a 1-soliton potential of the form in equation (1.13) the position of the zero ζ (x, t) of the corresponding Schrödinger wave function as a function of x and t is governed by the 2-time Hamiltonian system in equations (1.11a) and (1.11b) with H taken from equation (1.4). The aim of the present paper is to lift the above scheme to the case of a Schrödinger operator with an elementary wave function ψ(x, z) that is the product of a plane wave exp(zx) and a monic polynomial in the spectral parameter zN + sN−1 (x)zN−1 + · · · + s0 (x) of arbitrary degree N ∈ N. It is shown in Section 2 that the zeros of a Schrödinger eigenfunction of this type satisfy the equations of motion of a rational Ruijsenaars-Schneider system with harmonic term (see [RSc], [Sc], and [BR]). (Here the spatial variable of the Schrödinger operator plays the role of the time variable of the Ruijsenaars-Schneider system; cf. above.) The relevant equations of motion are then integrated in Section 3 via an explicit construction of the action-angle transform (cf. also [R2] for a construction of the action-angle transform for the RuijsenaarsSchneider systems without harmonic term). In Section 4 we utilize the explicit construction of the zeros as functions of the spatial variable to arrive at closed expressions for both the potential and the wave function of our Schrödinger operator (cf. equations (1.10a) and (1.10b)). This way we find a determinantal formula for the reflectionless potentials and wave functions in terms of hyperbolic Ruijsenaars-Schneider Lax matrices. This ties in with Shiota’s and Wilson’s recently found determinantal representation in terms of Calogero-Moser Lax matrices (see [M2], [M3], and [P]) for, respectively, the potential and wave function of the linear problem associated to the Kadomtsev-Petviashvili equation in the rational regime (see [Sh] and [Wi]). That is to say, our determinantal formulas may be viewed as KdV soliton counterparts of Shiota’s formula for the rational KP tau function and Wilson’s formula for the rational KP Baker function. From the determinantal representations we furthermore recover Hirota’s formula for the tau function of the reflectionless Schrödinger potential (see [H], [SCM], and [N]) as well as the solitonic Sato formula for the corresponding wave function (see [S], [SS], [DKJM], [JM], [SW], [OSTT], [Mo], [BBS], and [DK]). The explicit relation between the zeros of the wave function on the one hand and the action-angle variables for the rational Ruijsenaars-Schneider system with harmonic term on the other hand determines a Bethe-type system of algebraic equations for these zeros that generalizes equation (1.8). (The fact that for N = 1 the Bethe equation (1.8) turns out to be linear in ζ is very deceptive: For N > 2 the Bethe system becomes highly nonlinear.) The action-angle variables of the Ruijsenaars-Schneider system correspond actually to the spectral data of the reflectionless Schrödinger operator. The
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Bethe system may, in other words, be interpreted as an algebraic equation for the zeros of the wave function in terms of the spectral data. Such a Bethe system has already appeared before in the physics literature in connection with a quantization problem for the sine-Gordon solitons [BBS]. In our approach we find that the projection onto the configuration space of the curve swept out by the Hamiltonian flow of our RuijsenaarsSchneider system describes the unique (with fixed spectral data) solution curve of the Bethe system. (Here the initial conditions for the Ruijsenaars-Schneider flow are in one-to-one correspondence with the spectral data via the action-angle transform.) From the action-angle transform it is possible to deduce very precise information pertaining to the qualitative behavior of the zeros of the wave function as one varies the position variable x along the real axis. A notable feature is, for example, that the zeros do not cross each other (i.e., they always remain simple). Section 5 is devoted to the study of such dynamical issues with an emphasis on the scattering properties of the zeros. In particular, it is seen that the zeros have a constant asymptotics for x → ±∞, which is a manifestation of the fact that, also for N > 1, we are dealing with reflectionless Schrödinger operators. The scattering of the zeros turns out to decompose into 1-particle and 2-particle processes (i.e., the relevant phase shifts are built of 1- and 2-particle phase shifts.) We wrap up in Section 6 with a brief explanation of how to encode the N-soliton solutions for the KdV hierarchy in terms of the flows corresponding to the higher integrals of the rational Ruijsenaars-Schneider system with harmonic term. Specifically, we find that—for a Schrödinger potential constituting an N-soliton solution of the KdV hierarchy—the zeros of the wave function move in accordance with the rational Ruijsenaars-Schneider hierarchy with harmonic term. The results in this section fit within the framework of a long line of research on the characterization of special solutions to integrable nonlinear partial differential equations in terms of (integrable) particle systems, starting with the pioneering works of Kruskal, Thickstun, and, especially, Airault, McKean, and Moser (see [Kr], [Th], [AMM], [CC], [Ca], [K], [M3], [M4], [RSc], [P], [Sh], [Wi], and [KWZ]). Our approach should be viewed as an alternative route to tie in the KdV soliton solutions with integrable particle systems, which is different from the previous approaches due to Moser [M4] (using Neumann systems on spheres) and Ruijsenaars-Schneider [RSc] (using hyperbolic RuijsenaarsSchneider systems). 2. Reflectionless Schrödinger operators 2.1. The equations of motion for the zeros. We consider the eigenvalue problem associated to a Schrödinger operator on the line: 2 ∂x + u(x) ψ(x, z) = z2 ψ(x, z),
−∞ < x < ∞.
(2.1)
Here the variable z denotes the (complex) spectral parameter, and the function u(x) represents the (real-valued) potential of the problem. It seems quite a natural question
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to ask oneself for which potentials u(x) the Schrödinger equation (2.1) admits elementary solutions that factorize as a product of an exponential plane wave exp(zx) and a monic polynomial in the spectral parameter with coefficients depending on x: ψ(x, z) = exp(zx)
N
sm (x) zN−m ,
with s0 (x) ≡ 1.
(2.2)
m=0
For u ≡ 0, solutions of this type exist, of course, trivially. Indeed, in this simple situation any (monic) polynomial in z with constant coefficients sm gives rise to an eigenfunction. The simplest choice corresponds to the unit polynomial (i.e., the monic polynomial of degree N = 0), which produces the standard solution ψ(x, z) = exp(zx). This trivial example with u ≡ 0 illustrates very well the more general phenomenon that—starting from a solution to the Schrödinger equation (2.1) having the elementary structure in equation (2.2)—one may pass to other solutions of the same type via multiplication by an arbitrary (monic) polynomial in z with constant coefficients. However, since the solutions thus obtained just differ by an x-independent normalization factor, they are in essence equivalent. Without restriction we may therefore assume from now on that the polynomial part of our wave function is minimal in the sense that overall factors with constant coefficients are divided out. More specifically, we assume that the wave function is of the form ψ(x, z) = exp(zx)
N
z − ζj (x) ,
(2.3)
j =1
with the zeros ζ1 (x), . . . , ζN (x) being nonconstant functions of x. For technical reasons we furthermore restrict to the situation in which the zeros ζj (x) are real and analytic in the (real) variable x (i.e., for every point x0 on the real line the Taylor series of ζj (x) around x0 exists and converges in a sufficiently small neighborhood). The following proposition provides precise criteria to be satisfied by the zeros ζ1 (x), . . . , ζN (x) and the potential u(x) so as to guarantee that the wave function ψ(x, z) of equation (2.3) solves the Schrödinger equation (2.1). Proposition 2.1 (The motion of zeros). Let the zero functions ζj : R → R, j = 1, . . . , N , be analytic and nonconstant. Then the wave function ψ(x, z) of equation (2.3) solves the Schrödinger equation (2.1) if and only if the zeros ζ1 (x), . . . , ζN (x) are simple and satisfy the coupled nonlinear system of ordinary differential equations ζj + 2ζj ζj =
1≤k≤N, k=j
2ζj ζk ζj − ζ k
,
j = 1, . . . , N,
(2.4a)
and, furthermore, the potential u(x) is related to the zeros via the compatibility condition u(x) = 2 ζ1 (x) + · · · + ζN (x) . (2.4b)
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(Here the primes refer to the differentiation with respect to the spatial variable x.) Proof. We have from equation (2.3) that
ζk , ∂x ψ(x, z) = ψ(x, z) z − z − ζk 1≤k≤N
ζ + 2zζ 2ζk ζl k k 2 2 + . ∂x ψ(x, z) = ψ(x, z) z − z − ζk (z − ζk )(z − ζl ) 1≤k≤N 1≤k 0). We conclude that the potential u(x) = ζ1 (x) + · · · + ζN (x) → 0 and that the wave function ψ(x, z) = exp(zx)(z − ζ1 (x)) · · · (z − ζN (x)) ∼ exp(zx) as x → ±∞. That is to say, we are indeed dealing with Schrödinger operators with reflectionless potentials as indicated by the title of this section. 2.3. Lax representation and integrals of motion. The flow associated to the Ruijsenaars-Schneider Hamiltonian (2.8) turns out to be integrable. A Lax pair can be gleaned from Bruschi and Ragnisco [BR]: Lj,k = (ηj ηk )1/2 + ζj2 δj,k , Mj,k =
1 ≤ j, k ≤ N,
(ηj ηk )1/2 1 − δj,k , ζj − ζ k
1 ≤ j, k ≤ N.
(2.13a) (2.13b)
(Here δj,k represents the Kronecker delta.) This Lax pair amounts—up to a gauge transformation—to the rational degeneration of a Lax pair for the hyperbolic generalization of the Hamiltonian H from equation (2.8) due to Schneider [Sc]. The matrices L of equation (2.13a) and M of equation (2.13b) should be thought of as real-analytic matrix-valued functions on the manifold ᏹ of equation (2.7) or—upon employing the transformation (2.12)—on the phase space of equation (2.9). It is not difficult to check that the Hamilton equations (2.11a) and (2.11b) are equivalent to the Lax equation d L = [M, L]. dx Indeed, we have that Lj,k =
1 ηk 1/2 1 ηj 1/2 ηj + ηk + 2ζj ηj δj,k , 2 ηj 2 ηk
(2.14)
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[M, L]j,k =
N
Mj,l Ll,k − Lj,l Ml,k
l=1
=
ηk ηj
1/2 1≤l≤N l=j
1/2 ηj ηl ηj ηk ηl + ζj − ζ l ηk ζk − ζ l 1≤l≤N l=k
+ (ζj + ζk )(ηj ηk )1/2 (δj,k − 1). Comparison of the respective matrix elements reduces the Lax equation (2.14) to the equations of motion (2.6a) and (2.6b) (and thus to equations (2.11a) and (2.11b)). The Lax equation is in fact the first of a hierarchy of equations: m−1
m−1
d m n m−1−n n L = L LL = L [M, L]Lm−1−n dx n=0
m
= [M, L ],
n=0
(2.15)
m = 1, . . . , N.
It is immediate from this Lax hierarchy that the evolution of the Lax matrix is isospectral: d Tr(Lm ) = 0, dx
m = 1, . . . , N.
(2.16)
This means, in particular, that the coefficients of the characteristic polynomial of the Lax matrix are integrals for the Ruijsenaars-Schneider Hamiltonian H from equation (2.8). We can compute these integrals explicitly: N det(L − λ 1) = ζ12 − λ · · · ζN2 − λ + ζk2 − λ ηj j =1
=
N
(2.17a)
1≤k≤N k=j
(−λ)N−m Hm ,
(2.17b)
m=0
with N ηj σm−1 ζ12 , . . . , ζj2 , . . . , ζN2 , Hm = σm ζ12 , . . . , ζN2 +
(2.18)
j =1
m = 1, . . . , N (and H0 ≡ 0). Here σm denotes the mth elementary symmetric function yj1 · · · yjm = yj (2.19) σm (y1 , . . . , yM ) = 1≤j1 0 for j = 1, . . . , N , 1 for j = 1, . . . , N − 1, (iii) #({ζ1 , . . . , ζN } ∩ {ζˆj }) = #({ζ1 , . . . , ζN } ∩ {−ζˆj }) ≤ 0 for j = N;
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and let ᏺ0 be the open dense submanifold of ᏺ determined by the conditions (i ) ζ1 < · · · < ζN and 0 < ζˆ1 < · · · < ζˆN , (ii ) #({ζ1 , . . . , ζN } ∩ Ij ) = 1 for j = 1, . . . , N . ( Proposition 3.3 (Action variables). (i) The action (ζ , η) − → (ζ , ζˆ ) defines a realanalytic map from ᏹ of equation (2.7) into ᏺ. (ii) The map ( restricts to a real-analytic diffeomorphism between the open dense submanifolds ᏹ0 of equation (3.2) and ᏺ0 . The (real-analytic) inverse mapping −1
( (ζ , ζˆ ) −−→ (ζ , η) from ᏺ0 onto ᏹ0 is determined explicitly by 2 2 ˆ 1≤l≤N ζl − ζj 2 , j = 1, . . . , N. ηj = 2 1≤l≤N, l=j ζl − ζj
(3.7)
(
Proof. The mapping (ζ , η) − → (ζ , ζˆ ) is real analytic on ᏹ by Lemma 3.1. To see that ( maps ᏹ into ᏺ it is noted that ζ12 , . . . , ζN2 < ζˆN2 . (This is because the right-hand side of equation (3.3) varies between −∞ and +1 when λ runs from max1≤j ≤N {ζj2 } to +∞.) But then, since for (ζ , η) ∈ ᏹ0 the poles and the zeros of equation (3.3) are simple and interlacing each other (cf. above), it follows that on ᏹ0 each interval union Ij contains exactly one of the elements of {ζ1 , . . . , ζN }. This proves that the open dense submanifold ᏹ0 gets mapped into ᏺ0 ⊂ ᏺ. As we move from a point in ᏹ0 to a point in ᏹ \ ᏹ0 , the position ζj may reach the boundary of its interval union Im (say). In fact, this should happen for at least two positions, say, ζj and ζk (j < k). The movement to the boundary corresponds to the coalescence of the poles ζj2 and ζk2 with a zero ζˆl2 in equation (3.3) (l ∈ {1, . . . , N − 1}). It means that in this situation ζk = −ζj = ζˆl . One concludes that the positions ζ1 , . . . , ζN can only reach the endpoints of the interval unions in even pairs, whence ( maps ᏹ into ᏺ. To determine the inverse of (|ᏹ0 it is noted that for (ζ , ζˆ ) ∈ ᏺ0 the poles of equation (3.3) are simple. A calculation of the residue at λ = ζj2 then produces the inversion relation (3.7). It is furthermore not difficult to infer that the right-hand side of equation (3.7) is positive for (ζ , ζˆ ) ∈ ᏺ0 . Indeed, the interlacing property (ii ) of Definition 3.2 guarantees that for each j ∈ {1, . . . , N } the number of ζl2 and ζˆl2 smaller than ζj2 is equal (and we stay away from zeros and poles). Hence, the map ( restricts to a real-analytic diffeomorphism between ᏹ0 and ᏺ0 , with the inverse map being governed by equation (3.7). It is convenient to think of the pair (ζ , ζˆ ) ∈ ᏺ0 as real-analytic coordinates for the open dense submanifold ᏹ0 ⊂ ᏹ. These coordinates are related to the standard coordinates (ζ , η) ∈ ᏹ0 via the real-analytic diffeomorphism ( : ᏹ0 → ᏺ0 (see equation (3.4)). Remarkably, in the local (ζ , ζˆ )-coordinates we can perform the diagonalization of the Lax matrix completely explicitly. Lemma 3.4 (Diagonalization). Let L(ζ , ζˆ ) be given by equation (2.13a) with η1 , . . . , ηN taken from equation (3.7). Then one has for (ζ , ζˆ ) ∈ ᏺ0 that
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ˆ 2 U−1 , L = UZ ˆ = diag(ζˆ1 , . . . , ζˆN ) and with Z 1/2 2 1/2 2 ˆ ζˆl2 − ζj2 1 1≤l≤N ζk − ζl 2 , 2 2 ˆ2 ˆ ζj2 − ζˆk2 1≤l≤N, l=j ζl − ζj 1≤l≤N, l=k ζk − ζl
Uj,k =
1≤l≤N
1 ≤ j , k ≤ N. Furthermore, the matrix U is real analytic in (ζ , ζˆ ) ∈ ᏺ0 and orthogonal: U−1 = t U. Proof. It follows from the interlacing property (ii ) of Definition 3.2 that all the matrix elements Uj,k (ζ , ζˆ ) are real analytic for (ζ , ζˆ ) ∈ ᏺ0 . Specifically, we stay away from zeros in numerator and denominator, and the arguments of the square roots are positive (as #{1 ≤ l ≤ N | ζˆl2 < ζj2 } = #{1 ≤ l ≤ N | ζl2 < ζj2 } and #{1 ≤ l ≤ N | ζl2 > ζˆk2 } = #{1 ≤ l ≤ N | ζˆl2 > ζˆk2 }). In Appendix B (cf. Corollary B.4 and Proposition B.6), it is shown that the stated matrix U is orthogonal and that it diagonalizes the Lax matrix L(ζ , ζˆ ). It is clear from Proposition 3.3 and Lemma 3.4 that the matrix elements Uj,k are real analytic when viewed as functions on ᏹ0 of equation (3.2). From the explicit expressions for the matrix elements, one might be inclined to think that U becomes singular on ᏹ \ ᏹ0 . This, however, turns out to be not true: The matrix elements Uj,k extend to real-analytic functions on ᏹ of equation (2.13b) by Lemma 3.1 and by the second result from the analytic perturbation theory cited at the beginning of this section. Similarly, the right-hand side of equation (3.7) extends from a real-analytic function on ᏹ0 to a real-analytic function on ᏹ (because the left-hand side does so). Notice, on the other hand, that for N > 1 the inverse mapping (−1 : ᏺ0 → ᏹ0 does not extend analytically to ᏺ. (The expressions for ηj in equation (3.7) become singular on ᏺ \ ᏺ0 unless N = 1.) 3.2. Linearization. The coordinates ζˆ1 , . . . , ζˆN serve as the action variables for our Ruijsenaars-Schneider Hamiltonian H of equation (2.8). To construct the corresponding angle variables, it is convenient to consider the matrix-valued function ˆ −1/2 U−1 ZUZˆ −1/2 1 + Z ˆ −1/2 U−1 ZUZˆ −1/2 −1 N = 1−Z
(3.8)
ˆ and the orthogonal matrix U are defined as on ᏹ, where the diagonal matrices Z, Z in Subsections 2.3 and 3.1. The following lemma provides an explicit expression for the matrix elements of N in the local (ζ , ζˆ )-coordinates. Lemma 3.5 (Angle variables). (i) The symmetric matrix N from equation (3.8) is real analytic and positive definite on ᏹ of equation (2.7).
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(ii) In the coordinate patch (ᏺ0 , (−1 ) of the open dense subset ᏹ0 from equation (3.2) of ᏹ, the matrix elements of N are given explicitly by Nj,k =
1/2 1/2 2 ζˆj ζˆk
ζˆj + ζˆk
1/2 1/2
ηˆ j ηˆ k ,
1 ≤ j, k ≤ N,
(3.9a)
with ηˆ j =
ζˆj − ζl ζˆj + ζˆl , ζˆj + ζl 1≤l≤N ζˆj − ζˆl 1≤l≤N
j = 1, . . . , N.
(3.9b)
l=j
Proof. By the analyticity of Z, Zˆ 1/2 , and U, it is clear that the matrix N from equation (3.8) is analytic in ᏹ outside the locus
(ζ , η) ∈ ᏹ | det 1 + Zˆ −1/2 U−1 ZUZˆ −1/2 = 0 . We now show that this locus is in fact empty. Let us first consider the case (ζ , η) ∈ ᏹ0 . With the aid of the explicit expressions for U in the local (ζ , ζˆ )-coordinates from Proposition 3.3, we can compute the matrix elements of N from equation (3.8) explicitly. This entails the formulas in (3.9a) and (3.9b). For the details of the calculation the reader is referred to Appendix B (see Proposition B.8). It is not difficult to infer that the interlacing property (ii ) of Definition 3.2 guarantees that on ᏹ0 the quantities ηˆ j of equation (3.9b) are analytic and positive. Indeed, we have that ηˆ j =
1≤l≤N
ζˆj2 − ζl2 2 ζˆj + ζl 1≤l≤N l=j
2 ζˆj + ζˆl ζˆ 2 − ζˆ 2 j
l
with #{l = 1, . . . , N | ζl2 > ζˆj2 } = #{l = 1, . . . , N | ζˆl2 > ζˆj2 }, and we stay away from zeros in the numerator and denominator. It thus follows that N is real analytic and positive definite on ᏹ0 of equation (3.2). (To decide on the positivity of N we have used at this point that 0 < ζˆ1 < · · · < ζˆN and that the principal minors are easily computed using the Cauchy determinant formula (see [We, p. 202]); cf. also Corollary B.3). To generalize the result to the whole manifold ᏹ of equation (2.13b), it suffices to show that the ηˆ j of equation (3.9b) extend to real-analytic positive functions on ᏹ, or, equivalently, that the functions ζˆj − ζl , ζˆj + ζl
j = 1, . . . , N,
(3.10)
1≤l≤N
extend to real-analytic functions on ᏹ without zeros. This is seen by recalling that the matrix U, with elements Uj,k given by Lemma 3.4, extends analytically to an
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orthogonal matrix on ᏹ (by Lemma 3.1 and Result 2). Indeed, after squaring Uj,k and dividing by ηj of equation (3.7) (> 0), we conclude that for all j, k ∈ {1, . . . , N } the expression 2 −1 ζˆj − ζk2 ζˆj2 − ζl2 (3.11) 1≤l≤N, l=k
extends analytically to ᏹ. Since on ᏹ with a ζi → ζˆj there always corresponds a ζi → −ζˆj and vice versa (see Subsection 3.1), it follows from the analyticity of the expressions in equation (3.11) that in such a situation the fractions (ζˆj +ζi )/(ζˆj −ζi ) and (ζˆj − ζi )/(ζˆj + ζi ) extend analytically to the point of confluence. Hence, the expressions in equation (3.10) extend to real-analytic functions on ᏹ without zeros, which completes the proof. Let us define the action-angle manifold
ˆ = ζˆ , ηˆ ∈ RN × RN | 0 < ζˆ1 < · · · < ζˆN , ηˆ 1 , . . . , ηˆ N > 0 ᏹ
(3.12)
ˆ defined by the mapping and consider the action-angle transformation : ᏹ → ᏹ (ζ , η) −→ ζˆ , ηˆ
(3.13)
(where ζˆ = (ζˆ1 , . . . , ζˆN ) contains the ordered square roots of the Lax matrix from equation (2.13a) and where ηˆ = (ηˆ 1 , . . . , ηˆ N ) is determined by equation (3.9b)). Proposition 3.6 (Action-angle transformation). The action-angle transformation ˆ of equaˆ is a real-analytic map from ᏹ of equation (2.7) into ᏹ → (ζˆ , η) (ζ , η) − tion (3.12). Proof. This is immediate from Lemmas 3.1 and 3.5. The mapping of equation (3.13) turns out to linearize the equations of motion (2.6a) and (2.6b). Proposition 3.7 (Linearization). Let us assume that a continuously differentiable curve (ζ , η)(x), x ∈ R, lies in the manifold ᏹ of equation (2.7) and provides a solution to the equations of motion in equations (2.6a) and (2.6b). Then the image ˆ of the form (ζˆ , η)(x), ˆ of this curve with respect to the action-angle map : ᏹ → ᏹ x ∈ R, satisfies the linear system of differential equations ζˆj = 0, j = 1, . . . , N. (3.14) ηˆ = −2ζˆj ηˆ j , j
Proof. The equation ζˆj (x) = 0 is immediate from the isospectrality (2.16) of the Lax matrix. It is furthermore clear that for generic x (i.e., except for values in a
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discrete subset of R) we have (ζ , η)(x) ∈ ᏹ0 of equation (3.2). (This is because ζj = ηj > 0.) It means that for such generic values of x the vector (ζ , ζˆ )(x) lies in ᏺ0 given Proposition 3.3. The differentiation of ηˆ j (x) from equation (3.9b) and the invoking of the equations of motion then produces ηˆ j
= −ηˆ j
N l=1
= −2ζˆj ηˆ j
(2.6a)
= −2ζˆj ηˆ j
(3.7)
ζl
ζˆj − ζl
+
ζl
ζˆj + ζl
N
ηl 2 ζˆ − ζl2 l=1 j
N l=1
ζˆk2 − ζl2 2 2 1≤k≤N, k=l ζk − ζl
1≤k≤N, k=j
= −2ζˆj ηˆ j . In the last step we used the identity of Lemma B.5 to infer that the summation on the right-hand side of the third line produces 1. This proves the stated differential equation for ηˆ j (x) for the generic values of x such that (ζ , η)(x) ∈ ᏹ0 . The extension to all x ∈ R then follows by analyticity. 3.3. The integral curve and action-angle diffeomorphism. The linearized equations (3.14) are solved trivially: ηˆ j (x) = ηˆ j (0) exp − 2x ζˆj (0) , j = 1, . . . , N. (3.15) ζˆj (x) = ζˆj (0), Proposition 3.7 thus tells us that if (ζ , η)(x) satisfies the equations of motion given by equations (2.6a) and (2.6b), then the matrix N of equation (3.8) evolves as ˆ ˆ Inversion of the relation in equation (3.8) allows us to exp(−x Z)N(0) exp(−x Z). reconstruct Z from N, ˘ −1 Z = UZU
ˆ 1/2 (1 − N)(1 + N)−1 Z ˆ 1/2 , with Z˘ = Z
(3.16)
and thus solve the equations of motion. Proposition 3.8 (Integral curve). Let (ζ0 , η0 ) ∈ ᏹ of equation (2.13b), and let ˆ of equation (3.12) be the image of (ζ0 , η0 ) with respect to the action-angle ˆ ∈ᏹ (ζˆ , η) ˆ = diag(ζˆ1 , . . . , ζˆN ), and let N be the map of equation (3.13). Furthermore, let Z 1/2 ˆ ˆ ˆ matrix with entries Nj,k = 2(ηˆ j ηˆ k ζj ζk ) /(ζj + ζˆk ), 1 ≤ j , k ≤ N . Then we have that the ordered eigenvalues ζ1 (x) ≤ · · · ≤ ζN (x) of the matrix ˆ 1/2 (3.17) ˆ N exp − x Zˆ 1 + exp − x Z ˆ N exp − x Zˆ −1 Z Zˆ 1/2 1 − exp − x Z define a real-analytic curve (ζ , ζ )(x), x ∈ R, that lies in ᏹ of equation (2.13b) and solves the system of differential equations (2.6a) and (2.6b) subject to the initial condition (ζ , ζ )(0) = (ζ0 , η0 ).
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289
Proof. By Lemma 3.5 one has that N is positive definite. Hence, the symmetric matrix in equation (3.17) is analytic in x. But then so are its eigenvalues ζ1 (x), . . . , ζN (x), by the first result from the analytic perturbation theory cited at the beginning of this section. This shows that our curve (ζ , ζ )(x), x ∈ R, is real analytic in x. The general theory of ordinary differential equations guarantees that locally (i.e., for x sufficiently small) the (Hamiltonian) system (2.6a) and (2.6b) with initial condition (ζ0 , η0 ) has a unique continuously differentiable integral curve in ᏹ. It is clear from Proposition 3.7 and the inversion formula (3.16) that this local solution coincides with the local restriction of the curve (ζ , ζ )(x). This proves that the curve (ζ , ζ )(x) solves the differential equations (2.6a) and (2.6b) with initial condition (ζ0 , η0 ) for x sufficiently small. But then, by analyticity, the differential equations must actually hold for all x ∈ R. The fact that the integral curve (ζ , ζ )(x) indeed stays within ᏹ for all x ∈ R is now a consequence of Lemmas 2.2 and 2.3. Notice that Proposition 3.8 says, in particular, that the eigenvalues of the matrix in equation (3.17) are simple: ζ1 (x) < · · · < ζN (x) for x ∈ R. The following corollary is an immediate consequence of Proposition 3.8. Corollary 3.9 (Completeness). The flow generated by the Ruijsenaars-Schneider Hamiltonian with harmonic term H of equation (2.8) is complete on the phase space of equation (2.9) and analytic in the flow parameter x ∈ R. ˆ of equation ˆ ∈ᏹ The map of equation (3.13) associates an image point (ζˆ , η) (3.12) to a point (ζ , η) ∈ ᏹ of equation (2.13b). We recall that to compute the image point one first constructs the Lax matrix L of equation (2.13a) and determines its (simple) eigenvalues ζˆ1 < · · · < ζˆN . The components of ηˆ are next derived from equation (3.9b) (first for (ζ , η) in the submanifold ᏹ0 of equation (3.2) by directly evaluating the stated expression for ηˆ j , and then for (ζ , η) ∈ ᏹ by analytic continuation). It is not difficult to see that is an injection. Indeed, the inversion formula (3.16) ˆ For (ζˆ , η) ˆ in (ᏹ0 ) (cf. equation provides the components of ζ as functions of (ζˆ , η). (3.2)) the components of η now follow from equation (3.7). More generally one ˆ as the vector tangent to the real-analytic curve ζ (x), obtains η in terms of (ζˆ , η) x ∈ R—swept out by the ordered eigenvalues of the matrix (3.17)—at the special point x = 0. The inverse mapping −1 from (ᏹ) onto ᏹ turns out to be real analytic. To see this, one employs Result 2 to infer that the diagonal ζ of the matrix ˆ For (ζˆ , η) ˆ in the open dense Z from equation (3.16) is analytic in the variables (ζˆ , η). subset (ᏹ0 ) ⊂ (ᏹ) the analyticity of η is now immediate from equation (3.7). The ˆ ∈ (ᏹ) follows because the poles in ηj of equation (3.7) extension to general (ζˆ , η) originating from the denominator are compensated by zeros in the numerator. (This is clear from the above-mentioned alternative characterization of η as the tangent vector to the curve ζ (x) at x = 0.) The following proposition says that the map constitutes a real-analytic diffeoˆ (i.e., (ᏹ) = ᏹ ˆ ). morphism between the manifolds ᏹ and ᏹ
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Proposition 3.10 (Action-angle diffeomorphism). The action-angle transformation of equation (3.13) defines a real-analytic diffeomorphism between ᏹ of equaˆ of equation (3.12). tion (2.13b) and ᏹ Proof. In view of the Proposition 3.6 and the paragraph preceding the current ˆ is surjective. To this end we consider proposition, it suffices to show that : ᏹ → ᏹ ˆ of equation (3.12). The positivity of the corresponding ˆ in ᏹ an arbitrary point (ζˆ , η) matrix N (cf. Proposition 3.8) leads us to conclude (by a perturbation argument) that, for x sufficiently large, the j th eigenvalue ζj (x) of the matrix in equation (3.17) tends monotonically from below to ζˆj as x → +∞. Hence, we have that (ζ , ζ )(x) ∈ ᏹ of equation (2.13b) for x sufficiently large. But then, by Proposition 3.8, it follows that ˆ = (ζ , ζ )(x) stays in ᏹ for all x ∈ R. In particular, (ζ , ζ )(0) ∈ ᏹ and thus (ζˆ , η) ((ζ , ζ )(0)), which proves the surjectivity. 3.4. Symplectic structure. The variables (ζ , ρ) in of equation (2.9) determine a system of real-analytic canonical coordinates on ᏹ of equation (2.13b) via the identification in equation (2.12). We can easily compute the standard symplectic form ω of equation (2.22) in the (ζ , η)-coordinates (using equation (2.12)) and in the local (ζ , ζˆ )-coordinates (using equation (3.7)): ω=
dρj ∧ dζj
(3.18a)
1≤j ≤N
=
1≤j ≤N
ηj−1 dηj ∧ dζj +
1≤j =k≤N
dζj ∧ dζk ζj − ζk
dζˆj ∧ dζk dζˆj ∧ dζk = . + ζˆj − ζk ζˆj + ζk 1≤j,k≤N
(3.18b)
(3.18c)
ˆ of equation (3.12) with canonical Similarly, we can equip the action-angle manifold ᏹ coordinates by setting
ηˆ j = exp ρˆj
1≤l≤N, l=j
ζˆj + ζˆl , ˆ ζj − ζˆl
j = 1, . . . , N,
(3.19)
ˆ in with (ζˆ , ρ) ˆ =
ζˆ , ρˆ ∈ RN × RN | 0 < ζˆ1 < · · · < ζˆN .
(3.20)
ˆ of equation (3.20) reads in the (ζˆ , ρ)ˆ The standard symplectic form coming from ˆ ˆ coordinates, the (ζ , η)-coordinates, and the local (ζ , ζˆ )-coordinates:
DYNAMICS OF ZEROS AND THE SATO FORMULA
ωˆ =
dρˆj ∧ dζˆj
291 (3.21a)
1≤j ≤N
=
1≤j ≤N
ηˆ j−1 dηˆ j ∧ dζˆj +
1≤j =k≤N
dζˆj ∧ dζˆk ζˆj − ζˆk
ˆ dζj ∧ dζk dζˆj ∧ dζk = , + ζˆj − ζk ζˆj + ζk 1≤j, k≤N
(3.21b)
(3.21c)
respectively. We see that in the local (ζ , ζˆ )-coordinates both ω and ωˆ read the same. This proves that the action-angle mapping of equation (3.13) is a symplectomorphism. Proposition 3.11 (Canonicity). The action-angle mapping of equation (3.13) is a symplectomorphism (i.e., a symplectic diffeomorphism) between (ᏹ, ω) and ˆ , ω). ˆ (ᏹ It is instructive to compute the Poisson brackets between the coordinate functions of the various coordinate systems. (1) The (ζ , ρ)-coordinates (with values in of equation (2.9)) give {ζj , ζk } = 0,
{ζj , ρk } = δj,k ,
{ρj , ρk } = 0.
(3.22)
(2) The (ζ , η)-coordinates (with values in ᏹ of equation (2.7)) give {ζj , ζk } = 0,
{ζj , ηk } = ηj δj,k ,
{ηj , ηk } =
2ηj ηk 1 − δj,k . ζj − ζk
(3.23)
(3) The local (ζ , ζˆ )-coordinates (with values in ᏺ0 , cf. Definition 3.2) give 2 2 2 ˆ2 1 ˆ
1 1≤l≤N ζj − ζl 1≤l≤N ζk − ζl ˆ 2 ζj , ζk = − 2 , 2 ˆ ˆ2 ζj + ζˆk ζj − ζˆk 1≤l≤N, l=j ζj − ζl 1≤l≤N, l=k ζk − ζl (3.24a)
{ζj , ζk } = 0, ζˆj , ζˆk = 0. (3.24b) ˆ of equation (3.12)) give ˆ (4) The (ζˆ , η)-coordinates (with values in ᏹ
ζˆj , ζˆk = 0,
ζˆj , ηˆ k = ηj δj,k ,
2ηˆ j ηˆ k ηˆ j , ηˆ k = 1 − δj,k . ζˆj − ζˆk
ˆ of equation (3.20)) give ˆ (5) The (ζˆ , ρ)-coordinates (with values in
ζˆj , ζˆk = 0, ρˆj , ρˆk = 0. ζˆj , ρˆk = δj,k ,
(3.25)
(3.26)
As a consequence of Proposition 3.11 we arrive at the integrability of the RuijsenaarsSchneider Hamiltonian H of equation (2.8).
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Proposition 3.12 (Integrability). The functions H1 , . . . , HN given by equation (2.18) are in involution with respect to the symplectic form ω: {Hm , Hn } = 0,
1 ≤ m, n ≤ N.
Proof. The Poisson commutativity follows from the canonicity of the action-angle ˆ map in Proposition 3.11 and from the fact that in the action-angle coordinates (ζˆ , ρ) the Hamiltonians become Hm = σm ζˆ12 , . . . , ζˆN2 , m = 1, . . . , N (where σm is the mth elementary symmetric function). Indeed, in action-angle coordinates the vanishing of the Poisson brackets between the integrals Hm is immediate from equation (3.26). 4. The wave function and the tau function. In this section we return to our original problem of characterizing the 1-dimensional Schrödinger operators with eigenfunctions that are the product of a plane wave and a polynomial in the spectral parameter. We use the solution to the equations of motion for the rational RuijsenaarsSchneider system with harmonic term, from the previous section, to derive closed expressions for the potential and the eigenfunction of the Schrödinger operators under consideration. ˆ be a point in the manifold 4.1. A Wilson-type determinantal formula. Let (ζˆ , η) ˆ of equation (3.12). We consider the matrix ᏹ ˜ ˆ 1 − exp − 2x Z ˆ N 1 + exp − 2x Zˆ N −1 , Z(x) =Z (4.1) with Zˆ = diag(ζˆ1 , . . . , ζˆN ) and Nj,k = 2(ηˆ j ηˆ k ζˆj ζˆk )1/2 /(ζˆj + ζˆk ), 1 ≤ j, k ≤ N. From ˜ Proposition 3.8 it follows that the eigenvalues of Z(x) from equation (4.1) constitute a real-analytic solution to the differential equations in equation (2.4a). Hence, by Proposition 2.1, it is clear that the wave function ˜ ψ(x, z) = exp(zx) det z1 − Z(x) (4.2a) solves the Schrödinger equation (2.1) with potential ˜ u(x) = 2∂x Tr Z(x) .
(4.2b)
In equation (4.2a) the wave function is normalized such that the polynomial factor is monic. Often, though, it is somewhat more comfortable to work with the (Jost) wave function ˜ det z1 − Z(x) (x, z) = exp(zx) (4.3) , ˆ det z1 − Z which is normalized such that (x, z) → exp(zx) for x → +∞.
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293
ˆ of equation ˆ ∈ᏹ Theorem 4.1 (Wilson-type formula). Let us assume that (ζˆ , η) (3.12). We have that the wave function ˆ N ˆ z1 − Zˆ −1 exp − 2x Z det 1 + z1 + Z (x, z) = exp(zx) ˆ N det 1 + exp − 2x Z solves the Schrödinger equation 2 ∂x + u(x) ψ(x, z) = z2 ψ(x, z),
−∞ < x < ∞,
with a potential of the form ˆ N . u(x) = 2∂x2 log det 1 + exp − 2x Z ˆ and N are as defined above and z ∈ C \ {ζˆ1 , . . . , ζˆN }. Here the matrices Z Proof. The main point is the above-mentioned observation that (x, z) of equation (4.3) solves the Schrödinger equation with potential u(x) of equation (4.2b). Elementary manipulations then reveal that ˜ det z1 − Z(x) (x, z) = exp(zx) ˆ det z1 − Z ˆ 1 − exp − 2x Z ˆ N 1 + exp − 2x Zˆ N −1 det z1 − Z = exp(zx) ˆ det z1 − Z ˆ 1 − exp − 2x Z ˆ N det z 1 + exp − 2x Zˆ N − Z = exp(zx) ˆ det 1 + exp − 2x Z ˆ N det z1 − Z ˆ − z1 + Z ˆ exp − 2x Z ˆ N det z1 − Z = exp(zx) ˆ N det z1 − Zˆ det 1 + exp − 2x Z ˆ z1 − Z ˆ −1 exp − 2x Z ˆ N det 1 + z1 + Z = exp(zx) ˆ N det 1 + exp − 2x Z and that ˜ u(x) = 2∂x Tr Z(x) ˆ 1 − exp − 2x Zˆ N 1 + exp − 2x Z ˆ N −1 = 2∂x Tr Z ˆ N = 2∂x2 Tr log 1 + exp − 2x Z ˆ N , = 2∂x2 log det 1 + exp − 2x Z which completes the proof.
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From the determinantal formula for the potential given in the theorem, it is read off that we are dealing with the well-known class of reflectionless Schrödinger potentials (or Bargmann potentials) (see [DT], [AS], and [M4]). The parameters 0 < ζˆ1 < · · · < ζˆN and ηˆ 1 , . . . , ηˆ N > 0 correspond to the so-called spectral data of the potential. Specifically, for the spectral values z = −ζˆj , j = 1, . . . , N , the wave function (x, z) is square integrable. These spectral values constitute the discrete spectrum of our Schrödinger operator. The parameters ηˆ j , j = 1, . . . , N , govern the associated normalization constants: ∞ 1 2 x, −ζˆj dx = ηˆ j , j = 1, . . . , N. (4.4) ˆ 2ζj −∞ As it turns out, the determinantal formula for the reflectionless wave function from Theorem 4.1 can also be derived in a completely different way using finite-gap machinery (see [DMKM]) or inverse scattering theory (see [DT], [ELZ], and [DK]). When rewritten in appropriate coordinates our matrix N, with elements Nj,k = 2(ηˆ j ηˆ k ζˆj ζˆk )1/2 /(ζˆj + ζˆk ), 1 ≤ j , k ≤ N , becomes equal to the Lax matrix of a hyperbolic Ruijsenaars-Schneider system [RSc]. Its symmetric functions, which are easily computed with the aid of the Cauchy determinantal formula (see [We, p. 202]; cf. also Corollary B.3), form a complete set of commuting integrals for this system. More explicitly, we have that det(N − λ1) =
N
(−λ)N−m Sm ζˆ , ηˆ ,
(4.5)
m=0
with Sm ζˆ , ηˆ =
J ⊂{1,...,N} j ∈J #J =m
ηˆ j
ζˆj − ζˆk 2 , ζˆj + ζˆk
m = 1, . . . , N
(4.6)
j,k∈J j 0), which is possible if and only if σj ≥ σ1 > σj +1 , now manifests itself as a transposition of the form σ1 · · · σj σj +1 · · · σN −→ σ1 · · · σj +1 σj · · · σN . (5.6) By varying x along the real axis, the consecutively ordered configurations of the Dirichlet eigenvalues give rise to a path of permutations from (N N − 1 · · · 2 1) (for x 0) to (1 2 · · · N − 1 N) for (x ! 0). Two subsequent elements in a path are related by a transposition of the form in equation (5.6) with σj ≥ σ1 > σj +1 (or by a composition of such transpositions in the case of simultaneous crossings). In Figure 4 the possible permutation paths for N = 3 and N = 4 are given, assuming simultaneous crossings do not occur. The cases of simultaneous crossings are obtained via contraction; for example, “(3 2 1) → (2 3 1)” + “(2 3 1) → (2 1 3)” = “(3 2 1) → (2 1 3)”. For general N a rich branching structure of the possible paths emerges, corresponding to the different orders in which the sequence of level crossings can occur. For given N, the total number of distinct configurations generated by these paths is given by N N −1 = 2N−1 . (5.7) M −1 M=1
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(3 2 1)
(4 3 2 1)
/ (3 4 2 1)
/ (2 3 1)
/ (2 1 3)
/ (1 2 3)
(2; 3 4 1) HH HH vv v HH v v HH v v H# v v / (3 2 4 1) (2 3 1 4) HH v; HH v v HH vv HH vv H# v v (3 2 1 4)
/ (2 1 3 4)
/ (1 2 3 4)
Figure 4. Permutation paths for N = 3 and N = 4, encoding the possible consecutively ordered configurations of the Dirichlet eigenvalues ζj2 (x) as they appear when x varies along the real line in the positive direction. Here the permutation (σ1 σ2 · · · σN ) corresponds to the configuration ζσ21 (x) < ζσ22 (x) < · · · < ζσ2N (x), and the arrows represent the respective level crossings σj ↔ σj +1 .
The above dynamical considerations condense into the following properties of the Dirichlet spectrum of reflectionless Schrödinger operators. Proposition 5.3 (Location of zeros). Let (x, z) be the Jost eigenfunction of a Schrödinger operator with a Bargmann potential on the line, whose discrete spectrum is characterized by the data 0 < ζˆ1 < · · · < ζˆN (cf. Theorems 4.1 and 4.2 for explicit parametrizations of the potentials and Jost functions under consideration). Then the roots z = ζj (x), j = 1, . . . , N , of the equation (x, z) = 0 are distributed over the open interval ] − ζˆN , ζˆN [ in such a way that each of the interval unions I˜1 , . . . , I˜N in equation (5.4) contains precisely one root. Furthermore, the roots ζj (x) increase monotonically as a function of x. As a consequence of Proposition 5.3, we recover a well-known result pertaining to the interlacing of the Dirichlet eigenvalues and the discrete spectrum (see, e.g., [CL], [DT], and [ELZ]). Corollary 5.4 (Dirichlet eigenvalues). For the Schrödinger operators of Proposition 5.3, the Dirichlet eigenvalues ζ12 (x0 ) ≤ · · · ≤ ζN2 (x0 ) interlace the discrete spectrum ζˆ12 < · · · < ζˆN2 : ζ12 (x0 ) ≤ ζˆ12 ≤ ζ22 (x0 ) ≤ ζˆ22 ≤ · · · ≤ ζN2 (x0 ) ≤ ζˆN2 , with equality holding if and only if ζj2 (x0 ) = ζˆj2 = ζj2+1 (x0 ). Notice that in Corollary 5.4 we have renumbered the Dirichlet eigenvalues such that they are ordered from small to large.
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6. The Korteweg–de Vries hierarchy. It is well known that there is a close relation between Schrödinger equations with reflectionless potentials and the soliton solutions of the Korteweg–de Vries equation (see [DT], [AS], [N], [NMPZ], and [DMKM]). In this section we exploit this connection to characterize the motion of the zeros of the Jost function for a Schrödinger operator with a Korteweg–de Vries soliton potential in terms of Ruijsenaars-Schneider dynamics. It follows from Proposition 3.12 that the Hamiltonians Ᏼm =
1 Tr(Lm ), m
m = 1, . . . , M,
(6.1)
Poisson-commute amongst themselves. As a consequence, the Hamilton flows generated by Ᏼ1 , . . . , ᏴM commute. In other words, the multitime Hamiltonian system ∂ζj ∂ Ᏼm = , ∂tm ∂ρj
∂ρj ∂ Ᏼm =− , ∂tm ∂ζj
(6.2)
with 1 ≤ j ≤ N and 1 ≤ m ≤ M, is compatible and has a unique solution ζj (t1 , . . . , tM ) and ρj (t1 , . . . , tM ) given the initial condition ζj (0, . . . , 0) = ζj (0) and ρj (0, . . . , 0) = ρj (0). In action-angle coordinates the Hamiltonians from equation (6.1) become N
Ᏼm =
1 2m ζˆj , m
m = 1, . . . , M.
(6.3)
j =1
The transformed equations ∂tm ζˆj = ∂ρˆj Ᏼm and ∂tm ρˆj = −∂ζˆj Ᏼm are thus linear: ∂ ζˆj = 0, ∂tm
∂ ρˆj = −2ζˆj2m−1 , ∂tm
m = 1, . . . , M,
(6.4)
and have as solution ζˆj (t1 , . . . , tM ) = ζˆj (0), ρˆj (t1 , . . . , tM ) = ρˆj (0) − 2
(6.5a) tm ζˆj2m−1 (0),
(6.5b)
1≤m≤M
j = 1, . . . , N . Transforming back with the inverse of the action-angle transformation in equation (3.13) readily produces the solution to the system in equation (6.2) (cf. Subsection 3.3). For the associated Schrödinger potential u(t1 , . . . , tM ) = 2∂t1 (ζ1 (t1 , . . . , tM ) + · · · + ζN (t1 , . . . , tM )) we then obtain in the same way as before (cf. Theorems 4.1 and 4.2) u(t1 , . . . , tM ) = 2∂t21 log τ (t1 , . . . , tM )
(6.6a)
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with
τ (t1 , . . . , tM ) = det 1 + exp −2
M
ˆ 2m−1 N tm Z
(6.6b)
m=1
ζˆj + ζˆk exp ρˆj t1 , . . . , tM , = ˆ ζ − ζˆk j ∈J J ⊂{1,...,N} j ∈J, k∈J j
(6.6c)
where ρˆj (t1 , . . . , tM ) = ρˆj − 2
M m=1
tm ζˆj2m−1 .
(6.6d)
(Here the notation is the same as in Theorems 4.1 and 4.2 and we have suppressed the argument of ζˆj (0) and ρˆj (0).) Furthermore, the corresponding wave function, solving the Schrödinger equation (∂t21 + u(t1 , . . . , tM ) − z2 ) = 0 subject to the boundary 2m−1 ) for t → +∞, is given by (cf. condition (t1 , . . . , tN ; z) → exp( M 1 m=1 tm z equation (4.3)) M M z − ζj (t1 , . . . , tM ) 2m−1 tm z . (6.7a) (t1 , . . . , tM ; z) = exp z − ζˆj m=1
j =1
Invoking of our solution to the Hamiltonian system (6.2) entails (cf. Theorems 4.1 and 4.2) (t1 , . . . , tM ; z) = exp
M
M 2m−1 det 1 + z1 + Z ˆ z1 − Z ˆ −1 exp − 2 ˆ N tm Z
tm z2m−1
m=1
= exp
M
det 1 + exp − 2
×
m=1
M m=1
ˆ 2m−1 N tm Z (6.7b)
tm z2m−1
m=1
J ⊂{1,...,N} j ∈J
z+ζˆj
z−ζˆj
J ⊂{1,...,N} j ∈J, k∈J
ζˆj +ζˆk ρˆj (t1 , . . . , tM ) ζˆj −ζˆk exp j ∈J, k∈J
j ∈J
ζˆj + ζˆk ρˆj (t1 , . . . , tM ) ζˆj − ζˆk exp
.
j ∈J
(6.7c) The above formula for u(t1 , . . . , tM ) is precisely the Hirota formula for the celebrated N -soliton solution of the KdV hierarchy written in terms of the tau function
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309
(see [H], [SCM], and [N]). In particular, for M = 2 with t1 = x and t2 = t, we have that u(x, t) of equations (6.6a)–(6.6d) amounts to the N-soliton solution of KdV equation 3 1 ut = uux + uxxx . 2 4
(6.8)
The corresponding wave function (t1 , . . . , tN ; z) is often referred to as the so-called Baker function of the associated linear problem. It is well known from the work of the Kyoto school that this Baker function can also be expressed in terms of the tau function by means of a formula originally due to Sato (see [S], [SS], [DKJM], [JM], [SW], [OSTT], and [Mo]). Our formulas in equations (6.7b) and (6.7c) are equivalent to the Sato formula (upon specialization to the solitonic regime of the KdV hierarchy) (see [BBS] and [DK]). In summary, we have derived the following theorem. Theorem 6.1 (Zero motion for the KdV hierarchy). Let u(t1 , . . . , tM ) be an N soliton solution of the KdV hierarchy. Then the Schrödinger equation 2 ∂t1 + u(t1 , . . . , tM ) − z2 ψ = 0 (6.9a) has a solution of the form M tm z2m−1 z − ζ1 (t1 , . . . , tM ) · · · z − ζN (t1 , . . . , tM ) , ψ(t1 , . . . , tN ; z) = exp m=1
(6.9b)
with the zeros ζ1 (t1 , . . . , tM ), . . . , ζN (t1 , . . . , tM ) moving in accordance with the equations of motion (6.2) for the rational Ruijsenaars-Schneider hierarchy with harmonic term. Reversely, if a Schrödinger equation (6.9a) has a solution of the form (6.9b) with the zeros ζ1 (t1 , . . . , tM ), . . . , ζN (t1 , . . . , tM ) moving in accordance with the rational Ruijsenaars-Schneider hierarchy with harmonic term (6.2), then the potential u(t1 , . . . , tN ) (= 2∂t1 (ζ1 (t1 , . . . , tN ) + · · · + ζN (t1 , . . . , tN ))) constitutes an N-soliton solution of the Korteweg–de Vries hierarchy. Theorem 6.1 describes a relation between the KdV hierarchy and the rational Ruijsenaars-Schneider system with harmonic term. Different relations between the motion of KdV solitons and finite-dimensional integrable systems have been considered previously by Moser [M4] (who connects the KdV solitons to the motion of a Neumann system on a sphere) and by Ruijsenaars and Schneider [RSc] (who encode the KdV solitons in terms of the motion of a hyperbolic Ruijsenaars-Schneider system). Our considerations produce a formula for the solitonic tau function and Baker function of the KdV hierarchy in terms of the Lax matrix of the hyperbolic Ruijsenaars-Schneider system (cf. Subsection 4.1). In the case of the tau function,
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such a formula was obtained previously by Ruijsenaars and Schneider [RSc]. Similar formulas for the tau function and the associated Baker function for the rational solutions of the full KP hierarchy in terms of rational Calogero-Moser Lax matrices were presented recently by Shiota [Sh] (for the tau function) and Wilson [Wi] (for the Baker function). Appendices Appendix A. Proof of the no-crossing Lemma 2.2. In this appendix we prove Lemma 2.2, which says that a real-analytic, nonconstant solution to the differential system ζj + 2ζj ζj
2ζj ζk
=
1≤k≤N, k=j
ζj − ζ k
,
j = 1, . . . , N,
(A.1)
is such that ζj (x) = ζk (x) for all x ∈ R and j = k. In other words, the solution functions ζ1 (x), . . . , ζN (x) do not cross each other. To prove the lemma we demonstrate that the assumption that crossings do occur entails a contradiction. Without restriction one may assume that the crossing occurs for x = 0 and that the solution functions ζj (x) are distinct for x ∈] − ., .[\{0} with . > 0 sufficiently small. (Two (or more) solution functions cannot be identical as analytic functions of x because then the differential system (A.1) would require them to be constant.) More precisely, we may assume that ζj (x) = ζk (x),
1 ≤ j < k ≤ N, for x ∈] − ., .[\{0}
and, furthermore, that there is an index set J ⊂ {1, . . . , N } containing at least two elements such that ζj (0) = c,
j ∈ J,
and
ζj (0) = c,
j ∈ J.
By analyticity there exists an i ∈ J such that for all j ∈ J , |ζi (x) − c| ≤ |ζj (x) − c|
for x ∈] − δ, δ[
(A.2)
with 0 < δ < . sufficiently small. Basically, ζi (x) is the solution function that converges fastest to the crossing point c as x → 0. (Notice, however, that such a solution function may not be unique; indeed, in principle it might happen that we have two solution functions ζi (x), ζi (x) satisfying the estimate (A.2), which symmetrically approach the crossing point c from above and below such that ζi (x) + ζi (x) = 2c.) By analyticity and nonconstancy we have for j = 1, . . . , N that ζj (x) = ζj (0) +
x mj (mj ) ζ (0) mj ! j
+ o x mj ,
mj ≥ 1,
DYNAMICS OF ZEROS AND THE SATO FORMULA (mj )
with ζj
(0) = 0, and we have for j ∈ J that x nj (nj ) (n ) ζj (x) − ζi (x) = ζj (0) − ζi j (0) nj !
(n )
311
+ o x nj ,
(n )
with ζj j (0) = ζi j (0) and nj ≥ mi ≥ mj (cf. equation (A.2)). Armed with these Taylor expansions we determine the asymptotic behavior of the ith equation of the system (A.1). We get for the left-hand side: + o(1) if mi = 1, ζi (0) + 2ζi (0)ζi (0) ζi (x) + 2ζi (x)ζi (x) = (A.3) mi −2 (mi ) x ζi (0)/(mi − 2)! + o x mi −2 if mi ≥ 2, and for the right-hand side: 2 ζ (x)ζ (x) i
1≤k≤N, k=i
k
ζi (x) − ζk (x)
(mi ) (m ) m +m −2 ζi (0) ζk k (0) 2 x mi +mk −2 i k = +o x (mi − 1)!(mk − 1)! ζi (0) − ζk (0) k∈J 2 mk x mi −2 (m ) m −2 i i ζ (0) + o x − (mi − 1)! i
(A.4)
k∈J mk <mi
2 nk ! x 2mi −2−nk ζ (mi ) (0) ζ (mk ) (0) 2m −2−n i k i k +o x + . (n ) (n ) ((mi − 1)!)2 ζ k (0) − ζ k (0) k∈J, k=i mk =mi
i
k
We observe that it follows from our definition of ζi that (mk )
ζk
(0)
(n ) (n ) ζi k (0) − ζk k (0)
0 and ζi k (0) < (n ) (m ) (n ) ζk k (0), (ii) ζk (x) < ζi (x) < c or ζk (x) < c < ζi (x) ⇒ ζk k (0) < 0 and ζi k (0) > (n ) ζk k (0). By comparing the lowest-order behavior of the left-hand side and right-hand side for x → 0—using the inequality (A.5) and nk ≥ mi —one concludes that the righthand side contains nonzero terms of order O(x mi −2 ) and O(x 2mi −2−nk ) originating from the third and fourth lines of equation (A.4). These terms do not match with the O(1) or O(x mi −2 ) terms on the left-hand side given by equation (A.3). (The signs of the O(x mi −2 ) terms on both sides are not compatible.) We conclude that the set J must be empty, that is, that the solution functions cannot cross each other without violating our analyticity or nonconstancy assumption.
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VAN DIEJEN AND PUSCHMANN
Appendix B. Some Cauchy matrix identities. In this appendix we have collected some matrix identities associated to the well-known Cauchy matrix. These identities were essential in the construction of the action-angle transformation for the rational Ruijsenaars-Schneider system with harmonic term presented in Section 3. Most of the identities hinge on the following partial fraction decomposition for rational functions with simple poles that are regular at infinity. Lemma B.1 (Partial fractions decomposition). Let a1 , . . . ,aM and b1 , . . . , bN , with M ≤ N, be M + N points in the complex plane such that bn = bl if n = l. Then one has that N 1≤m≤M (x − am ) 1≤m≤M (bl − am ) = δN,M + (x − bl ) 1≤n≤N, n=l (bl − bn ) 1≤n≤N (x − bn ) l=1
as a rational identity in x. Proof. This is immediate from Liouville’s theorem upon inferring that the residues of the simple poles at x = bn , n = 1, . . . , N , and the asymptotics for x → ∞ are the same on both sides of the equation. Let α = (α1 , . . . , αN ) and β = (β1 , . . . , βN ) denote (complex) vectors with components subject to the genericity conditions: αj = αk , βj = βk for 1 ≤ j < k ≤ N, and αj = −βk for 1 ≤ j , k ≤ N. We form the matrices C(α, β) and D(α, β) with elements given by 1 1≤l≤N (αj + βl ) Cj,k (α, β) = , Dj,k (α, β) = δj,k , (B.1) αj + β k 1≤l≤N, l=j (αj − αl ) 1 ≤ j , k ≤ N . The matrix C(α, β) is commonly referred to as the Cauchy matrix. The following proposition provides a formula for the inverse of the Cauchy matrix. Proposition B.2 (Inverse of Cauchy matrix). We have that C−1 (α, β) = D(β, α)C(β, α)D(α, β) as a rational identity in the components of α and β. Proof. In components, the relation C(α, β)D(β, α)C(β, α)D(α, β) = 1 reads N l=1
(αk + βn ) 1 1 1≤m≤N (βl + αm ) 1≤n≤N = δj,k , αj + βl 1≤n≤N, n=l (βl − βn ) βl + αk 1≤m≤N, m=k (αk − αm )
1 ≤ j , k ≤ N. This identity follows from Lemma B.1 with M = N −1 via the substitutions x = αk , am = αm for m = 1, . . . , j − 1 and am = αm+1 for m = j, . . . , N − 1, and bn = −βn for n = 1, . . . , N .
DYNAMICS OF ZEROS AND THE SATO FORMULA
313
From the formula for the inverse of C(α, β) one obtains the famous Cauchy determinantal formula (and vice versa) (see [We, p. 202]). Corollary B.3 (Cauchy determinant). We have that 1≤j 0 in the sum should be deleted, and the factor of 1/2 in front of the sum should be replaced by 1/4. Acknowledgments. I would like to thank D. Allcock, E. Freitag, S. Kondo, I. Dolgachev, and D. Zagier for their help. This paper was written in part at the MaxPlanck-Institut für Mathematik in Bonn. 1. Notation and terminology. ˜ A metaplectic double cover of a group. √ √ The principal value of the square root, with −π/2 < arg( ∗) ≤ π/2. The dual of a lattice. A A discriminant form. An The An root lattice or the elements of order n in a discriminant form A. An The nth powers of elements of A. 2 (R). A discrete subgroup of SL Dn The Dn root lattice. The Dedekind delta function η24 . eγ An element of a basis of C[L /L]. e e(x) = exp(2π ix), en (x) = exp(2πix/n). Ek An Eisenstein series (see Section 10) or the Ek root lattice. η The Dedekind eta function. θL A theta function of a lattice L. g The genus of a subgroup of SL2 (R). I Im,n The even unimodular lattice of dimension m + n and signature m − n. L An even lattice. N The level of a modular form or discriminant form. q n e2πinτ or a discriminant form. Q The rational numbers. R The real numbers. Rj The primitive elliptic element fixing the point j . 2 (Z). ρL A representation of SL sign The signature of a lattice or discriminant form. SL A special linear group. Ta/c The primitive parabolic element fixing the cusp a/c. Tr The trace of something. τ A complex number with positive imaginary part or the number of orbits of cusps. χ A character. For χθ and χn , see Section 5. Z The integers.
322
RICHARD E. BORCHERDS
2. Modular forms. In this section we recall the definition of a vector-valued modular form and set up notation for the rest of the paper. We define e(x) to be exp(2πix), and we define en (x) to be exp(2πx/n). Recall 2 (R), whose elements can that the group SL2 (R) has a (metaplectic) double cover SL be written in the form √ a b , ± cτ + d , c d where ac db ∈ SL2 (R). The multiplication is defined so that the usual formulas for the transformation of modular forms work for half-integer weights, which means that
A, f (·) B, g(·) = AB, f B(·) g(·)
2 (Z) is for A, B ∈ SL2 (R) and for f, g as suitable functions on H . The group SL the inverse image of SL2 (Z) ⊂ SL (R) in SL (R). The group SL (Z) is generated 2 2 1 1 2 0 −1 , √τ , with S 2 = (ST )3 = Z, by S and T, where T = , 1 and S = 01 1 0 0 Z = −1 0 1 , i . The center is cyclic of order 4 and is generated by Z. The quotient by Z 2 is the group SL2 (Z). 2 (R) that contains Z and is cofinite (this Suppose that is a discrete subgroup of SL means that the quotient space has finite volume). Suppose that ρ is a representation of on a finite-dimensional complex vector space Vρ . Choose k ∈ Q. We define a modular form of weight k and type ρ to be a holomorphic function f on the upper half-plane H with values in the vector space Vρ such that f
aτ + b cτ + d
= (cτ + d)k ρ
a c
b √ , cτ + d f (τ ) d
√ for elements ac db , cτ + d of . The expression (cτ + d)k means, of course, √ 2k √ cτ + d with the principal value of . (We allow singularities at cusps.) A modular form has a Fourier expansion at the cusp at infinity as follows. The Fourier coefficients cn,γ ∈ C of f are defined by f (τ ) =
cn,γ q n eγ ,
n∈Q γ
where q n means e(nτ ) and where the sum runs over a basis eγ of Vρ consisting of eigenvectors of T . Note that n is not necessarily integral; more precisely, cn,γ is nonzero only if n ≡ λγ mod 1, where the eigenvalue of T on eγ is e(λγ ). We say that f is meromorphic at the cusp i∞ if cn,γ = 0 for n 0, and we say f is meromorphic at the cusp a/c if f ((aτ + b)/(cτ + d)) is meromorphic at i∞ for ac db ∈ SL2 (Z). We say that f is holomorphic at cusps if the coefficients of the Fourier expansions at all cusps vanish for n < 0.
REFLECTION GROUPS OF LORENTZIAN LATTICES
323
3. Discriminant forms and the Weil representation. In this section we recall the definition of the Weil representation of a discriminant form and prove some results about it that are used in Section 11. We let L be a nonsingular even lattice of dimension dim(L) and signature sign(L), with dual L . The quotient L /L is a finite group whose order is the absolute value of the discriminant of the lattice L. We define a discriminant form A to be a finite abelian group with a Q/Z-valued quadratic form γ → γ 2 /2. If L is an even lattice, then L /L is a discriminant form, with the quadratic form of L /L given by the mod 1 reduction of γ 2 /2, and conversely every discriminant form can be constructed in this way. (For the theory of the discriminant form of a lattice, see [Nik1].) This quadratic form on L /L determines the signature mod 8 of L, by Milgram’s formula γ 2 L sign(L) = e . e 2 L 8 γ ∈L /L
We define the signature sign(A) ∈ Z/8Z of a discriminant form to be the signature mod 8 of any even lattice with that discriminant form. We let the elements eγ for γ ∈ L /L be the standard basis of the group ring C[L /L], so that eγ eδ = eγ +δ . 2 (Z), called A particularly important example ρA of a unitary representation of SL the Weil representation of the discriminant form A, can be constructed as follows. The underlying space of ρA is the group ring C[A] of A, and the action is defined by (γ , γ ) eγ , ρA (T ) eγ = e 2 e − sign(A)/8 ρA (S) eγ = e − (γ , δ) eδ , √ |A| δ∈A 2 (Z). The representation ρA factors where S and T are the standard generators of SL 2 (Z/NZ) of the finite group SL2 (Z/NZ), where N is a through the double cover SL positive integer such that Nγ 2 /2 is an integer for all γ ∈ L . The smallest such integer N is called the level of A. In particular, the representation ρA factors through a finite 2 (Z). If L is an even lattice, then we define ρL to be the representation quotient of SL ρL /L . We summarize some results about discriminant forms from [CS, Chapter 15, Section 7]; for more details see [Nik1] or [CS]. We use a minor variation of the notation of [CS] for discriminant forms. We recall that every discriminant form can be written as a sum of Jordan components (not uniquely if p = 2), and every Jordan component can be written as the sum of indecomposable Jordan components (usually not uniquely). The possible nontrivial Jordan components are as follows. We let q > 1 be a power of a prime p and n a positive integer and t ∈ Z/8Z. We define antisquare by antisquare(q ±n ) = 0 if q is a square or the exponent is +n, and antisquare(q ±n ) = 1 if q is not a square and the exponent is −n. (See [CS, page 370].)
324
RICHARD E. BORCHERDS
For q odd, the nontrivial Jordan components of exponent q are q ±n for n ≥ 1. The indecomposable components are q ±1 , generated by an element γ with qγ = 0, γ 2 ≡ a/q mod 2, where a is an even integer with pa = ±1. The component q ±n is a sum of copies of q +1 and q −1 , with an even number of copies of q −1 if ±n = +n and an odd number if ±n = −n. These components all have level q. The signature is given by sign(q ±n ) = −n(q − 1) + 4 antisquare(q ±n ). For q even, the odd Jordan components of exponent q are qt±n . If n = 1, then t ≡ ±1 mod 8 if ± = + and t ≡ ±3 mod 8 if ± = −. If n = 2, then t ≡ 0, ±2 mod 8 if ± = + and t ≡ 4, ±2 mod 8 if ± = −. For any n, we have t ≡ n mod 2. The indecomposable components are qt±1 for 2t = ±1, and they are generated by an element γ with qγ = 0, γ 2 ≡ t/q mod 2. (Note that some of these are isomorphic to each other.) These components all have level 2q. The signature is given by sign(qt±n ) = t + 4 antisquare(q ±n ). For q even, the nontrivial even Jordan components of exponent q are q ±2n = qI±2n I . ±2 The indecomposable even Jordan components are q , which are generated by two elements γ and δ with qγ = qδ = 0, (γ , δ) = 1/q, γ 2 ≡ δ 2 ≡ 0 mod 2 if ± = +, γ 2 ≡ δ 2 ≡ 2/q mod 2 if ± = −. These components all have level q. The signature is given by sign(q ±n ) = 4 antisquare(q ±n ). The sum of two Jordan components with the same prime power q can be worked out as follows: We add the ranks, multiply the signs in the exponent, and if any components have a subscript t, we add together all subscripts t. If A is a discriminant form, then we define An to be the elements of order n. We define An to be the nth powers of elements of A, so that we have an exact sequence 0 −→ An −→ A −→ An −→ 0, and An is the orthogonal complement of An . We define An∗ to be the set of elements δ ∈ A such that (γ , δ) ≡ nγ 2 /2 mod 1 for all γ ∈ An , so that An∗ is a coset of An . We easily see that An is the same as An∗ if and only if the Jordan block of type 2k (where 2k || n) is even. In any case, An∗ always contains an element δ with 2δ = 0. Lemma 3.1. Suppose that A is a discriminant form. Then nγ 2 e (γ , δ) − 2
γ ∈A
is zero unless δ ∈ An∗ (in which case, it has absolute value
√
|A||An |).
Proof. The square of the absolute value of this sum is γ1 ,γ2 ∈A
nγ 2 nγ 2 e (γ1 , δ) − 1 − (γ2 , δ) + 2 2 2
=
γ1 ,γ2 ∈A
nγ 2 e (γ1 , δ) − 1 − n(γ1 , γ2 ) 2
REFLECTION GROUPS OF LORENTZIAN LATTICES
= |A|
325
nγ 2 e (γ1 , δ) − 1 . 2
γ1 ∈An
The map taking γ1 to e((γ1 , δ) − nγ12 /2) is a character of An , so this sum is zero unless this is the trivial character, in other words, unless δ ∈ An∗ . This proves Lemma 3.1. 2 (Z) has image a b ∈ SL2 (Z). Then ρA (e0 ) is Lemma 3.2. Suppose that g ∈ SL c d a linear combination of the elements eγ for γ ∈ Ac∗ . Proof. It is sufficient to prove this when g is of the form T m ST n S for some m, n ∈ Z with (N, n) = (N, c), because g is a product of an element of this form by an element of ˜ 0 (N ) (where N is the level of A) and because e0 is an eigenvalue of ˜ 0 (N ). 2 (Z). We We calculate the image of 1 = e0 ∈ C[A] under these elements of SL know that e − sign(A)/8 eγ . S(e0 ) = √ |A| γ ∈L /L
Applying T n shows that e − sign(A)/8 n(γ , γ ) eγ . e T S(1) = √ 2 |A| γ ∈A n
Applying S again shows that e − sign(A)/4 n(γ , γ ) ST S(1) = eδ . e (γ , δ) + |A| 2 n
δ∈A γ ∈A
Using Lemma 3.1, we see that the coefficient of eδ in this expression is zero unless δ ∈ Ac∗ . As all the elements eδ are eigenvectors of T m , the same is true for T m ST n S(e0 ). This proves Lemma 3.2. 4. The singular theta correspondence. We summarize some of the results from [B5]. The main idea is that we can use modular forms with poles at cusps to construct some automorphic forms with singularities. In particular, we can often use this to construct piecewise linear functions on hyperbolic space with singularities along the reflection hyperplanes of a reflection group, and this gives the connection between modular forms with singularities and nice hyperbolic reflection groups. If L is a lattice, then we define the Grassmannian G(L) to be the set of maximal positive definite subspaces of L⊗R. It is a symmetric space acted on by the orthogonal group OL (R).
326
RICHARD E. BORCHERDS
The Siegel theta function θL+γ of a coset L + γ of L in L is defined by
θL τ ; v
+
=
e
τ λ2v + 2
λ∈L+γ
+
τ¯ λ2v −
2
for τ ∈ H , v + ∈ G(L). We write 5L for the C[L /L]-valued function eγ θL+γ (τ ; v). 5L (τ ; v) = γ ∈L /L
2 (Z) Siegel’s transformation formula for 5L under SL given by b− /2 aτ + b a b+ /2 5L ρL cτ¯ + d v = (cτ + d) c cτ + d
(see [B5, Theorem 4.1]) is b √ , cτ + d 5L (τ ; v). d
We define 6(v, F ) by 6(v, F ) =
SL2 \H
¯ L (τ ; v)F (τ )y b+ /2−2 dx dy, 5
as in [B5, Section 6]. By [B5, Theorem 6.2], 6(v, F ) is an automorphic function of v ∈ G(L) whose only singularities are on points of the form γ ⊥ , for γ ∈ L , γ 2 < 0, where there is a nonzero coefficient cγ 2 /2,γ of F . Theorem 4.1. Suppose L is an even lattice of signature (2, b− ), and suppose F is a modular form of weight 1−b− /2 and of representation ρL which is holomorphic on H , meromorphic at cusps, and whose coefficients cλ (m) are integers for m ≤ 0. Then there is a meromorphic function 9L (ZL , F ) for Z ∈ P with the following properties. (1) 9L (ZL , F ) is an automorphic form of weight c0 (0)/2 for the group Aut(L, F ), with respect to some unitary character χ of Aut(L, F ). (2) The only zeros or poles of 9L lie on the rational quadratic divisors λ⊥ for λ ∈ L, λ2 < 0 and are zeros of order 0<x∈R xλ∈L
cxλ
x 2 λ2 2
(or poles, if this number is negative). (3) 9L is a holomorphic function if the orders of all zeros in (2) above are nonnegative. If, in addition, L has dimension at least 5 or if L has dimension 4 and contains no two-dimensional isotropic sublattice, then 9L is a holomorphic automorphic form. If, in addition, c0 (0) = b− −2, then 9L has singular weight so the only nonzero Fourier coefficients of 9L correspond to vectors of K of norm 0.
REFLECTION GROUPS OF LORENTZIAN LATTICES
327
This follows from [B5, Theorem 13.3]. If L is Lorentzian (in other words, if sign(L) = 2 − dim(L)), then the set of all one-dimensional positive definite subspaces of L is a copy of hyperbolic space of dimension dim(L) − 1. Theorem 4.2. Suppose M is a Lorentzian lattice of dimension 1 + b− . Suppose that F is a modular form of type ρM and of weight (1/2−b− /2, 0) which is holomorphic on H , meromorphic at cusps, and all of whose Fourier coefficients cλ (m) are real for m < 0. Finally, suppose that if cλ (λ2 /2) = 0 and λ2 < 0, then reflection in λ⊥ is in Aut(M, F, C). Then Aut(M, F, C) is the semidirect product of a reflection subgroup and a subgroup fixing the Weyl vector ρ(M, W, F ) of a Weyl chamber W . This is a special case of [B5, Theorem 12.1]. Both Theorems 4.1 and 4.2 depend on integrating the vector-valued modular form against a vector-valued theta function over a fundamental domain of SL2 (Z). In this paper, we usually start with a complex-valued modular form for 0 (N) rather than a vector-valued form as used in Theorems 4.1 and 4.2. There are two more or less equivalent ways to use these theorems on complex-valued forms of level N. First, instead of integrating a vector-valued form times the vector-valued theta function of a lattice over a fundamental domain of SL2 (Z), we can integrate a scalar-valued modular form times the theta function of a lattice over a fundamental domain of 0 (N ). Alternatively, we can first induce the complex-valued modular form for 0 (N) up to a vector-valued modular form for SL2 (Z) and then apply the theorems directly to part of this vector-valued form. For these constructions to work, it is necessary and sufficient for the complex-valued form to be a modular form for some character χ of ˜ 0 (N ), where the scalar-valued theta function of the lattice is a modular form of character χ and level sign(L)/2 for ˜ 0 (N). Several sections of this paper describe how to find such modular forms. Note that the singularities of the automorphic form associated to a level N modular form depend on all poles at all cusps of this form, not just the poles at i∞. Theorem 4.2 is very useful in practice for finding Lorentzian lattices with interesting reflection groups, because we just find lattices together with modular forms satisfying the conditions of the theorem. However, there is a problem with using it for theoretical purposes: It seems hard to give useful general conditions under which the Weyl vector is nonzero or has positive norm. If the Weyl vector happens to be zero, then of course Theorem 4.2 does not say anything. In practical examples, this does not matter because we can just check in each case to see whether the vector is zero (which does happen occasionally). Note the rather curious fact that in this paper we do not need to use the fact that Theorem 4.2 has been proved (or even that it is true!) because we are only using it to suggest interesting places to look for lattices, and whenever we find a lattice using Theorem 4.2, we still have to prove its properties directly because of the possibility that the Weyl vector is zero.
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RICHARD E. BORCHERDS
5. Theta functions. In this section, we work out the level and character of theta functions of even lattices. Most of the results are known, but there seems to be no convenient reference giving the results in the generality we require. Lemma 5.1. Suppose that N is a positive integer. If 4 N, then two characters of 0 (N ) are the same, provided that they have the same values on the elements such that c > 0, d > 0, and d ≡ 1 mod 4. If 4 | N, then two characters of ˜ 0 (N) are the same, provided that they have the same values on Z and on the elements such that c > 0, d > 0, and d ≡ 1 mod 4. Proof. It is sufficient to show that the images of the elements mentioned above generate 0 (N ). Suppose that ac db ∈ 0 (N). We show how to multiply it by powers of elements of the generating set above so that it becomes an element of the generating set, which proves the lemma. We first note that T is in the group generated by the set above, because the generating set is closed under left multiplication by T and is nonempty. If d is even, then c is odd so we can multiply it on the right by T so that d is odd; hence, we can assume that d is odd. Next we arrange that d ≡ 1 mod 4. If 4 | N, we multiply by Z if necessary so that d ≡ 1 mod 4. If 4 N, we multiply on the right by N1 01 , if necessary, to make c not divisible by 4 and then multiply on the right by a suitable power of T so that d ≡ 1 mod 4. We now have to make c and d positive 1 0 (without changing d mod 4). We multiply on the right by a suitable power of 4N 1 to make c positive without changing d. Finally, we multiply on the right by a suitable power of 01 41 to make d positive without changing c. The result is in the generating set, so this proves Lemma 5.1. We define the symbol dc for all pairs of coprime integers c and d as follows. The symbol is multiplicative in both c and d. If d is an odd prime, it is just the usual Legendre symbol. If d is 2, it is 1 if c ≡ ±1 mod 08 and−1 otherwise. If d = −1, it is 1 if c > 0 and −1 if c < 0. Finally, we define ±1 = ±1 0 = 1. We now define some characters χn (for n a positive integer) and χθ of ˜ 0 (N). We suppose that if p is an odd prime occurring an odd number of times in the prime factorization of n, then it divides N. Also suppose that if 2 occurs an odd number of times in the prime factorization of n, then 8 divides N. We define the character χn of 0 (N ) by d a b = . χn c d n
2 Lemma 5.2. Let θA1 = n∈Z q n be the theta function of the A1 lattice. There is a (unique) character χθ of the metaplectic double cover of 0 (4) such that θA1
aτ + b cτ + d
= χθ
a c
√ b √ , cτ + d cτ + dθA1 (τ ). d
REFLECTION GROUPS OF LORENTZIAN LATTICES
329
(In other words, θA1 is a modular form for ˜ 0 (4) of weight 1/2 and character χθ .) The values are given by c if d ≡ 1 mod 4, ± √ d a b , ± cτ + d = χθ c d c if d ≡ 3 mod 4. ±(−i) d In particular, χθ (Z) = −i. Proof. This follows from the theorem on page 148 of [Ko]. Lemma 5.3. Suppose 4 | N . Then the kernel of the character χθ of 0 (N) maps isomorphically onto 1 (4)∩0 (N), and if we identify the kernel with this image, then 0 (N ) is the product (1 (4) ∩ 0 (N)) × Z/4Z (where Z/4Z is its center, generated by Z). The lifting of 1 (4) ∩ 0 (N) to ˜ 0 (N) is given by c √ a b a b , −→ cτ + d . c d c d d Proof. This follows immediately from Lemma 5.2 because χθ is a character whose values are ±1 or ±i, χθ (Z) = −i, and 0 (N) is the product of its center of order 2 (generated by Z) and the subgroup 1 (4) ∩ 0 (N). This proves Lemma 5.3. We define the group 02 (N ) to be the subgroup of 0 (N) of elements whose diagonal entries are squares in Z/nZ. If 4 N, then 02 (N) is the intersection of the kernels of the characters χp for p an odd prime dividing N . If 4 | N, then 02 (N) can be lifted to a subgroup of ˜ 0 (N ), as in Lemma 5.3, and is the intersection of the kernels of the characters χθ and χp of ˜ 0 (N) for p a prime dividing N/4. Theorem 5.4. Suppose that A is adiscriminant form of level dividing N. If b √ 2 (Z) acts on the Weil and c are divisible by N , then g = ac db , cτ + d ∈ SL representation C[A] by g eγ = χA (g)eaγ , where χA is the character of ˜ 0 (N) given by −1 sign(A)+(|A| )−1 χ|A|2sign(A) χ θ χA = χ |A|
if 4 | N, if 4 N.
Proof. First assume that A has even signature. Choose an even lattice in a positive definite space with discriminant form A. Then [Eb, Corollary 3.1] and the discussion on [Eb, page 94] show that (−1)sign(A)/2 |A| a b eγ = eaγ , c d d
330
RICHARD E. BORCHERDS
provided that d is odd and positive. By Lemma 5.1, it is sufficient to check that this is equal to the value of |A|χA when d g = Z and when d ≡ 1 mod 4, d > 0. But in the latter case −1 = 1 and = d d |A| , so this has the same character values as χ|A| . Also if sign(A)
(−1/A)−1
d ≡ 1 mod 4 and sign(A) is even, then χ2sign(A) , χθ , and χθ are all 1 on the element g. Therefore, the two characters coincide on elements with d ≡ 1 mod 4. As Z(eγ ) = (−i)sign(A) e−γ , we see that χA (Z) = (−i)sign(A) . We now check that −1 = the characters are equal on the element Z. If 4 N, this follows from χ|A| (Z) = |A| sign(A)
(−1)sign(A)/2 . If 4 | N, this follows from χ2 (Z) = 1, χθ (Z) = (−i)sign(A) , and −1 −1 −1 1−( ) = (−i)1−(|A|) = χθ |A| (Z). This proves Theorem 5.4 when A has χ|A| (Z) = |A| even signature. We can do the case of odd signature very quickly by reducing it to the case of even signature as follows. If A has odd signature, then 4 | N , and the discriminant form A ⊕ 2 has determinant 2|A| (where 2 is the discriminant form of the A1 lattice and has order 2). Theorem 5.4 for the element γ ∈ A now follows from Lemma 5.2 and Theorem 5.4 applied to the element γ +0 ∈ A⊕ 2 . This proves Theorem 5.4.
6. Eta quotients. In this section we work out the levels and characters of some eta quotients. We use these results in Section 12 to construct examples of modular forms of given characters. The function η(tτ ) has a zero of order (t, c)2 /24t at the cusp a/c. 1/24 n Lemma 6.1 (Rademacher). Recall that η(τ ) = q n>0 (1−q ) is the Dedekind a b eta function. Suppose that c d ∈ SL2 (Z) with c > 0. Then √ aτ + b a b √ , cτ + d η = χη cτ + d η(τ ), c d cτ + d 2 (Z) with values given as follows: where χη is a character of SL √ a b , ± cτ + d χη c d d ± e24 − 3c + bd 1 − c2 + c(a + d) , c odd, c > 0; c −d 2 c odd, c < 0; ± −c e24 3c − 6 + bd 1 − c + c(a + d) , = c d odd, c ≥ 0; ± e24 3d − 3 + ac 1 − d 2 + d(b − c) , d −c ± e24 − 3d − 9 + ac 1 − d 2 + d(b − c) , d odd, c < 0. d
331
REFLECTION GROUPS OF LORENTZIAN LATTICES
Proof. The cases with c ≥ 0 follow from the theorem in [R, page 163]. The cases with c < 0 follow easily from the cases with c > 0 and the fact that χη (Z) = −i. This proves Lemma 6.1. Theorem 6.2. Suppose rδ for that we are given a positive integer N and integers
δ | N and |A| with |A|/ δ|N δ rδ a rational square. Suppose that (1/24) δ|N rδ δ and
(N/24) δ|N rδ /δ are both integers. Then
η(δτ )rδ
δ|N
is a modular form for ˜ 0 (N ) of weight k = 2k+(−1)−1 4 N and to χθ |A| χ22k |A| if 4 | N.
δ rδ /2
and of character equal to χ|A| if
Proof. By Lemma 5.1, it is enough to check that the characters are equal whenever c > 0 and d ≡ 1 mod 4 and also that they are equal on Z if 4 | N. If c > 0 and d ≡ 1 mod 4, then by Lemma 6.1 the character value is given by
e24
δ|N
rδ c/δ c ac 2 1 − d + d bδ − 3d − 3 + δ δ d
= e24 db
δ|N
rδ + 3(d − 1) δrδ + a − d − ad 2 c rδ δ
δ|N
δ|N
δ|N rδ rδ c δ|N δ × d d 2k c |A| = i (d−1)k d d 2k 2k d c d = . 2 d |A| If 4 N, then 2k is even so this is the value of the character χ|A| . If 4 | N , then this is 2k+(−1)−1 the value of χθ2k χ22k |A| , which is the same as the value of χθ |A| χ22k |A| because χθ = ±1 whenever d ≡ 1 mod 4. So both characters have the same value whenever c > 0 and d ≡ 1 mod 4. Finally, we have to check that both characters are equal on Z whenever 4 | N. This follows by the same argument used in Theorem 5.4. This proves Theorem 6.2. Theorem 6.2 generalizes some theorems of Newman (see [N1], [N2]), who did the case of weight 0 and trivial character.
332
RICHARD E. BORCHERDS
7. Dimensions of spaces of modular forms. In this section we recall the formulas for the dimensions of some spaces of modular or cusp forms associated to a 2 (R). For weight at least 2, the representation ρ of a discrete cofinite subgroup of SL dimension is given by either the Riemann-Roch theorem or the Selberg trace formula. More generally, if G is a group acting on A, then it also acts on the spaces of cusp forms and we calculate the character of these representations. These results are used in Sections 9 and 12. For weight 1/2 forms, Serre and Stark described an explicit basis as follows. Theorem 7.1. Suppose that χ is an even Dirichlet character mod N. Then a basis for the space of modular forms of weight 1/2 and character χθ χ for 0 (N) is given by the forms 2 ψ(n)q tn , n∈Z
where ψ is a primitive even character of conductor r(ψ), t is a positive integer such that 4r(ψ)2 t divides N, and χ(n) = ψ(n) Dn for all n coprime to N, where D is the √ discriminant of the quadratic field Q[ t]. (Note that ψ is determined by t and χ.) Proof. This is [SS, Theorem A, page 34]. The dimensions of spaces of holomorphic modular forms can all be worked out as follows. For weight less than zero, there are no nonzero forms, and weight 0 is trivial as these are just constants. For weight greater than 2, we can work out the dimension using the Selberg trace formula (see below) or the Riemann-Roch theorem; with a bit more care this also works for weight 2 (there are extra correction terms coming from weight 0 forms in this case). For weight 1/2, the Serre-Stark theorem gives an explicit basis, which can be used to do the case of weight 3/2 because the Selberg trace formula gives the difference of dimensions for weights k and 2 − k. This leaves the case of weight 1, which seems to be the hardest case to do. In general, weight 1 forms are closely related to odd two-dimensional complex representations of the Galois group of Q. Fortunately, for the low-level cases we are interested in, the weight 1 forms are usually easy to construct explicitly using Eisenstein series and theta series of two-dimensional lattices (mainly because the exotic Galois representations only occur for higher levels). Now we use the Selberg trace formula to find the dimensions of spaces of forms of weight at least 2. If X is a finite-order automorphism of a finite-dimensional complex vector space V with eigenvalues e(−βj ) for 1 ≤ j ≤ dim(V ) and 0 ≤ βj < 1, then we define
δ∞ (X) to be (1/2 − βj ) and we define δN (X) to be δ∞ (X) − dim(V )/2N. More generally, if g is an endomorphism of V commuting with the action of G, then we define δρ,∞ (X, g) to be 1 − βj Tr g | V e(βj )X , 2
REFLECTION GROUPS OF LORENTZIAN LATTICES
333
where the sum is over the distinct eigenvalues e(−βj ) of X, and we put δN (X, g) = δ∞ (X, g) − Tr(g)/2N. Lemma 7.2. If ρ is a representation of a group containing X on a finite-dimensional complex vector space and X N = 1, then
δN (X, g) =
δ∞ (X, g) =
1 N
Tr ρ X j g 0<j 2, then Ek is a modular form of weight k for 0 (1). If
d|N ad E2 (τ ) is a modular form of weight 2 for 0 (N).
d|N ad /d
= 0, then
Proof. These are just the usual Eisenstein series for SL2 (Z). Lemma 10.2. Assume that k ≥ 2 is integral, and let χ be a nonprincipal Dirichlet character mod N with χ (−1) = (−1)k . Then Ek (τ, χ) =
n≥1
q
n
d
k−1
d|n
n χ d
is a modular form of weight k and character χ for 0 (N). Proof. See [Mi, Theorems 7.1.3 and 7.2.12 and Lemma 7.1.1]. Lemma 10.3. Let χ be a nonprincipal Dirichlet character mod N with χ(−1) = −1. Then n 2 E1 (τ, χ ) = 1 + qn χ , L(0, χ) d n≥1
where L(0, χ) = −B1,χ = −
d|n
χ(n)
0 2 appear in the moduli space of K3 surfaces, possibly with some extra structure such as a B-field. The existence of a reflective form for these lattices
REFLECTION GROUPS OF LORENTZIAN LATTICES
345
appears to be significant in the corresponding moduli spaces as it is usually necessary to discard points of the Grassmannian that are orthogonal to norm −2 vectors. For signatures −8 and −16, we find the lattices I I1,9 and I I1,17 whose (arithmetic) reflection groups were first described by Vinberg [V1]. For signature zero we take f to be j (τ ) − 744 = q −1 + 196884q + · · · . We get an automorphic function for the lattice I I2,2 , which is more or less j (σ ) − j (τ ) in suitable coordinates. The corresponding Lie algebra is the monster Lie algebra. The reflection group is not very interesting as it is of order 2 (and is the Weyl group of the monster Lie algebra). N = 2. The discriminant form A has order 22n for some nonnegative integer n. The character χA is always trivial as A always has square order and signature divisible by 4. The group 0 (2) = 02 (2) has two cusps that can be taken as i∞ (of width 1) and 0 (of width 2). It has one elliptic point of order 2, which can be take as the point (1 + i)/2, fixed by 21 −1 −1 . Table 3 group
index ν2
0 (2)
3
1
ν3
ν∞
genus
0
2
0
Table 4 cusps width
η
0
2
116 2−8
i∞
1
1−8 216
zero weight 1
4
1
4
The ring of modular forms for 02 (2) = 0 (2) is a polynomial ring on the generators −E2 (τ ) + 2E2 (2τ ) = θD4 (τ ) = 1 + 24q + 24q 2 + O(q 3 ) of weight 2 with a zero at the elliptic point, and E4 (τ ) of weight 4. The Hilbert function is 1/(1 − x 2 )(1 − x 4 ). All poles of order at most 1 at cusps are reflective by Lemma 11.2 as N = 2 is square-free. If A = I I (2−2 ), then A has no nonzero elements of norm 1 mod 2, so a pole of order 2 at the cusp 0 is also reflective. (There are also other possible reflective singularities for lattices of high 2-rank.) By looking at the form 2+ (τ )−1 , with order 1 poles at all cusps, we see that all level 2 even lattices of signature at least −16 have reflective modular forms. The Lorentzian lattices I I1,17 (2+8 ) and I I1,17 (2+10 ) have norm 0 Weyl vectors; their reflection groups are not arithmetic but are similar to the case of I I1,25 . The remaining Lorentzian lattices of dimension at most 18 have positive norm Weyl vectors, so their reflection groups are arithmetic. The lattice I I1,17 (2+2 ) is the even sublattice of an odd unimodular lattice (see [V1]). The arithmetic reflection group of the lattice
346
RICHARD E. BORCHERDS
I I1,17 (2+4 ) appeared recently in Kondo and Keum’s work [KK] as the Picard lattice of the Kummer surface of a generic product of elliptic curves, and it can be obtained as the orthogonal complement of a D42 in I I1,25 . The reflection group of I I1,17 (2+6 ) seems to be the highest dimension of a “new” example of an arithmetic reflection group in this paper. This lattice has an unusually complicated fundamental domain, with 896+64 sides. It can be described as follows. Let K be the 16-dimensional even lattice in the genus I I0,16 (2+6 ) that has root system A16 1 . It can be constructed by applying construction A of [CS, Chapter 5] to the first-order Reed-Muller code of length 16 with 32 elements (see [CS, page 129], which uses construction B applied to this code to construct the Barnes-Wall lattice). Then L = I I1,17 (2+6 ) can be constructed as K ⊕ I I1,1 . The fundamental domain of the reflection group R of L has two norm 0 vectors z, z of type K (together with many other norm 0 vectors of other lattices). The generalized Weyl vector is given by ρ = (z + z )/2. The fundamental domain has 64 walls corresponding to norm −2 roots of L and 896 walls corresponding to norm −4 roots of L (or equivalently to norm −1 roots of L ). The norm −2 simple roots split into two groups of 32. The first group of 32 have inner product zero with z and −1 with z and correspond to the sixteen coordinate vectors of K and their negatives. The other 32 have inner product −1 with z and zero with z and correspond to the 32 elements of the Reed-Muller code. (There is, of course, an automorphism of the fundamental domain exchanging z and z and the two groups of 32 norm −2 simple roots.) The 896 norm −1 simple roots of L all have inner product 1 with both z and z and correspond to the elements of the dual of the Reed-Muller code whose weight is 2 mod 4. The Reed-Muller code has Hamming weight enumerator x 16 +30x 8 y 8 +y 16 ; so by the MacWilliams identity (see [CS, page 78]), the dual code has weight enumerator (x + y)16 + 30(x + y)8 (x − y)8 + (x − y)16 32 = x 16 + 140y 4 x 12 + 448y 6 x 10 + 870y 8 x 8 + 448y 10 x 6 + 140y 12 x 4 + y 16 , and therefore there are 448 + 448 = 896 elements of length 2 mod 4. Alternatively, L can be constructed as the orthogonal complement of a certain A81 in the Dynkin diagram of I I1,25 . (Note that the Dynkin diagram of I I1,25 contains more than one orbit of subsets isomorphic to A81 . The orbit we use has the special property that it is not contained in an A71 A2 subdiagram.) The 64 norm −2 roots correspond to the 64 ways to extend the A81 to an A91 , and the 896 norm −1 simple roots correspond to the 896 ways of extending it to an A3 A61 diagram. These A3 A61 diagrams are the same as those used by Kondo in [Kon] to describe the automorphism group of a generic Jacobian Kummer surface. The automorphism group of the Dynkin diagram of L has a normal subgroup of order 210 , and the quotient is the alternating group A8 . Unfortunately, I I1,17 (2+6 ) cannot be the Picard lattice of a K3 surface: Kondo pointed out to me that its 2-rank (6) is larger than its codimension (4) in I I3,19 .
REFLECTION GROUPS OF LORENTZIAN LATTICES
347
The lattices I In,n+20 (2−2 ) also have a reflective modular form θD4 (τ )/(τ ). In particular, the Lorentzian lattice I I1,21 (2−2 ) has an arithmetic reflection group (see [B1, page 149] for a description of its fundamental domain). Esselmann [Es] showed that this is essentially the only example of a cofinite reflection group of a Lorentzian lattice of dimension at least 21. (Of course, we can find trivial variations of this example by taking the Atkin-Lehner conjugate I I1,21 (2−20 ) or by multiplying all norms by a constant.) We can find some automorphic forms of singular weight, corresponding to lattices L and reflective modular forms f as follows. Case 1: L = I I2,18 (2+10 ), f = η1−8 2−8 . This is the denominator function of a generalized Kac-Moody algebra of rank 18. This example is related to the element 2A of Aut(J), of cycle shape 18 28 . There are 24 lattices in the genus I I0,16 (2+8 ) by [SV], one of which is the Barnes-Wall lattice, and the others all have root systems of rank 16. Case 2: L = I I2,10 (2+2 ), f = 24 η1−16 28 . This example is related to the element −2A of Aut(J). Case 3: L = I I2,10 (2+10 ), f = η18 2−16 . Case 1 is closely related to the reflection group of the lattice I I1,17 (2+8 ), whose fundamental domain has a nonzero norm 0 vector fixed by its automorphism group, as in the lattice I I1,25 . For more about this lattice and its reflection group, see [B2]. The last two cases are really the same, since the lattices are Atkin-Lehner conjugates of each other, and the automorphic forms we get are more or less the same. This automorphic form is the denominator function of two generalized Kac-Moody superalgebras, and it is also closely related to the moduli space of Enriques surfaces. See [B5, Example 13.7], [Sch], and [B4] for more details. If R is the reflection group of the lattice I I1,17 (2+2 ) generated by the reflections of norm −2 vectors and if D is its fundamental domain, then Aut(D) has a finite-index subgroup isomorphic to Z and fixes a nonzero norm 0 vector z. However, there seems to be no reflective modular form for 0 (2) corresponding to this reflection group. The remaining level 2 cases of signatures −4, −8, and −12 are left to the reader; they all give known arithmetic reflection groups, often associated to unimodular lattices as in [V1]. Table 5 group 0 (3)
index ν2 4
0
ν3
ν∞
genus
1
2
0
N = 3. The character χ3 is trivial for forms of even weight and nontrivial for forms of odd weight. The forms of integer weights and arbitrary character are the same as the forms for 1 (3) and trivial character. Note that 0 (3) is the product of 1 (3) and its center of order 2 is generated by Z.
348
RICHARD E. BORCHERDS
Table 6 cusps width
η
0
3
19 3−3
i∞
1
1−3 39
zero weight 1
3
1
3
The ring of modular forms for 02 (3) = 1 (3) is a polynomial ring generated by θA2 (τ ) = E1 (τ, χ3 ) = 1 + 6q + 6q 3 + 6q 4 + O(q 7 ) of weight 1 and by E3 (τ, χ3 ) = η(τ )−3 η(3τ )9 = q + 3q 2 + 9q 3 + 13q 4 + 24q 5 + O(q 6 ) of weight 3. The Hilbert function is 1/(1 − u3 x)(1 − u3 x 3 ). Next we find some reflective singularities. At the cusp i∞, the singularity q −1 is reflective. At the cusp 0, the singularity q3−1 is reflective. If the discriminant form A is I I (3±1 ) or I I (3−2 ), then A has no nonzero elements of norm 0 mod 2, so the singularities q3−1 , q3−3 are reflective at the cusp zero. The forms 5A2 (τ )n /3+ show that all level 3 even lattices of signature at least −12 have nonzero reflective modular forms. There are also some other examples of reflective modular forms for lattices of small 3-rank. If we take the signature to be −18 and take A to be I I (3+1 ), then the form 5E6 (τ )/(τ ) is reflective. This can be used to show that the reflection group of the lattice I I1,19 (3+1 ) is arithmetic. This reflection group was first found by Vinberg [V2] in his investigations of the “most algebraic” K3 surfaces. The lattice can also be constructed as the orthogonal complement of an E6 in I I1,25 , and this gives another proof that the reflection group is arithmetic (see [B1]). Next take the signature to be −14, and take A to be I I (3−1 ). We let f be the form E1 (τ, χ3 )5 − 270E1 (τ, χ3 )2 η(τ )−3 η(3τ )9 = q −1 − 216 − 9126q + O q 2 . (τ ) The constant 270 is chosen so that the coefficient of q −2/3 = q3−2 of f (−1/τ ) = −i3−5/2 τ 3 (−9q −1 + 810q −1/3 + 1944 + 53136q 2/3 + O(q)) vanishes. So the automorphic forms with singularities constructed from f have all their singularities orthogonal to roots. However, something unexpected now happens: The C[A]-valued modular form induced from f is identically zero! So the piecewise linear automorphic forms, constructed from f as in Theorem 4.2, have no singularities and are also zero. In spite of this, the reflection group of L = I I1,15 (3−1 ) still has a nonzero vector (of norm 0) in the fundamental domain fixed by the automorphism group of the fundamental domain. To see this, we represent L as the orthogonal complement of an A2 in I I1,17 . Then the quotient of Aut + (L) by the reflection group can be worked out using [B6, Theorem 2.7], and it turns out to be an infinite dihedral group, which has an index 2 subgroup isomorphic to Z. Next we can classify the primitive norm 0 vectors z of L, and we find that there are just two orbits, with the lattice z⊥ /z having
349
REFLECTION GROUPS OF LORENTZIAN LATTICES
root systems E8 E6 and D13 . Fix ρ to be a primitive norm 0 vector corresponding to a lattice with root system D13 . As D13 has a rank of one less than the corresponding lattice, there is a group Z of automorphisms of the fundamental domain fixing z. This group Z has finite index in the full automorphism group of the fundamental domain, so the full automorphism group of the fundamental domain must fix z. However, z is not quite a Weyl vector, as it has zero inner product with some simple roots (forming an affine D13 Dynkin diagram) and has inner product 1 with the others. There are 10 lattices in the genus I I0,12 (3+6 ) (see [SV]). One is the Coxeter-Todd lattice with no roots, and the others all have root systems of rank 12. There are also some automorphic forms of singular weight corresponding to the following lattices and reflective forms: (1) I I2,14 (3−8 ), η1−6 3−6 ; (2) I I2,8 (3+7 ), η13 3−9 ; (3) I I2,8 (3+3 ), 32 η1−9 33 . The lattice I I2,8 (3+5 ) appears in [ACT], where it is the underlying integral lattice of a unimodular Eisenstein lattice. The automorphic forms for this case have been studied in great detail by Freitag in [AF] and [Fr]. In particular, there is one of weight 12 (coming from the function 27η1−9 33 ) whose restriction to the complex hyperbolic space CH 4 vanishes (to order 3) exactly along the reflection hyperplanes of a certain complex reflection group related to the moduli space of cubic surfaces. So its cube root is an automorphic form of weight 4 with order 1 zeros along all the reflection hyperplanes (see [A]). Table 7 group 0 (4)
index ν2 6
0
ν3
ν∞
genus
0
3
0
Table 8 cusps
width characters
η
zero weight character
4
18 2−4
1
2
1/2
1
1−2 25 4−2
1/4
1/2
1/4 = i∞
1
2−4 48
1
2
1=0
χθ = −i
χθ
We have seen above that sometimes the Weyl vector of a reflective form unexpectedly vanishes because all the singularities just happen to cancel out. Another way that the Weyl vector can unexpectedly vanish is if the vectors corresponding to the singularities happen not to exist (usually when the p-rank of A is small). For example, for the lattice L = I I2,8 (3−1 ), the automorphic form is constant even though the vector-valued modular form has nontrivial singularities. The singularities of the
350
RICHARD E. BORCHERDS
vector-valued modular form imply that the automorphic form has zeros corresponding to all norm 4/3 vectors of L , but L happens to have no such vectors so the automorphic form is constant. N = 4. The group 0 (4) has three cusps that can be taken as i∞ (of width 1), 0 (of width 4), and 1/2 (of width 1). It has no elliptic points and has genus 0. The double cover of 0 (4) is the product of its center of order 4 (generated by Z) and a subgroup that can be identified with 1 (4). The ring of modular forms of integral or half-integral weight for 1 (4) is a polynomial ring generated by θA1 (τ ) = 1 + 2q + 2q 4 + O(q 9 ) of weight 1/2 and η(τ )8 η(2τ )−4 = 1 − 8q + · · · of weight 2. The Hilbert function is 1/(1 − uθ x 1/2 )(1 − x 2 ). The ideals of cusp forms vanishing at i∞, 0, or 1/2 are generated by η(2τ )−4 η(4τ )8 , η(τ )8 η(2τ )−4 , η(τ )−2 η(2τ )5 η(4τ )−2 . Note that the last function has a zero of order 1/4 at 1/2. The ideal of cusp forms of even weight is generated by 4+ (τ ) = η(2τ )12 of weight 6. If L is a unimodular positive definite lattice, then θL (2τ ) is a modular form for 1 (4). Next we find some reflective singularities. To reduce the number of cases to consider, we assume that A = L /L has exponent 2, so that A is I I (2t+n ) for some t and n. As usual, poles of order 1 are reflective singularities at the cusps i∞ and 0. At the cusp 1/2, poles of order 1/4 and 1/2 are reflective, because all elements of A2∗ have order 1 or 2. If the parity vector of A does not have norm 0 mod 2, then a pole of order 1 at 1/2 is also reflective. Finally, at the cusp 0, poles of order 1 or 2 are reflective, and if A has no nonzero vectors of norm 0 mod 2, then poles of order 4 are reflective. The form η(τ )−12 η(2τ )−2 η(4τ )4 shows that all level 4 exponent 2 lattices of signature −14 have nonzero reflective modular forms. By multiplying this form by θA1 (τ )n for n ≥ 1, we see that the level 4 lattices of signature at least −14 have nonzero reflective modular forms. We can find many examples of eta quotients that are eigenforms of Hecke operators by finding eta quotients with poles of order at most 1 at all cusps. This gives fifteen nonconstant examples as follows: η1−2 25 4−2 , η1−4 210 4−4 , η1−8 220 4−8 , η18 2−4 , η16 21 4−2 , η14 26 4−4 , η216 4−8 , η2−4 48 , η1−2 21 46 , η1−4 26 44 , η1−8 216 , η1−8 28 4−8 , η1−6 23 4−6 , η1−4 2−2 4−4 , η2−12 . The inverses of these forms are often reflective forms for various lattices. Note that the forms η1−6 215 4−6 , η12 211 4−6 , η1−6 211 42 , η12 27 42 are eigenfunctions of Hecke operators, but as they have a zero of order 3/4 at 1/2, their inverses do not usually give reflective automorphic forms (except for rather special discriminant forms). The form η12 27 42 is the highest-weight eta product I know of that is an eigenform and has nonintegral weight. Most of the time lattices of positive signature with reflective forms do not seem to be interesting, but there are some exceptions. For example, there is a reflective form for the lattice I I2,1 (21+1 ). The corresponding automorphic form is essentially
REFLECTION GROUPS OF LORENTZIAN LATTICES
351
E6 , which is the denominator function of a generalized Kac-Moody algebra. See [B3, Section 15, Example 2] for more details. The lattices I I1,19 (26+2 ), I I1,15 (22+2 ), I I1,11 (26+2 ) have reflective forms of type θDn /. They are the even sublattices of odd unimodular lattices and have cofinite reflection groups, as was first found by Vinberg [V1]. +1 The function θE7 / is a reflective form for the lattices I In,n+17 (2−1 ). In particular, +1 ) whose reflection group we find Nikulin’s example of the Lorentzian lattice I I1,18 (2−1 is arithmetic. This example can also be constructed as the orthogonal complement of an E7 in I I1,25 . Yoshikawa told me that he has used the automorphic forms coming from the modular forms η(τ )−8 η(2τ )8 η(4τ )−8 θA1 (τ )k to construct automorphic products (for odd unimodular lattices). These automorphic products are the squares of discriminant forms of various moduli spaces of “generalized Enriques surfaces” and can also be constructed using analytic torsion. Table 9 group
index ν2
0 (5)
6
2
ν3
ν∞
genus
0
2
0
Table 10 cusps width
η
zero weight
0
5
15 5−1
1
2
i∞
1
1−1 55
1
2
N = 5. The ring of modular forms for 02 (5) is not a polynomial ring but is generated by the three-dimensional space of weight 2 forms, which is spanned by η(τ )5 η(5τ )−1 , η(τ )−1 η(5τ )5 (of nontrivial character) and E2 (τ )−5E2 (5τ ) (of trivial character). The Hilbert function is (1 + x 2 )/(1 − u5 x 2 )2 . Remark. The ring of all modular forms of integral weight for 1 (5) is a polynomial ring generated by the weight 1 Eisenstein series 1+(3+i)(q +(1−i)q 2 +(1+i)q 3 − iq 4 + q 5 + O(q 6 )) and its complex conjugate. These correspond to the two complex conjugate order 4 characters of Z/4Z, and each of them has a simple zero at one of the elliptic points and no other zeros. The subring of forms of even weight is the ring of modular forms for 02 (5). Even lattices of level 5 all have signature divisible by 4. The form 5 (τ )−1 shows that all level 5 even lattices of signature −8 and even 5-rank have nonzero reflective modular forms. (So does the lattice of 5-rank 1; see below.) If we multiply this form by products of powers of E2 (τ ) − 5E2 (5τ ) and η(τ )5 η(5τ )−1 , we also see that all even level 5 lattices of signature at least −4 have nonzero reflective modular forms.
352
RICHARD E. BORCHERDS
For the discriminant forms A = I I (5±1 ) or I I (5+2 ), the only norm 0 element is zero, so q5−1 and q5−5 are all reflective singularities at the cusp 0. If we take A to be I I (5−1 ) and take the signature to be −8, then there is a reflective form. This gives a Lorentzian lattice I I1,9 (5−1 ) with a reflection group of finite index. If we take L to be I I1,17 (5−1 ), then there is a reflective automorphic form. (This is slightly surprising as the space of forms with a pole of order at most 1 at i∞ and a pole of order at most 5 at zero is two-dimensional, so we would normally expect there to be no such forms satisfying the two conditions that the coefficients of q5−2 and q5−3 both vanish. However, it turns out that these two conditions are not independent; in fact the modular form we get has “complex multiplication” (see [Ri]), meaning that the coefficient of q5n is zero whenever n ≡ 2, 3 mod 5.) In spite of the existence of a nonzero reflective modular form, the reflection group of I I1,17 (5−1 ) is not cofinite and does not even have virtually free abelian index. (In particular, this lattice is a counterexample to several otherwise plausible conjectures about Lorentzian lattices with nonzero reflective modular forms.) As a substitute for this, the lattice is very closely related to Bugaenko’s largest example of a cocompact hyperbolic reflection group. In fact I I1,17 (5−1 ) can be made into a lattice over Z[φ], and Bugaenko [Bu] showed that the corresponding hyperbolic reflection group was cocompact. The relationship between Bugaenko’s reflection group and the reflective form is rather mysterious. The lattice has five orbits of primitive norm 0 vectors, corresponding to the five elements of the genus I I0,16 (5−1 ), which have root systems A2 A14 , E7 A9 , E6 D9 , E8 E7 5A1 , D14 A1 5A1 . It is possible to produce some examples of cocompact hyperbolic reflection groups from level 5 lattices as follows. Lemma 12.1. Suppose that L is an even Lorentzian lattice of level 5, and suppose that there is a self-adjoint endomorphism φ of L such that φ 2 = φ +1. Let H φ be the hyperbolic space of the Lorentzian eigenspace (L ⊗ R)φ . Then the subgroup of the reflection of L acting on H φ is a hyperbolic reflection group of H φ . If W is cofinite, then W φ is cocompact. Proof. Let H be the hyperbolic space of L, and let H φ be the subspace of it fixed by φ. The main point is that the intersection of any reflection hyperplane of W with H φ is a reflection hyperplane of the group W φ acting on H φ . To see this, recall that a reflection of W is the reflection of a norm −2 vector of L or a norm −2/5 vector of L . First suppose that v is a norm −2 vector of L. As v ⊥ intersects H φ , v φ⊥ and v must generate a negative definite space. This easily implies that v and v ⊥ span a lattice isomorphic to A21 , and the product of two reflections of this lattice is the automorphism −1, which commutes with φ. This is a reflection of W φ acting on H φ whose reflection hyperplane is v ⊥ ∩H φ . The argument when v is a norm −2/5 vector of L is similar. It now follows that W φ is a reflection group acting on H φ whose fundamental domain is the intersection of H φ with a fundamental domain of W acting on H .
REFLECTION GROUPS OF LORENTZIAN LATTICES
353
Finally, if W is cofinite, then all norm 0 vectors in the fundamental domain of W are rational and therefore cannot be fixed by φ, so the fundamental domain of W φ has no norm 0 vectors in it and is therefore compact. This proves Lemma 12.1. Unfortunately, this lemma does not give the largest examples found by Bugaenko. If we take A to be I I (5+1 ) and take the signature to be −12, then q5−4 is a reflective singularity at zero as A has no nonzero elements of norm −4/5 mod 2, and q5−5 is reflective as any norm 0 element of A is zero. So A has a reflective modular form of weight −6, level 5, and character χ5 whose singularity at i∞ is a multiple of q −1 and whose singularity at zero is a linear combination of q5−1 , q5−4 , and q5−5 . (There is a two-dimensional space of such forms.) This gives a Lorentzian lattice L = I I1,13 (5+1 ) with Aut + (L)/R infinite dihedral. It is the orthogonal complement of an A4 in I I1,17 . This case is similar to I I1,15 (3−1 ). The (level 1) form E6 / has a singularity at zero of the form q −1 = q5−5 so it is a reflective modular form for the lattice L = I I1,13 (5−2 ). The corresponding vectorvalued modular form is zero. The lattice L is a module over Z[φ]. It may be the lattice of the orthogonal complement of an I2 (5) in Bugaenko’s lattice, which would imply that it has a cocompact reflection group. As in the case N = 3, we also get a few examples of automorphic forms of singular weight coming from the reflective forms η1−4 5−4 , η1−1 55 , and η15 5−1 . There are five lattices in the genus I I0,8 (5+4 ) corresponding to the case I I2,10 by [SV]. One has no roots and by [SH, page 744], the root systems of the other four are A41 5A41 , A22 5A22 , A4 5A4 , D4 5D4 . Problem. Does the lattice I I2,6 (5+3 ) correspond to some nice moduli space in the same way that the corresponding lattices I I2,10 (2+2 ) and I I2,8 (3+5 ), for levels 2 and 3, correspond to the moduli spaces of Enriques surfaces or cubic surfaces? Table 11 group 0 (6)
index ν2 12
0
ν3
ν∞
genus
0
4
0
Table 12 width
η
6
16 2−3 3−2 61
1
1
1/2
3
1−3 26 31 6−2
1
1
1/3
2
1−2 21 36 6−3
1
1
1/6 = i∞
1
11 2−2 3−3 66
1
1
cusps 1=0
zero weight
354
RICHARD E. BORCHERDS
N = 6. The group 0 (6) is the product of 1 (6) and its center of order 2 generated by Z. The forms of trivial character have even weight, and those of nontrivial character have odd weight. The ring of modular forms of integral weight for 0 (6) is a polynomial ring generated by E1 (τ, χ3 ) and E1 (2τ, χ3 ). The Hilbert function is 1/(1 − u3 x)2 . The ideal of cusp forms is generated by 6+ (τ ) = η(τ )2 η(2τ )2 η(3τ )2 η(6τ )2 of weight 4. We can find eta quotients with a given integral-order pole at the cusps. In particular, we find the following fifteen nonconstant holomorphic eta quotients with zeros of order at most 1 at all cusps: η16 2−3 3−2 61 , η1−3 26 31 6−2 , η1−2 21 36 6−3 , η11 2−2 3−3 66 , η13 23 3−1 6−1 , η1−1 2−1 33 63 , η14 2−2 34 6−2 , η1−2 24 3−2 64 , η17 2−5 3−5 67 , η1−5 27 37 6−5 , η11 24 35 6−4 , η14 21 3−4 65 , η15 2−4 31 64 , η1−4 25 34 61 , η12 22 32 62 . Their inverses give numerous examples of reflective forms for various lattices. For example, η1−2 2−2 3−2 6−2 is a reflective form for all even level 6 lattices of signature −8, and by multiplying by a power of E1 (τ, χ3 ) we get reflective forms whenever the signature is at least −8. The other eta quotients give many examples where roots of certain norms are excluded. We can also find many examples of signature less than −8 if we restrict the 2-rank or 3-rank to be at most 2. One example of such a lattice with an arithmetic reflection group is the orthogonal complement I I1,15 (2−2 3−1 ) of a D4 E6 root system in I I1,25 . Table 13 group
index ν2
0 (7)
8
0
ν3
ν∞
genus
2
2
0
Table 14 cusps width
η
0
7
17 7−1
i∞
1
1−1 77
zero weight 2
3
2
3
N = 7. The ring of modular forms of integral weight for 02 (7) is generated by E1 (τ, χ7 ) (of weight 1, which vanishes at both elliptic points), 7+ (τ ) (of weight 3, which vanishes at both cusps), and the two weight 3 Eisenstein series. The Hilbert function is (1 + u7 x 3 )/(1 − u7 x)(1 − u7 x 3 ). The ideal of cusp forms is generated by 7+ (τ ) = η(τ )3 η(7τ )3 of weight 3. Note that the ideal of forms vanishing at i∞ is not principal. The function η(τ )4 η(7τ )−4 is a Hauptmodul for 0 (7). Remark. The ring of modular forms for 1 (7) of integral weight has a simpler structure: it is generated by the three weight 1 forms E1 (τ, χ7 ), E1 (τ, χ), E1 (τ, χ), ¯ where χ is a character of Z/7Z of order 6 and χ¯ is its complex conjugate. The ideal of relations between these generators is generated by E1 (τ, χ7 )2 −E1 (τ, χ)E1 (τ, χ). ¯ We can even embed this into a polynomial ring of modular forms: All the zeros of
355
REFLECTION GROUPS OF LORENTZIAN LATTICES
the forms E1 (τ, χ ) and E1 (τ, χ) ¯ have order 2, so their square roots are also modular forms (of half-integral weight for a strange character of 0 (7)), and they generate a polynomial ring whose elements of integral weight are the modular forms for 1 (7). Any even lattice of level 7 and signature at least −6 has a reflective form of the form E1 (τ, χ7 )n η1−3 7−3 . The automorphic form associated to η1−3 7−3 and to the lattice I I2,8 (7+5 ) has singular weight and is the denominator function of a generalized Kac-Moody algebra. The reflection group of I I1,7 (7−3 ) has a norm 0 Weyl vector. The corresponding genus I I0,6 (7−3 ) has three elements [SH, Proposition 3.4a], and by [SH, Table 1] there is one with no roots (corresponding to the norm 0 Weyl vector), one with root system A3 7A3 , and one with root system A31 7A31 . For the discriminant form I I (7−1 ), the singularities q7−1 and q7−7 are reflective. The lattice I I1,11 (7−1 ) has a reflective form and is the orthogonal complement of an A6 in I I1,17 . The quotient Aut+ (L)/R is infinite dihedral and fixes a norm 0 vector corresponding to a lattice in the genus I I0,10 (7−1 ) with root system D9 . There is a second lattice in this genus, isomorphic to the sum of E8 , and a two-dimensional definite lattice of determinant 7, so its root system is E8 A1 7A1 . Table 15 group 0 (8)
index ν2 12
0
ν3
ν∞
genus
0
4
0
Table 16 cusps
width
1=0
8
1/2
2
1/4
1
1/8 = i∞
1
characters
η 14 2−2
zero weight character 1
1
χθ2
χ2 = χθ = −1 1−2 25 4−2
1/2
1/2
χθ
2−2 45 8−2
1/2
1/2
χ θ χ2
4−2 84
1
1
χθ2
χ2 = −1
N = 8. The double cover of 0 (8) is the product of its center of order 4 (generated by Z) and a subgroup that can be identified with its image 0 (8)∩1 (4). The ring of modular forms of integral or half-integral weight for 02 (8) = 1 (8) is a polynomial ring generated by η(2τ )−2 η(4τ )5 η(8τ )−2 (of weight 1/2 and character χ2 χθ ) and η(τ )−2 η(2τ )5 η(4τ )−2 (of weight 1/2 and character χθ ). The Hilbert function is 1/(1 − uθ u2 x 1/2 )(1 − uθ x 1/2 ). There are many level 8 discriminant forms and many possible reflective singularities. Together they give a bewildering number of examples of level 8 lattices with reflective forms; they are probably best left to a computer to classify. As examples, we
356
RICHARD E. BORCHERDS
just mention I I1,18 (47+1 ), I I1,16 (41+1 ), I I1,14 (43−1 ), I I1,12 (45−1 ). These are the even sublattices of some of the odd unimodular lattices with cofinite reflection groups found by Vinberg and Kaplinskaja [VK]. Table 17 group 0 (9)
index ν2 12
0
ν3
ν∞
genus
0
4
0
Table 18 width
η
1=0
9
13 3−1
1
1
1/3, 2/3
1
1−3 310 9−3
1,1
2
1
3−1 93
1
1
cusps
1/9 = i∞
zero weight
N = 9. The group 0 (9) is the product of its center of order 2 and the group so a form has nontrivial character if and only if it has odd weight. The ring of modular forms of integral weight for 02 (9) is a polynomial ring generated by η(τ )3 η(3τ )−1 and η(9τ )3 η(3τ )−1 . The Hilbert function is 1/(1 − x)2 . For the sake of completeness, we also describe generators for the ring of all integral weight modular forms for 1 (9). This is a three-dimensional free module over the ring of modular forms for 02 (9), with a basis consisting of 1 and the two weight 1 Eisenstein series for the two order 6 characters of Z/6Z. Each of these Eisenstein series has zeros of order 1/3 and 2/3 at the cusps 1/3 and 2/3 (not necessarily in that order). Note that the character of a modular form can be read off from the parity of its weight and the fractional part of the order of the zero at 1/3. We can embed this ring in a polynomial ring, generated by the cube roots of the two weight 1 modular forms with poles of order 1 at 1/3 or 2/3. The form η(τ )−3 η(3τ )2 η(9τ )−3 is a reflective form for all even level 9 lattices of signature −4, and by multiplying it by a suitable power of, say, θA2 (τ ), we get reflective forms whenever the signature is at least −4. A few examples of modular forms that might correspond to automorphic forms of singular weight on some lattices are η3−8 , η1−3 31 , η31 9−3 , and η1−3 32 9−3 .
02 (9),
N = 10. The ring of modular forms of integral weight for 02 (10) is generated by the seven-dimensional space of forms of weight 2, and the ideal of relations between them is generated by fifteen quadratic relations. The Hilbert function is (1 + 5x 2 )/(1 − x)2 . Remark. We can embed the ring of modular forms for 02 (10) in a polynomial ring as follows. The ring of modular forms of integral weight for 1 (10) is generated
REFLECTION GROUPS OF LORENTZIAN LATTICES
357
Table 19 index ν2
group 0 (10)
18
ν3
ν∞
genus
0
4
0
2
Table 20 width
η
10
110 2−5 5−2 101
3
2
1/2
5
1−5 210 51 10−2
3
2
1/5
2
1−2 21 510 10−5
3
2
1
11 2−2 5−5 1010
3
2
cusps 1=0
1/10 = i∞
zero weight
by the four-dimensional space of forms of weight 1. We can find two of these forms that have zeros of order 3 at the two elliptic points of order 2. Their cube roots are modular forms of weight 1/3 for characters of order 3 of 1 (10), and they generate a polynomial ring in two variables. The space of modular forms for 02 (10) can be identified with the polynomials of degree divisible by 6. The cube root of the product of any three of the forms above with zeros only at one cusp is a form with a zero of order 1 at three of the four cusps, and these four forms are a basis of the space of weight 2 forms with nontrivial character. Their inverses are the functions η1−1 2−2 5−3 102 , η1−2 2−1 52 10−3 , η1−3 22 5−1 10−2 , η12 2−3 5−2 10−1 . Any one of these four functions (of nontrivial character and weight −2) shows that any even level 10 lattice of signature −4 and odd 5-rank has a reflective form. There is also a weight −2 form of trivial character whose poles and zeros are a pole of order 1 at each cusp and order 1 zeros at the two elliptic points. (Construction: take a linear combination of the weight 2 Eisenstein series with trivial character that vanishes at two cusps (this automatically vanishes at the two elliptic points), then divide it by an eta product with order 2 zeros at these cusps and order 1 zeros at the other two cusps.) This is a reflective form for the even level 10 lattice of signature −4 with even 5-rank. So every even level 10 lattice of signature greater than or equal to −4 has a reflective form. N = 11. The ring of modular forms of integral weight for 02 (11) is generated by E1 (τ, χ11 ) (of weight 1, which vanishes at both elliptic points), 11+ (τ ) (of weight 2, which vanishes at both cusps), and the two weight 3 Eisenstein series. The Hilbert function is (1 + x 3 )/(1 − x)(1 − x 2 ). The ideal of cusp forms is generated by 11+ (τ ) = η(τ )2 η(11τ )2 of weight 2. The ideal of forms vanishing at i∞ is not principal. The function η(τ )12 η(11τ )−12 has a pole of order 5 at i∞, a zero of order 5 at zero, and is a modular function for 0 (11), showing that zero is a torsion point
358
RICHARD E. BORCHERDS
of order 5 on the modular elliptic curve of 0 (11). (In fact this point generates the subgroup of rational points on this elliptic curve.) Table 21 index ν2
group 0 (11)
12
ν3
ν∞
genus
0
2
1
0
Table 22 η
cusps width
zero weight
0
11
111 11−1
i∞
1
1−1 1111
5
5
5
5
The forms E1 (τ, χ11 )n η1−2 11−2 show that all even lattices of level 11 and signature at least −4 are reflective. The lattice I I1,7 (11−1 ) has a reflection group of infinite dihedral index in its automorphism group. The corresponding genus I I0,6 (11−1 ) contains just one lattice, which has root system D5 . Table 23 index ν2
group 0 (12)
24
0
ν3
ν∞
genus
0
6
0
Table 24 cusps
width characters
η
zero weight
12
16 2−3 3−2 61
2
1
1/2
3
1−6 215 32 4−6 6−5 122
2
1
1/3
4
1−2 21 36 6−3
2
1
1/4
3
2−3 46 61 12−2
2
1
1/6
1
12 2−5 3−6 42 615 12−6
2
1
1/12 = i∞
1
21 4−2 6−3 126
2
1
1=0
χθ = i
χθ = −i
N = 12. The ring of modular forms is generated by the forms θ(τ ), θ(3τ ), E1 (τ, χ3 ), and E1 (2τ, χ3 ), and the Hilbert function is (1 + x 1/2 + 2x)/(1 − x 1/2 )(1 − x). There are quite a lot of modular forms whose zeros are all zeros of order at most 1 at cusps: We can find forms with zeros of order 1/2 at 1/2 and 1/6 and an odd
REFLECTION GROUPS OF LORENTZIAN LATTICES
359
number of zeros at the other cusps, or we can find forms whose zeros at 1/2 and 1/6 have orders (0, 0), (1/4, 3/4), (3/4, 1/4), or (1, 1) and that have an even number of zeros at the other cusps. The maximum weight of these forms is 3, attained by the form η(2τ )3 η(6τ )3 with a zero of order 1 at every cusp. The inverses of these forms are reflective forms for many lattices of signature up to −6. Table 25 index ν2
group 0 (13)
14
ν3
ν∞
genus
2
2
0
2
Table 26 cusps width 0 i∞
η
zero weight
13
113 13−1
7
6
1
1−1 1313
7
6
N = 13. The Hilbert function is (1+2x 2 +6x 4 +5x 6 )/(1−x 2 )(1−x 6 ). The space of weight 2 forms for 02 (13) is three-dimensional, spanned by E2 (τ ) − 13E3 (13τ ), E2 (τ, χ13 ), and the cusp form E1 (τ, χ)2 − E2 (τ, χ) ¯ 2 , where χ is an order 4 char∗ acter of (Z/13Z) . The ring of modular forms for 1 (13) is generated by the sixdimensional space of forms of weight 1, which has a basis of the six forms E1 (τ, χ) as χ runs through the six odd characters of (Z/13Z)∗ . Each of these weight 1 Eisenstein series has a zero at an elliptic point of order 2, two zeros at elliptic points of order 3, and no other zeros. The function η12 13−2 is a Hauptmodul for 0 (13). There is also a Hauptmodul for 0 (13)+. These give automorphic forms for the lattice I I2,2 (13+2 ) which are the denominator functions for generalized Kac-Moody algebras related to elements of order 13 in the monster group. There are no modular forms of negative weight with poles of order at most 1 at the cusps, as can be seen from the relation index 7 number of zeros = weight × = weight × 12 6 and the fact that the weight is even and there are only two cusps. The space of cusp forms of weight 4 and character χ13 has dimension 2, and as this is the space of obstructions to finding a form of weight −2 and character χ13 with given singularities, we see that there is a nonzero form of weight −2 and character χ13 whose singularities are a pole of order 1 at i∞ and a singularity at zero with terms involving only q −1 and q −3 . This is a reflective form for the lattices I In,4+n (13+1 ). The lattice L = I I1,5 (13+1 ) is one of the lattices with Aut(L)/R(L) infinite dihedral. This is the orthogonal complement of an A12 in I I1,17 . There is a unique
360
RICHARD E. BORCHERDS
lattice in the genus I I0,4 (13+1 ), and it has root system D3 . This can be seen from the fact that all such lattices are the orthogonal complement of an A12 in an even 16-dimensional self-dual negative definite lattice. N = 14. The group 0 (14) is the product of 02 (14) and its center of order 2 generated by Z. Table 27 group
index ν2
0 (14)
24
0
ν3
ν∞
genus
0
4
1
Table 28 width
η
1=0
14
114 2−7 7−2 141
6
3
1/2
7
1−7 214 71 14−2
6
3
1/7
2
1−2 21 714 14−7
6
3
1/14 = i∞
1
11 2−2 7−7 1414
6
3
cusps
zero weight
The ring of modular forms of integral weight for 02 (14) is generated by E1 (τ, χ7 ), E1 (2τ, χ7 ), and 14+ (τ ). The Hilbert function is (1 + x 2 )/(1 − x)2 . The ideal of cusp forms is generated by 14+ (τ ) = η(τ )η(2τ )η(7τ )η(14τ ) of weight 2. We can construct some automorphic forms on lattices of signature −2 or −4 of singular weight from the modular forms η1−2 21 7−2 141 , η11 2−2 71 14−2 , and η1−1 2−1 7−1 14−1 . Table 29 group
index ν2
0 (15)
24
0
ν3
ν∞
genus
0
4
1
Table 30 width
η
15
115 3−5 5−3 151
8
4
1/3
5
1−5 315 51 15−3
8
4
1/5
3
1−3 31 515 15−5
8
4
1
11 3−3 5−5 1515
8
4
cusps 1=0
1/15 = i∞
zero weight
361
REFLECTION GROUPS OF LORENTZIAN LATTICES
N = 15. This case seems very similar to the case N = 14. The ring of modular forms of integral weight for 02 (15) is generated by E1 (τ, χ3 ), E1 (5τ, χ3 ), and 15+ (τ ). The Hilbert function is (1 + x 2 )/(1 − x)2 . The ideal of cusp forms is generated by 15+ (τ ) = η(τ )η(5τ )η(3τ )η(15τ ) of weight 2. We can construct some automorphic forms on lattices of signature −2 or −4 of singular weight from the modular forms η1−2 31 51 15−2 , η11 3−2 5−2 151 , and η1−1 3−1 5−1 15−1 . Table 31 group 0 (16)
index ν2 24
0
ν3
ν∞
genus
0
6
0
Table 32 η
zero
16
12 2−1
1
1/2
χθ
1/2
4
1−2 25 4−2
1
1/2
χθ
1/4, 3/4
1
2−2 45 8−2
1/2,1/2
1/2
χ 2 χθ
1
1/2
χθ
1
1/2
χθ
cusps 1=0
width characters
χ2 = −1
1/8
1
4−2 85 16−2
1/16 = i∞
1
8−1 162
weight character
N = 16. Note that the isomorphism class of a cusp a/c is no longer always determined by (c, 16). The ring of modular forms of integral or half-integral weight for 02 (16) is generated by the three-dimensional space of forms of weight 1/2, which is spanned by the five forms listed above. The Hilbert function is (1 + χ2 χθ x 1/2 )/(1 − χθ x 1/2 )2 . As in the case N = 8, there seem to be rather a lot of examples. Table 33 group 0 (17)
index ν2 18
2
ν3
ν∞
genus
0
2
1
N = 17. All modular forms for 02 (17) have even weight. The Hilbert function is (1 + (1 + 2u17 )x 2 + (3 + 4u17 )x 4 + x 6 )/(1 − x 2 )(1 − x 4 ). The group 0 (17)+ has genus 0, and its Hauptmodul q −1 + 7q + 14q 2 + O(q 3 ) has poles of order 1 at all cusps. The Hauptmodul with poles of order 1 at all cusps is a reflective modular form for lattices I In,n (17±2m ). For example, we get automorphic forms for the lattice
362
RICHARD E. BORCHERDS
I I2,2 (17+2 ). The automorphic form for I I2,2 (17+2 ) is the denominator function of a generalized Kac-Moody algebra associated with an element of order 17 of the monster group. We get similar statements if we replace 17 by any of the other primes p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, or 71 such that 0 (p)+ has genus 0. Table 34 group 0 (18)
index ν2 36
0
ν3
ν∞
genus
0
8
0
Table 35 width
η
1=0
18
16 2−3 3−2 61
3
1
1/2
9
1−3 26 31 6−2
3
1
1/3, 2/3
2
1−6 23 320 6−10 9−6 183
3,3
2
1/6, 5/6
1
13 2−6 3−10 620 93 18−6
3,3
2
1/9
2
3−2 61 96 18−3
3
1
1
31 6−2 9−3 186
3
1
cusps
1/18 = i∞
zero weight
N = 18. The Hilbert function is (1 + 2u3 x)/(1 − u3 x)2 , and the ring of modular forms is generated by the four-dimensional space of forms of weight 1, which is spanned by E1 (τ, χ3 ), E1 (2τ, χ3 ), E1 (3τ, χ3 ), and E1 (6τ, χ3 ). There are many eta quotients with poles of order 1 at all cusps: For any set of three or six cusps (with multiplicities) such that cusps with the same denominator have the same multiplicity, there is an eta quotient of weight 1 or 2 with these zeros. This gives twelve such eta quotients of weight 1 and eight of weight 2. There are four of weight 1 with no poles at the cusps 1/3, 2/3, 1/6, or 5/6, and the inverses of these are reflective forms for even level 18 lattices of signature −2. Table 36 group 0 (20)
index ν2 36
0
ν3
ν∞
genus
0
6
1
N = 20. The space of weight 1/2 forms is spanned by η1−2 25 4−2 (zero of order 5/4 at 1/2 and order 1/4 at 1/10) and η5−2 105 20−2 (zero of order 1/4 at 1/2 and order 5/4 at 1/10). The space of cusp forms of weight 2 and trivial character is spanned by η22 102 ; it has zeros of order 1 at all cusps.
363
REFLECTION GROUPS OF LORENTZIAN LATTICES
Table 37 cusps
η
width characters
zero weight
110 2−5 5−2 101
6
2
1−10 225 4−10 52 10−5 202
6
2
1=0
20
1/2
5
1/4
5
2−5 410 101 20−2
6
2
1/5
4
1−2 21 510 10−5
6
2
1/10
1
12 2−5 42 5−10 1025 20−10
6
2
21 4−2 10−5 2010
6
2
1/20 = i∞
χθ = −i
χθ = −i
1
The form 11 21 4−1 5−1 101 201 has zeros of order 1/2 at 1/2 and 1/10 and zeros of order 1 at 1 and 1/20. The form 1−1 21 41 51 101 20−1 has zeros of order 1/2 at 1/2 and 1/10 and zeros of order 1 at 1/4 and 1/5. Their inverses are reflective forms for even lattices of level 20, signature −2, and even 5-rank. Table 38 group
index ν2
0 (23)
24
ν3
ν∞
genus
0
2
2
0
Table 39 cusps width 0 i∞
η
zero weight
23
123 23−1
22
11
1
1−1 2323
22
11
N = 23. The ring of modular forms of integral weight for 02 (23) is generated by 23+ (τ ) = η(τ )η(23τ ), E1 (τ, χ23 ), and one of the two weight 3 Eisenstein series. The Hilbert function is (1 + x 3 )/(1 − x)2 . The ideal of cusp forms is generated by 23+ (τ ) = η(τ )η(23τ ) of weight 1. This is the lowest level for which there is a cusp form of weight 1. Remark. The modular function η(τ )12 η(23τ )−12 has a pole of order 11 at i∞ and a zero of order 11 at zero, and it shows that the cusp 0 gives a torsion point of order 11 on the modular abelian surface of 0 (23). See [Sh, page 197] for more about this. The forms E1 (τ, χ23 )n /23+ (τ ) show that all level 23 lattices of signature at least −2 have reflective forms. The automorphic form of the lattice I I2,4 (23+3 ) and of the function 1/23+ has singular weight and is the denominator function of a generalized Kac-Moody algebra. This generalized Kac-Moody algebra contains the Feingold-Frenkel rank 3
364
RICHARD E. BORCHERDS
Kac-Moody algebra as a subalgebra and can be used to explain why the root multiplicities of the Feingold-Frenkel algebra are often given by values of the partition function. See [Ni] for details. N = 28. The eta product 11 2−1 41 71 14−1 281 has weight 1, character χ7 , and its zeros are order 1 zeros at the cusps 1/1, 1/4, 1/7, and 1/28. (It is nonzero at the cusps 1/2 and 1/14.) So its inverse is a reflective form for any level 28 even lattice of signature −2 and odd 7-rank. N = 30. Here are some weight 1 eta quotients with order 1 zeros at six of the eight cusps: 21 31 51 6−1 10−1 301 (nonzero at 1/3, 1/5), 11 3−1 5−1 61 101 151 (nonzero at 1/6, 1/10), 1−1 21 31 51 15−1 301 (nonzero at 1/2, 1/30), 11 2−1 61 101 151 30−1 (nonzero at 1/1, 1/15). These are all modular forms for the character χ15 . So any even lattice of level 30 signature −2, and odd 5-rank has a reflective form. (The 3-rank is automatically odd for any such lattice.) The maximal order of an automorphism of the Leech lattice with fixed points is 30, and three of the eta quotients above occur as generalized cycle shapes of such order 30 automorphisms. There is no special reason for stopping at N = 30: There are hints that there might be examples for N up to a few hundred. 13. Open problems. We list a few suggestions for further research. Problem 13.1. Find some analogue of reflective forms for other sorts of hyperbolic reflection groups. In particular, explain why the complex hyperbolic reflection groups found by Allcock [A] have underlying integral lattices with nonzero reflective modular forms. Is this true of all complex hyperbolic reflection groups (except perhaps in small dimensions)? Is there a relation between the cocompact hyperbolic reflection groups found by Bugaenko [Bu] and lattices with reflective forms? Problem 13.2. Find some sort of converse theorem that implies that all “interesting” lattices of some sort have nonzero reflective forms. Bruinier [Br] has recently proved some related converse theorems, showing that certain sorts of automorphic infinite products always come from modular forms with singularities. Problem 13.3. Are there any other reflection groups of high-dimensional lattices with cofinite volume other than those listed in Section 12? Problem 13.4. Which of the rank 3 hyperbolic lattices classified by Nikulin [Nik2] have reflective forms? Note that in Part III of Nikulin’s papers, there are many examples where the Weyl vector has negative norm. Problem 13.5. Classify all holomorphic eta quotients whose zeros at cusps are all of order at most 1. This would be suitable for a computer. Problem 13.6. Write a computer program to classify the lattices such that the space of possible reflective singularities is greater than the dimension of the space of
REFLECTION GROUPS OF LORENTZIAN LATTICES
365
cusp forms that give obstructions to the existence of such a singularity. This would give a large class of examples of lattices with reflective forms, and the examples in Section 12 suggest that this would include most of them. Problem 13.7. The group 0 (242 ) has 48 cusps, all conjugate under its normalizer. The form 1/η(24τ ) has poles of order 1 at all cusps. Are there any lattices for which it is a reflective form? References [A] [ACT] [AF] [BBBCO]
[B1] [B2] [B3] [B4] [B5] [B6] [B7] [Br] [Bu]
[CS] [Eb] [Es] [F] [Fr] [KK]
D. Allcock, The Leech lattice and complex hyperbolic reflections, preprint, 1997, available from http://www.math.utah.edu/ ˜allcock. D. Allcock, J. A. Carlson, and D. Toledo, A complex hyperbolic structure for moduli of cubic surfaces, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 49–54. D. Allcock and E. Freitag, Cubic surfaces and Borcherds products, preprint, 1999, available from http://www.rzuser.uni-heidelberg.de/˜t91. C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier, User’s Guide to PARI-GP, guide and PARI programs available from ftp://megrez.math.ubordeaux.fr/pub/pari/. R. E. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), 133–153. , Lattices like the Leech lattice, J. Algebra 130 (1990), 219–234. , Automorphic forms on Os+2,2 (R) and infinite products, Invent. Math. 120 (1995), 161–213. , The moduli space of Enriques surfaces and the fake monster Lie superalgebra, Topology 35 (1996), 699–710. , Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491–562. , Coxeter groups, Lorentzian lattices, and K3 surfaces, Internat. Math. Res. Notices 1998, 1011–1031. , The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97 (1999), 219–233. J. H. Bruinier, Borcherds products and Chern classes of Hirzebruch-Zagier divisors, Invent. Math. 138 (1999), 51–83. V. O. Bugaenko, “Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices” in Lie Groups, Their Discrete Subgroups, and Invariant Theory, Adv. Soviet Math. 8, Amer. Math. Soc., Providence, 1992, 33–55. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 2d ed, Grundlehren Math. Wiss. 290, Springer, New York, 1993. W. Ebeling, Lattices and Codes: A Course Partially Based on Lectures by F. Hirzebruch, Adv. Lectures Math., Vieweg, Braunschweig, 1994. F. Esselmann, Über die maximale Dimension von Lorentz-Gittern mit coendlicher Spiegelungsgruppe, J. Number Theory 61 (1996), 103–144. J. Fischer, An Approach to the Selberg Trace Formula via the Selberg Zeta-Function, Lecture Notes in Math. 1253, Springer, Berlin, 1987. E. Freitag, Some modular forms related to cubic surfaces, preprint, 1999, available from www.rzuser.uni-heidelberg.de/˜t91. J. H. Keum and S. Kondo, The automorphism groups of Kummer surfaces associated with the product of two elliptic curves, preprint, 1998.
366 [Ko] [Kon] [M] [Mi] [N1] [N2] [Ni] [Nik1]
[Nik2]
[R] [Ri]
[SH] [SV] [Sch] [SS]
[Sh] [V1]
[V2] [VK]
RICHARD E. BORCHERDS N. Koblitz, Introduction to Elliptic Curves and Modular Forms, 2d ed., Grad. Texts in Math. 97, Springer, New York, 1993. ¯ The automorphism group of a generic Jacobian Kummer surface, J. Algebraic S. Kondo, Geom. 7 (1998), 589–609. Y. Martin, Multiplicative η-quotients, Trans. Amer. Math. Soc. 348 (1996), 4825–4856. T. Miyake, Modular Forms, Springer, Berlin, 1989. M. Newman, Construction and application of a class of modular functions, Proc. London. Math. Soc. (3) 7 (1957), 334–350. , Construction and application of a class of modular functions, II, Proc. London Math. Soc. (3) 9 (1959), 373–387. P. Niemann, Some generalized Kac-Moody algebras with known root multiplicities, Ph.D. thesis, Cambridge Univ., Cambridge, 1997. V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177, 238; English transl. in Math. USSR-Izv. 14 (1979), 103–167. , On the classification of hyperbolic root systems of the rank three, I, http://www.arXiv.org/abs/alg-geom/9711032; II, http://www.arXiv.org/abs/ alg-geom/9712033; III, http://www.arXiv.org/abs/math.AG/9905150. H. Rademacher, Topics in Analytic Number Theory, Grundlehren Math. Wiss. 169, Springer, New York, 1973. K. A. Ribet, “Galois representations attached to eigenforms with Nebentypus” in Modular Functions of One Variable, V (Bonn, 1976), Lecture Notes in Math. 601, Springer, Berlin, 1977; 17–51. R. Scharlau and B. Hemkemeier, Classification of integral lattices with large class number, Math. Comp. 67 (1998), 737–749. R. Scharlau and B. B. Venkov, The genus of the Barnes-Wall lattice, Comment. Math. Helv. 69 (1994), 322–333. N. R. Scheithauer, The fake monster superalgebra, http://www.arXiv.org/abs/math.QA/ 9905113, to appear in Adv. Math. J.-P. Serre and H. M. Stark, “Modular forms of weight 1/2” in Modular Functions of One Variable, VI (Bonn, 1976), Lecture Notes in Math. 627, Springer, Berlin, 1977, 27–67. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Kanô Memorial Lectures 1, Publ. Math. Soc. Japan 11, Iwanami Shoten, Tokyo, 1971. È. B. Vinberg, “Some arithmetical discrete groups in Lobaˇcevski˘ı spaces” in Discrete Subgroups of Lie Groups and Applications to Moduli (Bombay, 1973), Oxford Univ. Press, Bombay, 1975, 323–348. , The two most algebraic K3 surfaces, Math. Ann. 265 (1983), 1–21. È. B. Vinberg and I. M. Kaplinskaja, The groups O18,1 (Z) and O19,1 (Z) (in Russian), Dokl. Akad. Nauk SSSR 238 (1978), 1273–1275.
Department of Mathematics, University of California at Berkeley, Berkeley, California 94720-3840, USA; [email protected]; http://math.berkeley.edu/˜reb
Vol. 104, No. 3
DUKE MATHEMATICAL JOURNAL
© 2000
DIFFERENTIABILITY PROPERTIES OF ISOTROPIC FUNCTIONS MIROSLAV ŠILHAVÝ
1. Introduction. Let Sym denote the linear space of all symmetric second-order tensors on an n-dimensional real vector space Vect with scalar product. (If Vect is identified with Rn , then Sym may be identified with the set of all symmetric n-by-n matrices.) A function f : Sym → R is said to be isotropic if f (A) = f (QAQT ) for all A ∈ Sym and all Q proper orthogonals. An isotropic function has a representation f (A) = f˜(a), where f˜ is a symmetric function on Rn and a = (a1 , . . . , an ) are the eigenvalues of A with appropriate multiplicities. Clearly, f˜(a) = f (diag(a)) in any orthonormal basis, and thus if f is of class C r , r = 0, 1, . . . , ∞, then also f˜ is of class C r . Ball [1] showed that for r = 0, 1, 2, ∞, the converse is also true and conjectured that the converse is true for all r. This was subsequently proved by Sylvester [6] using complex techniques and detailed estimates of the derivatives of eigenvalues. Earlier, Chadwick and Ogden [2], [3] gave formulas for D r f , r = 1, 2, 3, in terms of f˜ and its derivatives assuming the differentiability (see also [1]). In this note, I derive the result of Sylvester by elementary means and give a recursive formula for D r f in terms of f˜ for arbitrary r. I also specialize these formulas to derive the forms of D r f , r = 1, 2, 3, which are equivalent to those by Chadwick and Ogden. 2. Notation. Throughout, the indices i, j, k range the interval {1, . . . , n}, unless stated otherwise. The direct vector notation is used in [4], [5]. In addition to the notation explained in the introduction, we recall that a second-order tensor A is a linear transformation from Vect into Vect, with the product of two tensors being the composition of the linear transformations. Furthermore, Orth+ denotes the proper orthogonal group, and Skew denotes the set of all skew tensors. By a basis in Vect, we always mean an orthonormal basis. Let Sn be the set of all real symmetric n-by-n matrices. Let ei be the canonical basis in Rr . All vector spaces are finite-dimensional and real. For a vector space X, we denote by Fr (X) the vector space of all symmetric rlinear forms F : X × · · · × X → R on X. The direct notation is used to denote the derivatives (differentials) of functions f defined on a vector space X with values in R. Thus for x ∈ X, the rth derivative D r f (x) is a symmetric r-form on X; that is,
Received 19 May 1999. Revision received 3 June 1999. 2000 Mathematics Subject Classification. Primary 74A20; Secondary 74B99. Author’s work supported by grant number 201/00/1516 of the Grant Agency of the Czech Republic. 367
368
MIROSLAV ŠILHAVÝ
D r f (x) ∈ Fr (X) and D r f : X → Fr (X). For each positive integer r and each class C r function f on X and x ∈ Rn , we denote by D [r] f (x) = (Df (x), . . . , D r f (x)). A function f : Rn → R is said to be symmetric if f (P w) = f (w) for every w ∈ Rn and every n-by-n permutation matrix P . We denote by CSr (Rn ) the set of all symmetric functions of class C r on Rn , and we denote by CIr (Sym) the set of all isotropic functions of class C r on Sym. Proposition 2.1. The function f : Sym → R is isotropic if and only if there exists a symmetric function f˜ : Rn → R, such that for each basis {ei } and each A ∈ Sym represented by A = diag(a), a ∈ Rn , f (A) = f˜(a).
(1)
The correspondence ˜: f → f˜ is one-to-one between isotropic functions on Sym and symmetric functions on Rn . This is well known and immediate. The function f˜ is called the representation of f . 3. The main result. For each i = j , we denote by W (ij ) the skew matrix with elements Mkl , where Mij = −Mj i = 1 and Mkl = 0 for all other pairs of indices. For B ∈ Sn , we denote B[ij ] = W (ij ) B −BW (ij ) ∈ Sn . Let (R) ⊂ Rn be an open ball of radius R with the center at the origin. Lemma 3.1. For each positive integer r and each f˜ ∈ CSr (Rn ), there exist functions F s (·, f˜) ∈ C r−s (Rn , Fr (Sn )), s = 1, . . . , r, such that the following hold. (a) If s = 1, . . . , r, [r−s] s D F ·, f˜ C 0 ((R)) ≤ C(r, R)D [r] f˜C 0 ((R)) , (2) where C(r, R) is a constant independent of f˜ and ·C 0 ((R)) is the supremum norm on (R). (b) If s = 1, n 1 ˜ F a, f B = Di f˜(a)Bii1 , 1
a ∈ Rn ,
B 1 ∈ Sn .
(3)
Hijs Bijs ,
(4)
i=1
(c) If 1 < s ≤ r, n 1 F s a, f˜ B 1 , . . . , B s = Gsi Biis + 2 i=1
1≤i=j ≤n
a ∈ Rn , B 1 , . . . , B s ∈ Sn , where for each i, Gsi = Di F s−1 a, f˜ B 1 , . . . , B s−1
(5)
DIFFERENTIABILITY PROPERTIES
369
and for each i, j , 1 ≤ i = j ≤ n, Hijs
=
1 s−1 0
Di F
s−1
k=1
1 k s−1 ˜ a , f B , . . . , B[ij ] , . . . , B dt
t
(6)
with the abbreviation a t := a + t (aj − ai )ei . Proof. For a fixed r by induction on s. For s = 1 we clearly have F 1 (·, f ) ∈ and (2) holds with C(r, R) = 1. Let 1 < s ≤ r be given, let F s be defined by (4) through (6), and let (2) hold with some C(r, R) for all values of s = s less than our s. Since F s−1 (·, f ) is of class C r−s+1 by the induction hypothesis, we see from (5) and (6) that Gsi , Hijs are all of class C r−s . Moreover, a differentiation, the chain rule, and the induction hypothesis (2) provide [r−s] s D Gi C 0 ((R)) ≤ M D [r−s+1] F s−1 ·, f˜ B 1 , . . . , B s−1 C 0 ((R))
≤ MC(r, R)D [r] f˜C 0 ((R)) B 1 · · · B s−1 , C s−1 ((R))
for some M ≥ 1 independent of f˜; that is, [r−s] s
D Gi C 0 ((R)) ≤ C (r, R)D [r] f˜C 0 ((R)) B 1 · · · B s−1 , and similarly [r−s] s
D Hij C 0 ((R)) ≤ C (r, R)D [r] f˜C 0 ((R)) B 1 · · · B s−1 , with possibly a larger value of C (r, R) = MC(r, R). Lemma 3.2. Let f : Sym → R be an isotropic function of class C r . Then (a) for each A, B1 , . . . , Br ∈ Sym and Q ∈ Orth+ we have D r f (A) B1 , . . . , Br = D r f QAQT QB1 QT , . . . , QBr QT ;
(7)
(b) for each A, B1 , . . . , Br−1 ∈ Sym and W ∈ Skew we have r
r−1
D f (A) [W, A], B , . . . , B 1
=−
r−1 k=1
D r−1 f (A) B1 , . . . , W, Bk , . . . , Br−1 . (8)
Here [A, B] = AB − BA. Proof. (a) Differentiate f (A) = f (QAQT ) r times in the directions B1 , . . . , Br . (b) In (7) we replace r by r − 1 and set Q = etW , t ∈ R. A differentiation with respect to t at t = 0 gives the result.
370
MIROSLAV ŠILHAVÝ
Lemma 3.3. For each r ≥ 1, each f ∈ CIr (Sym), each basis {ei }, and each A, ∈ Sym, represented by the matrices A = A = diag(a), a ∈ Rn , B 1 , . . . , B r ∈ Sn , we have D r f (A) B1 , . . . , Br = F r a, f˜ B 1 , . . . , B r . (9) B1 , . . . , Br
Proof. We prove this by induction on r. For r = 1, (9) and (3) represent a wellknown formula for the first derivative of an isotropic function (e.g., [5]). Suppose that the assertion of the lemma is true for some particular r −1 ≥ 1. In view of the linearity of D r f (A)[B1 , . . . , Br ] with respect to Br , it suffices to prove (9) only for some special choices of Br and for B1 , . . . , Br−1 ∈ Sym arbitrary. Namely, it suffices to take (i) Br ≡ B r = diag(ei ), i = 1, . . . , n, and (ii) Br ≡ B r = B (ij ) , 1 ≤ i = j ≤ n, where B (ij ) denotes the n-by-n symmetric matrix with elements Mkl , where Mij = Mj i = 1 and all other elements Mkl vanish. First let Br ≡ B r = diag(ei ). By the induction hypothesis, D r−1 f (A + λBr ) B1 , . . . , Br−1 = F r−1 a + λb, f˜ B 1 , . . . , B r−1 , and a differentiation combined with the fact that F r−1 (·, f˜) is of class C 1 (by Lemma 3.1) provides n D r f (A) B1 , . . . , Br = Di F r−1 a, f˜ B 1 , . . . , B r−1 Biir , i=1
which is (9) in this special case. Let Br ≡ B r = B (ij ) , where i, j is a fixed pair, 1 ≤ i = j ≤ n. Assume first that ai = aj . Set W = W = (ai − aj )−1 W (ij ) and note that [W, A] = −B (ij ) . The application of (8) and the induction hypothesis give 1 r−1 r−1 k , . . . , B r−1 D f (A) B , . . . , B k=1 [ij ] D r f (A) B1 , . . . , Br = ai − a j (10) r−1 r−1 k , . . . , B r−1 a, f˜ B 1 , . . . , B[ij k=1 F ] = , ai − a j where we have identified tensors with matrices. Let a t be as in Lemma 3.1 and note that if a = (. . . , ai , . . . , aj , . . . ), then a t for t = 1 equals a 1 = (. . . , aj , . . . , aj , . . . ). Then for A1 = A1 = diag(a 1 ), we have A1[ij ] = 0, and hence from (8) and the induction hypothesis, 0=
r−1 k=1
=
r−1 k=1
k r−1 D r−1 f A1 B 1 , . . . , B[ij ], . . . , B F
k r−1 a 1 , f˜ B 1 , . . . , B[ij . ], . . . , B
r−1
(11)
DIFFERENTIABILITY PROPERTIES
371
Thus the last expression in (10) can be rewritten as the right-hand side of (6), which is (9) in this case. Next assume that ai = aj . The preceding part of the proof shows: r−1 r−1 k , . . . , B r−1 a% , f˜ B 1 , . . . , B[ij 1 k=1 F ] r r D f (A% ) B , . . . , B = (12) % for each % = 0, where A% = A + % diag(ei ), a% = a + %ei , B 1 , . . . , B r−1 ∈ Sn , and B r = B (ij ) . The limit as % → 0 of the left-hand side of (12) is D r f (A)[B 1 , . . . , B r ]; the limit as % → 0 of the right-hand side exists and equals r−1 k=1
k r−1 Di F r−1 a, f˜ B 1 , . . . , B[ij ], . . . , B
by l’Hospital’s rule. Thus (9) also holds if ai = aj . Remark 3.4. We have proved the following alternative expression of H r : r−1 k , . . . , B r−1 F r−1 a, f˜ B 1 , . . . , B[ij k=1 ] if ai = aj , ai − a j r Hij = r−1 k r−1 Di F r−1 a, f˜ B 1 , . . . , B[ij if ai = aj . ], . . . , B
(13)
k=1
This formula is useful for calculating the derivatives, in contrast to (6), which is useful for examining the differentiability properties of Hijr . Theorem 3.5. Let f : Sym → R be an isotropic function. Then (a) f ∈ C r (Sym), r = 0, 1, . . . , ∞, if and only if f˜ ∈ C r (Rn ); (b) for each basis {ei } and each A, B1 , . . . , Br ∈ Sym, represented by the matrices A = A = diag(a), a ∈ Rn , B 1 , . . . , B r ∈ Sn , we have D r f (A) B1 , . . . , Br = F r a, f˜ B 1 , . . . , B r . Proof. (a) The direct implication is immediate. Let us prove the converse. Note first that (2) and (9) imply that for each r ≥ 1 and each R > 0, there exists a constant C(r, R) such that if f ∈ CI∞ (Sym), then (14) D r f C 0 ((R)) ≤ C(r, R)D [r] f˜C 0 ((R)) . Let f be such that f˜ ∈ C r (Rn ) and let R > 0. Let ϕ : Sym → R be a C ∞ mollifier of the form ϕ(A) = ψ(|A|), A ∈ Sym; set ϕ% (A) = ϕ(A/%)/% n(n+1)/2 , A ∈ Sym; and for each f ∈ CIr (Sym), r = 0, . . . , let f% (A) = f (B)ϕ% (A − B) dB,
372
MIROSLAV ŠILHAVÝ
where the integral extends over Sym and dB denotes the Lebesgue measure on Sym. Clearly, f% is isotropic. We denote by f˜% the representation of f% and note that f% −→ f,
f˜% −→ f˜,
D [r] f˜% −→ D [r] f˜ as % −→ 0+,
(15)
and the convergence is uniform on compact sets. By (14) and (15) we see that for each s, 1 ≤ s ≤ r, D s f% is a Cauchy sequence in C 0 ((R), Fs (Sym)) and thus there exist M s ∈ C 0 ((R), Fs (Sym)) such that D s f% −→ M s
as % −→ 0+
uniformly on compact sets. Since D s f% are the continuous derivatives of f% , we have f% (A)D s g(A) dA = (−1)s D s f% (A)g(A) dA for each g ∈ C0∞ (Sym). The limit % → 0+ gives f (A)D s g(A) dA = M s (A)g(A) dA. Thus M s are the distributional derivatives of f on Sym. Since M s are continuous functions, elementary considerations show that f ∈ C r (Sym) and D s f = M s , s = 1, . . . , r. (b) This follows from Lemma 3.3. 4. Low-order derivatives. Let {ei } be a basis and let A, B1 , B2 , B3 ∈ Sym be represented by the matrices A = A = diag(a), a ∈ Rn , B 1 , B 2 , B 3 ∈ Sn and let the components of a be distinct. Let f : Sym → R and let the subscripts attached to f˜ denote the partial derivatives of f˜. The application of (9), (3), (4), (5), and (13) provides n 1 f˜i Bii1 Df (A) B = i=1
if f
∈ C 1 (Sym);
further,
n 2 f˜ij Bii1 Bjj + D 2 f (A) B1 , B2 = i,j =1
1≤i=j ≤n
if f ∈ C 2 (Sym) (cf. [3], [1]), and n 2 3 f˜ij k Bii1 Bjj Bkk D 3 f (A) B1 , B2 , B3 = i,j,k=1
f˜i − f˜j 1 2 B B ai − aj ij ij
DIFFERENTIABILITY PROPERTIES n
+
k=1 1≤i=j ≤n
+2
373
f˜ik − f˜j k 1 2 3 2 3 1 2 3 Bij + Bkk Bij Bij Bij Bij Bkk + Bij1 Bkk ai − a j
n i,j,k=1 i=j =k=i
ai f˜j − f˜k + aj f˜k − f˜i + ak f˜i − f˜j 1 2 3 Bki Bkj Bij (ai − aj )(aj − ak )(ak − ai )
if f ∈ C 3 (Sym). The last formula is equivalent to the one given in [3]. References [1] [2] [3] [4] [5] [6]
J. M. Ball, Differentiability properties of symmetric and isotropic functions, Duke Math. J. 51 (1984), 699–728. P. Chadwick and R. W. Ogden, On the definition of elastic moduli, Arch. Rational Mech. Anal. 44 (1971/72), 41–53. , A theorem of tensor calculus and its application to isotropic elasticity, Arch. Rational Mech. Anal. 44 (1971/72), 54–68. M. E. Gurtin, An Introduction to Continuum Mechanics, Math. Sci. Engrg. 158, Academic Press, New York, 1981. M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media, Texts Monogr. Phys., Springer, Berlin, 1997. J. Sylvester, On the differentiability of O(n) invariant functions of symmetric matrices, Duke Math. J. 52 (1985), 475–483.
Mathematical Institute of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Prague 1, Czech Republic; [email protected]
Vol. 104, No. 3
DUKE MATHEMATICAL JOURNAL
© 2000
A NEW ELLIPSOID ASSOCIATED WITH CONVEX BODIES ERWIN LUTWAK, DEANE YANG, and GAOYONG ZHANG
Corresponding to each origin-symmetric convex (or more general) subset of Euclidean n-space Rn , there is a unique ellipsoid with the following property: The moment of inertia of the ellipsoid and the moment of inertia of the convex set are the same about every 1-dimensional subspace of Rn . This ellipsoid is called the Legendre ellipsoid of the convex set. The Legendre ellipsoid and its polar (the Binet ellipsoid) are well-known concepts from classical mechanics. See Milman and Pajor [MPa1], [MPa2], Lindenstrauss and Milman [LiM], and Leichtweiß [Le] for some historical references. It has slowly come to be recognized that alongside the Brunn-Minkowski theory there is a dual theory. The nature of the duality between the Brunn-Minkowski theory and the dual Brunn-Minkowski theory is subtle and not yet understood. It is easily seen that the Legendre (and Binet) ellipsoid is an object of this dual Brunn-Minkowski theory. This observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical Legendre ellipsoid in the Brunn-Minkowski theory. It is the aim of this paper to demonstrate the existence of precisely this dual object. In retrospect, one may well wonder why the new ellipsoid presented in this note was not discovered long ago. The simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of L2 -curvature in hand. However, the Brunn-Minkowski theory was only recently extended to incorporate the new notion of Lp -curvature (see [L2], [L3]). A positive-definite n × n real symmetric matrix A generates an ellipsoid (A), in Rn , defined by (A) = x ∈ Rn : x · Ax ≤ 1 , where x · Ax denotes the standard inner product of x and Ax in Rn . Associated with a star-shaped (about the origin) set K ⊂ Rn is its Legendre ellipsoid 2 K, which is generated by the matrix [mij (K)]−1 , where mij (K) =
n+2 V (K)
K
ei · x ej · x dx,
with e1 , . . . , en denoting the standard basis for Rn and V (K) denoting the n-dimensional volume of K. Received 30 September 1999. Revision received 4 January 2000. 2000 Mathematics Subject Classification. Primary 52A40. Authors’ work supported, in part, by National Science Foundation grant number DMS-9803261. 375
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We will associate a new ellipsoid −2 K with each convex body K ⊂ Rn . One approach to defining −2 K without introducing new notation is to first define it for polytopes and then use approximation (with respect to the Hausdorff metric) to extend the definition to all convex bodies. Suppose P ⊂ Rn is a polytope that contains the origin in its interior. Let u1 , . . . , uN denote the outer unit normals to the faces of P , let a1 , . . . , aN denote the areas (i.e., (n − 1)-dimensional volumes) of the corresponding faces, and let h1 , . . . , hN denote the distances from the origin to the corresponding faces. The ellipsoid −2 P is generated by the matrix [m ˜ ij (P )], where N
m ˜ ij (P ) =
1 al ei · ul ej · ul . V (P ) hl l=1
An alternate definition of the operator −2 is given after additional notation is introduced. The easily established affine nature of the operator 2 is formally stated in the following lemma. Lemma 1. If K ⊂ Rn is star-shaped about the origin, then for each φ ∈ GL(n), 2 (φK) = φ 2 K. While more difficult to see, we prove the following lemma. Lemma 1∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. Then for each φ ∈ GL(n), −2 (φK) = φ −2 K. The following theorem is fundamental and goes back, at least, to Blaschke [Bl], John [J], and Petty [P1] (see also Milman and Pajor [MPa1], [MPa2]). We give yet another proof in this paper. Theorem 1. If K ⊂ Rn is star-shaped about the origin, then V 2 K ≥ V (K), with equality if and only if K is an ellipsoid centered at the origin. For our new ellipsoids we establish the following. Theorem 1∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. Then V −2 K ≤ V (K), with equality if and only if K is an ellipsoid centered at the origin.
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The operator −2 has the following monotonicity property. Theorem 2∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If E is an ellipsoid centered at the origin such that E ⊂ K, then V −2 E ≤ V −2 K , with equality if and only if E = −2 K. Let S n−1 denote the unit sphere, centered at the origin, in Rn . Let B denote the unit ball, centered at the origin, in Rn , and let ωn = V (B). From Theorem 2∗ we obtain the following. Theorem 3∗ . Suppose K ⊂ Rn is a convex body that is origin-symmetric; then V −2 K ≥ 2−n ωn V (K), with equality if and only if K is a parallelotope. The analog of Theorem 3∗ for the operator 2 is one of the major open problems in the field: Finding the maximum of V ( 2 K)/V (K) as K ranges even over the class of origin-symmetric convex bodies (or even important small subclasses) is difficult (see, e.g., the survey of Lindenstrauss and Milman [LiM]). It is even difficult to show that there exists a√ c (independent of the dimension n) such that [V ( 2 K)/V (K)]1/n is bounded by c n as K ranges over the class of origin-symmetric convex bodies. This problem was first posed by Bourgain [Bo1]. The best-known bounds to date appear to be those of Bourgain [Bo2] (see also Dar [D] and Junge [Ju]). There is an important class of questions in the local theory of Banach spaces which are well known to be equivalent in that an answer to one will immediately provide an answer to the others. Bourgain’s problem is one member of this important class of equivalent problems. See Milman and Pajor [MPa2, Section 5]. We present a version of Theorem 3∗ for arbitrary convex bodies. We then present a classical characterization of the operator 2 and its obvious counterpart for the operator −2 . Finally, we present an analog of Milman’s important notion of “isotropic position” and explore some of its consequences. We have chosen to reprove all the classical results concerning the operator 2 for two reasons. First, we want to show the close connection and interrelationship between the operators 2 and −2 . Second, we believe that new proofs of classical results are almost always enlightening. A serious attempt has been made to present all arguments in a reasonably selfcontained manner. For quick reference, some basic properties of L2 -mixed and dual mixed volumes will be listed. Some recent applications of dual mixed volumes can be found in [G1], [Z1], [Z2], and [Z3]. The L1 -analogs of some of the identities presented may be found in [L1]. For general reference the reader may wish to consult the books of Gardner [G2], Schneider [S], and Thompson [T].
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Recall that if K ⊂ Rn is a convex body that contains the origin in its interior, then the polar of K, is defined by K ∗ = x ∈ Rn : x · y ≤ 1, for all y ∈ K .
K ∗,
From the definition it follows easily that for each convex body K, we have K ∗∗ = K.
(1)
From the definition of a polar body, it follows trivially that for each convex body K and φ ∈ GL(n) (φK)∗ = φ −t (K ∗ ),
(2)
where φ −t denotes the inverse of the transpose of φ. The radial function ρK = ρ(K, ·) : Rn \ {0} → [0, ∞) of a compact, star-shaped (about the origin) K ⊂ Rn , is defined for x = 0 by ρ(K, x) = max λ ≥ 0 : λx ∈ K . If ρK is positive and continuous, K is called a star body (about the origin). Two star bodies K and L are said to be dilates (of one another) if ρK (u)/ρL (u) is independent of u ∈ S n−1 . From the definition of radial function, it follows immediately that for a star body K, an x ∈ Rn \ {0}, and a φ ∈ GL(n), we have ρφK (x) = ρK φ −1 x ; (3) φK = {φx : x ∈ K} is the image of K under φ. If K ⊂ Rn is a convex body that contains the origin in its interior, then its support function hK = h(K, ·) : Rn → [0, ∞) is defined for x ∈ Rn by h(K, x) = max x · y : y ∈ K . Since it is assumed throughout that all of our convex bodies contain the origin in their interiors, all support functions are strictly positive on Rn \ {0}. From the definition of support function, it follows immediately that for a convex body K, an x ∈ Rn , and a φ ∈ GL(n), we have hφK (x) = hK φ t x , (3∗ ) where φ t denotes the transpose of φ. If K is a convex body, then it follows from the definitions of support and radial functions and the definition of polar body that hK ∗ =
1 ρK
and
ρK ∗ =
1 . hK
(4)
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−2 ε ·L is For star bodies K, L, and ε > 0, the L2 -harmonic radial combination K + the star body defined by −2 ε · L, · −2 = ρ(K, ·)−2 + ερ(L, ·)−2 . ρ K+
(5)
For convex bodies K, L, and ε > 0, the Firey L2 -combination K+2 ε ·L is defined as the convex body whose support function is given by 2 h K+2 ε · L, · = h(K, ·)2 + εh(L, ·)2 .
(5∗ )
Note that the “scalar” multiplications “ε · L” in (5) and (5∗ ) are different. The temptation to put a subscript under each “·” was resisted. From (4) we see that the relationship between the two types of combinations is that for convex bodies K, L, and ε > 0, −2 ε · L∗ ∗ . K+2 ε · L = K ∗ + The dual mixed volume V−2 (K, L) of the star bodies K, L can be defined by −2 ε · L − V (K) V K+ n V−2 (K, L) = lim . (6) −2 ε ε→0+ The L2 –mixed volume V2 (K, L) of the convex bodies K, L was defined in [L2] by V K+2 ε · L − V (K) n V2 (K, L) = lim . (6∗ ) 2 ε ε→0+ That this limit exists was demonstrated in [L2]. From the definitions (5) and (6), it follows immediately that for each star body K, V−2 (K, K) = V (K).
(7)
From the definitions (5∗ ) and (6∗ ), it follows immediately that for each convex body K, V2 (K, K) = V (K).
(7∗ )
From (3) and the definition of an L2 -harmonic radial combination (5), it follows immediately that for an L2 -harmonic radial combination of star bodies K and L, and φ ∈ GL(n), −2 ε · L = φK + −2 ε · φL. φ K+ This observation together with the definition of the dual mixed volume V−2 shows that for φ ∈ SL(n) and star bodies K, L we have V−2 (φK, φL) = V−2 (K, L), or equivalently, V−2 φK, L = V−2 K, φ −1 L . (8)
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From (3∗ ) and the definition of a Firey L2 -combination (5∗ ), it follows immediately that for a Firey combination of convex bodies K and L, and φ ∈ GL(n), φ K+2 ε · L = φK+2 ε · φL. This observation together with the definition of the L2 –mixed volume V2 shows that for φ ∈ SL(n) and convex bodies K, L we have V2 (φK, φL) = V2 (K, L) or equivalently V2 φK, L = V2 K, φ −1 L . (8∗ ) Definitions (5) and (6) and the polar coordinate formula for volume give the following integral representation of the dual mixed volume V−2 (K, L) of the star bodies K, L: 1 V−2 (K, L) = ρ n+2 (v)ρL−2 (v) dS(v), (9) n S n−1 K where the integration is with respect to spherical Lebesgue measure S on S n−1 . It was shown in [L2] that corresponding to each convex body K, there is a positive Borel measure S2 (K, ·) on S n−1 such that 1 V2 (K, L) = h2 (u) dS2 (K, u) (9∗ ) n S n−1 L for each convex body L. We require two basic inequalities regarding the mixed volumes V2 and the dual mixed volumes V−2 . The dual mixed volume inequality for V−2 is that for star bodies K, L, V−2 (K, L) ≥ V (K)(n+2)/n V (L)−2/n ,
(10)
with equality if and only if K and L are dilates. This inequality is an immediate consequence of the Hölder inequality and the integral representation (9). The L2 analog of the classical Minkowski inequality states that for convex bodies K, L, V2 (K, L) ≥ V (K)(n−2)/n V (L)2/n ,
(10∗ )
with equality if and only if K and L are dilates. This L2 -analog of the Minkowski inequality was established in [L2] by using the classical Minkowski mixed volume inequality. An immediate consequence of the dual mixed volume inequality (10) and identity (7) that we use is that if for star bodies K, L we have V−2 (Q, K) V−2 (Q, L) = V (Q) V (Q) for all star bodies Q, which belong to some class that contains both K and L, then in fact K = L.
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It is easy to verify that if A is a positive definite n × n real symmetric matrix, then the radial and support functions of the ellipsoid (A) = {x ∈ Rn : x · Ax ≤ 1} are given by −2 ρ(A) (u) = u · Au and h2(A) (u) = u · A−1 u for u ∈ S n−1 . Thus, for a star body K, h2 2 K (u) =
n+2 V (K)
K
(u · x)2 dx,
(11)
for u ∈ S n−1 . The normalization above is chosen so that for the unit ball B, we have 2 B = B. It must be emphasized that our normalization differs from the classical. For the polar of 2 K we write 2∗ K rather than ( 2 K)∗ . For each convex body K, we can define the ellipsoid −2 K by 1 −2 ρ −2 K (u) = (u · v)2 dS2 (K, v) (11∗ ) V (K) S n−1 for u ∈ S n−1 . Note that for the unit ball B, we have −2 B = B. For the polar of ∗ K rather than ( K)∗ , and thus −2 K we write −2 −2 1 (u · v)2 dS2 (K, v). h2 ∗ K (u) = −2 V (K) S n−1 It was shown in [L2] that the L2 –surface area measure S2 (K, ·) is absolutely continuous with respect to the classical surface area measure SK and that the RadonNikodym derivative 1 dS2 (K, ·) = . dSK hK Thus, if P is a polytope whose faces have outer unit normals u1 , . . . , uN , and ai denotes the area of the face with outer normal ui , and hi denotes the distance from the origin to this face, then the measure S2 (P , ·) is concentrated at the points u1 , . . . , uN ∈ S n−1 and S2 (P , {ui }) = ai / hi . Thus, for the polytope P , we have ρ −2 (u) = −2 P
N
1 al (u · ul )2 V (P ) hl l=1
for u ∈ S n−1 . If K is a convex body such that ∂K is C 2 and whose Gauss curvature is positive, then it is well known that the measure SK is absolutely continuous with respect to spherical Lebesgue measure (i.e., (n−1)-dimensional Hausdorff measure) S, and the Radon-Nikodym derivative dSK = fK , dS where fK : S n−1 → (0, ∞) is the reciprocal Gauss curvature of ∂K viewed as a
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function of the outer normals (i.e., fK (u), for u ∈ S n−1 , is the reciprocal Gauss curvature at the point of ∂K whose outer unit normal is u). Thus for u ∈ S n−1 , 1 (u) = (u · v)2 h−1 ρ −2 K (v)fK (v) dS(v), −2 K V (K) S n−1 or equivalently, (u) = ρ −2 −2 K
1 V (K)
∂K
2 u · ν(x) h−1 K ν(x) dx,
where ν(x) denotes the outer unit normal at x ∈ ∂K and the integration is with respect to the intrinsic measure on ∂K. A connection between the operators 2 and −2 is given in the following identity. Lemma 2. Suppose K, L ⊂ Rn . If L is a convex body that contains the origin in its interior and K is a star body about the origin, then V−2 K, −2 L V2 L, 2 K = . V (L) V (K) Proof. From the integral representation (9∗ ), definition (11), Fubini’s theorem, definition (11∗ ), and the integral representation (9), it follows that 1 V2 L, 2 K = h2 (u) dS2 (L, u) n S n−1 2 K 1 n+2 2 = (u · x) dx dS2 (L, u) n S n−1 V (K) K 1 n+2 (u · v)2 ρK (v) dS(v) dS2 (L, u) = nV (K) S n−1 S n−1 V (L) = ρ n+2 (v)ρ −2 (v) dS(v) −2 L nV (K) S n−1 K V (L) = V−2 K, −2 L . V (K) From the integral representation (9), definition (11), and Fubini’s theorem, we immediately see that if K and L are star bodies, then V−2 K, 2∗ L V−2 L, 2∗ K = . (12) V (K) V (L) From the integral representation (9∗ ), definition (11∗ ), (4), and Fubini’s theorem, we immediately see that if K and L are convex bodies, then ∗ L ∗ K V2 K, −2 V2 L, −2 = . (12∗ ) V (K) V (L)
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An immediate consequence of the definition of the L2 -centroid body (11) and the transformation rule for support function (3∗ ) is that for φ ∈ GL(n), 2 φK = φ 2 K. Since, for the unit ball, B, we have 2 B = B, it follows that if E is an ellipsoid centered at the origin, then 2 E = E. The following lemma shows that −2 is also an intertwining operator with the linear group GL(n). Lemma 1∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If φ ∈ GL(n), then −2 (φK) = φ −2 K. Proof. From Lemma 2, followed by (8∗ ), Lemma 1, Lemma 2 again, and (8), we have for each star body Q, V2 φK, 2 Q V2 K, φ −1 2 Q V−2 Q, −2 φK = = V (Q) V (φK) V (K) V2 K, 2 φ −1 Q V−2 φ −1 Q, −2 K = = V (K) V φ −1 Q V−2 Q, φ −2 K = . V (Q) But V−2 (Q, −2 φK)/V (Q) = V−2 (Q, φ −2 K)/V (Q) for all star bodies Q implies that −2 φK = φ −2 K. Since, for the unit ball B, we have −2 B = B, it follows from Lemma 1∗ that if E is an ellipsoid centered at the origin, then −2 E = E. Thus −2 2 K = 2 K for all K. In Lemma 2 take L = 2 K, use (7∗ ), and get: For each star body K, V−2 K, 2 K = V (K). (13) But (13) and the dual mixed volume inequality (10) immediately yield the following. Theorem 1. If K ⊂ Rn is a star body about the origin, then V 2 K ≥ V (K), with equality if and only if K is an ellipsoid centered at the origin.
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In Lemma 2 take K = −2 L, use (7), and get: For each convex body L, V2 L, −2 L = V (L).
(13∗ )
But (13∗ ) and the L2 –mixed volume inequality (10∗ ) immediately yield the following. Theorem 1∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. Then V −2 K ≤ V (K), with equality if and only if K is an ellipsoid centered at the origin. Theorem 2∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If E is an ellipsoid centered at the origin such that E ⊂ K, then V −2 E ≤ V −2 K , with equality if and only if E = −2 K. Proof. From the integral representation (9∗ ) we see that the mixed volume V2 (K, ·) is monotone with respect to set inclusion. Now from (7∗ ), the monotonicity of the mixed volume V2 (K, ·), Lemma 1∗ and (1), identity (12∗ ), and the L2 –mixed volume inequality (10∗ ), we have 1=
V2 (K, K) V (K)
∗ E∗ ∗ K V2 E ∗ , −2 V2 (K, E) V2 K, −2 ≥ = = V (K) V (K) V (E ∗ ) 2/n
∗ K 2/n V (E ∗ ) −2/n V −2 ωn −2/n ωn ≥ = , ωn ωn V (E) V −2 K where the last equality is a consequence of the fact that, by (2), the product of the volumes of polar reciprocal ellipsoids, which are centered at the origin, is ωn2 . Hence we have V −2 K ≥ V (E) = V −2 E , with equality (from the equality conditions of the L2 –mixed volume inequality (10∗ )) implying that E and −2 K are dilates, which in turn implies that E = −2 K. The infimum of V ( −2 K)/V (K) taken over all convex bodies that contain the origin in their interiors is zero. To get a positive lower bound, some restriction must be made on the positions of the bodies (relative to the origin). A fundamental result due to Ball [B] is that if K is an origin-symmetric convex body, then there exists an ellipsoid EK ⊂ K, centered at the origin, such that V (EK ) ≥ 2−n ωn V (K). Barthe [Br] improved Ball’s theorem by showing that for each origin-symmetric convex body K that is not a parallelotope, there exists an
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385
ellipsoid EK ⊂ K, centered at the origin, such that V (EK ) > 2−n ωn V (K). Combine this with Theorem 2∗ and the immediate result is the following. Theorem 3∗ . Suppose K ⊂ Rn is a convex body that is origin-symmetric; then ωn V −2 K ≥ n V (K), 2 with equality if and only if K is a parallelotope. Associated with each convex body K is an important affinely associated point called the John point, j (K) ∈ int K. This point is the center of the (unique) ellipsoid of maximal volume that is contained in the body K. (As an aside, the authors note that in their opinions it would be more appropriate to call this point the Löwner point.) The John point is an affinely associated point in that for each φ ∈ SL(n) we have j (φK) = φj (K). A fundamental result due to Ball [B] is that if K is positioned so that its John point is at the origin, then there exists an ellipsoid EK ⊂ K, centered at the origin, such that V (EK ) ≥ n!ωn n−n/2 (n+1)−(n+1)/2 V (K). Barthe [Br] improved Ball’s theorem by showing that if K is positioned so that its John point is at the origin, then unless K is a simplex, there exists an ellipsoid EK ⊂ K, centered at the origin, such that V (EK ) >
n!ωn V (K). n/2 n (n + 1)(n+1)/2
Together with Theorem 2∗ this immediately gives the following. Theorem 4∗ . If K ⊂ Rn is a convex body positioned so that its John point is at the origin, then n!ωn V (K), V −2 K ≥ n/2 n (n + 1)(n+1)/2 with equality if and only if K is a simplex. The volume-normalized version of the operator 2 is the operator that maps each star body K to [ωn /V ( 2 K)]1/n 2 K. A classical characterization of the volumenormalized version of the operator 2 is as the solution to the following problem: Given a fixed star body K, find an ellipsoid centered at the origin E that minimizes V−2 (K, E) subject to the constraint that V (E) = ωn . Existence, uniqueness, and characterization of the solution to the problem are all contained in the following. Theorem 5. Suppose K ⊂ Rn is a star body about the origin and E is an ellipsoid centered at the origin such that V (E) = ωn . Then
V 2 K 2/n V−2 (K, E) ≥ V (K) , ωn with equality if and only if E = λ 2 K where λ = [ωn /V ( 2 K)]1/n .
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Proof. From (2) and Lemma 1, followed by (12), and the dual Minkowski inequality (10), we have 2/n V−2 E ∗ , 2∗ K V−2 (K, E) V−2 K, 2∗ E ∗ −2/n = = ≥ ωn V 2 K , V (K) V (K) V (E ∗ ) with equality if and only if E and 2 K are dilates. Suppose K ⊂ Rn is a fixed convex body that contains the origin in its interior. Find an ellipsoid E, centered at the origin, which minimizes V2 (K, E) subject to the constraint that V (E) = ωn . The solution of the problem turns out to characterize the volume-normalized operator −2 . Existence, uniqueness, and characterization of the solution to the problem are all contained in the following. Theorem 5∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior and E is an ellipsoid centered at the origin such that V (E) = ωn . Then
V −2 K −2/n V2 (K, E) ≥ V (K) , ωn with equality if and only if E = λ −2 K, where λ = [ωn /V ( −2 K)]1/n . Proof. From (2) and Lemma 1∗ , followed by (12∗ ), and the L2 –Minkowski inequality (10∗ ), we have ∗ E∗ ∗ K V2 E ∗ , −2 2/n V2 (K, E) V2 K, −2 −2/n ∗ = = ≥ ωn V −2 K , ∗ V (K) V (K) V (E ) with equality if and only if E and −2 K are dilates. The L1 -analog of the problem solved by Theorem 5∗ was treated by Petty [P2]. Generalizations were considered by Clack [C] and Giannopoulos and Papadimitrakis [GiPap]. A star body K is said to be in isotropic position if 2 K is a ball and V (K) = 1. Note that for each star body there is a GL(n)-transformation that will map the body into one that is in isotropic position. If the star body K is in isotropic √ position, then the isotropic constant LK of K is defined to be the radius of (1/ n + 2) 2 K. If K is an arbitrary star body, then define its isotropic constant by
V 2 K 1/n 1 LK = √ . n + 2 ωn V (K) From Theorem 1, it immediately follows that for each star body K, −1/n
ωn LK ≥ √ , n+2 with equality if and only if K is an ellipsoid centered at the origin. An important
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question (previously mentioned) asks if sup LK : K is an origin-symmetric convex body in Rn in isotropic position is dominated by a real number independent of the dimension n. A convex body K is said to be in dual isotropic position if −2 K is a ball and V (K) = 1. Note that for each convex body there is a GL(n)-transformation that will map the body into one that is in isotropic position. If K is in dual isotropic position, then define the dual isotropic constant L∗K to be the radius of −2 K. If K is an arbitrary convex body, we can define its dual isotropic constant by
V −2 K 1/n ∗ . LK = ωn V (K) Theorems 1∗ and 3∗ immediately give the following. Theorem 6∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If K is origin-symmetric and in dual isotropic position, then 1 −1/n ≤ L∗K ≤ ωn . 2 Equality on the left-hand side holds if and only if K is a parallelotope and equality on the right-hand side holds if and only if K is an ellipsoid. Let v denote (n − 1)-dimensional volume. For u ∈ S n−1 , let u⊥ denote the 1codimension subspace of Rn that is orthogonal to u. Milman and Pajor [MPa2] showed that if K is origin-symmetric, then √ n n + 2 V (K) V (K) ≤ h 2 K (u) ≤ √ √ ⊥ 2(n + 2) v K ∩ u⊥ 2 3 v K ∩u for all u ∈ S n−1 . Equality on the left-hand side holds for K a right cylinder and u orthogonal to its base, and equality on the right-hand side holds for K a double right cone and u along its axis. For u ∈ S n−1 and a convex body K, let K | u⊥ denote the image of the orthogonal projection of K onto u⊥ . Theorem 7∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If K is origin-symmetric, then for every u ∈ S n−1 v K | u⊥ 2 v K | u⊥ ∗ K (u) ≤ 2 ≤ h −2 . √ V (K) n V (K) Equality on the left-hand side holds for K a double right cone and u along its axis, and equality on the right-hand side holds for K a right cylinder and u orthogonal to its base.
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Proof. From (11∗ ), together with the fact that dS2 (K, ·) = h−1 K dSK , and the Hölder inequality we have √
1/2 1 n|u · v| 2 −1 ρ −2 K (u) = hK (v) dSK (v) nV (K) S n−1 hK (v) √ 1 ≥ n|u · v| dSK (v) nV (K) S n−1 2v K | u⊥ = √ , nV (K) which gives the left inequality. To get the right-hand inequality, note that
1/2 1 |u · v| −1 ρ −2 K (u) = |u · v| dSK (v) V (K) S n−1 hK (v)
2v K | u⊥ |u · v| 1/2 ≤ max V (K) v∈S n−1 hK (v)
1/2 2v K | u⊥ ρK (u)−1 = V (K) 2v K | u⊥ , ≤ V (K) where the last inequality follows from the well-known and easily established fact that V (K) ≤ 2v(K | u⊥ )ρK (u). Theorem 8∗ . Suppose K ⊂ Rn is a convex body that contains the origin in its interior. If K is origin-symmetric and in dual isotropic position, then √ v K | u⊥ ≤ n, for all u ∈ S n−1 . Equality holds if and only if K is a cube and u is in the direction of one of its vertices. Proof. Suppose K is not a cube. From the left-hand inequality of Theorem 7∗ , the fact that [V ( −2 K)/ωn ]1/n is the radius of the ball −2 K, together with Theorem 3∗ , we have
2v K | u⊥ V ( −2 K) −1/n < 2V (K)−1/n . ≤ √ ωn nV (K) Ball conjectured that each origin-symmetric convex body can be GL(n)-transformed into a body for which the inequality of Theorem 8∗ holds. Giannopoulos and Papadimitrakis [GiPap] showed that this can be accomplished by making the body “surface isotropic.” Theorem 8∗ shows that this can also be done by making the body dual isotropic.
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Acknowledgment. The authors are most grateful to the referee for the extraordinary attention given to our paper. References [B] [Br] [Bl] [Bo1] [Bo2]
[BoM] [C] [D]
[F] [G1] [G2] [GiPap] [J] [Ju] [Le] [LiM]
[L1] [L2] [L3] [MPa1] [MPa2]
[P1]
K. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), 351–359. F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), 335–361. W. Blaschke, Affine Geometrie XIV, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.–Phys. Kl. 70 (1918), 72–75. J. Bourgain, On high-dimensional maximal functions associated to convex bodies, Amer. J. Math. 108 (1986), 1467–1476. , “On the distribution of polynomials on high-dimensional convex sets” in Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1469, Springer, Berlin, 1991, 127–137. J. Bourgain and V. Milman, New volume ratio properties for convex symmetric bodies in R n , Invent. Math. 88 (1987), 319–340. R. Clack, Minkowski surface area under affine transformations, Mathematika 37 (1990), 232–238. S. Dar, “Remarks on Bourgain’s problem on slicing of convex bodies” in Geometric Aspects of Functional Analysis (Israel, 1992–1994), Oper. Theory Adv. Appl. 77 (1995), 61–66. W. J. Firey, p-means of convex bodies, Math. Scand. 10 (1962), 17–24. R. J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. (2) 140 (1994), 435–447. , Geometric Tomography, Encyclopedia Math. Appl. 58, Cambridge Univ. Press, Cambridge, 1995. A. Giannopoulos and M. Papadimitrakis, Isotropic surface area measures, Mathematika 46 (1999), 1–13. F. John, Polar correspondence with respect to a convex region, Duke Math. J. 3 (1937), 355–369. M. Junge, Hyperplane conjecture for quotient spaces of Lp , Forum Math. 6 (1994), 617–635. K. Leichtweiß, Affine Geometry of Convex Bodies, J. A. Barth, Heidelberg, 1998. J. Lindenstrauss and V. D. Milman, “The local theory of normed spaces and its applications to convexity” in Handbook of Convex Geometry, ed. P. M. Gruber and J. M. Wills, North-Holland, Amsterdam, 1993, 1149–1220. E. Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), 365–391. , The Brunn-Minkowski-Firey theory, I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131–150. , The Brunn-Minkowski-Firey theory, II: Affine and geominimal surface areas, Adv. Math. 118 (1996), 244–294. V. D. Milman and A. Pajor, Cas limites dans des inégalités du type de Khinchine et applications géométriques, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), 91–96. , “Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space” in Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1376, Springer, Berlin, 1989, 64–104. C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535–1547.
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LUTWAK, YANG, AND ZHANG , Surface area of a convex body under affine transformations, Proc. Amer. Math. Soc. 12 (1961), 824–828. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl. 44, Cambridge Univ. Press, Cambridge, 1993. A. C. Thompson, Minkowski Geometry, Encyclopedia Math. Appl. 63, Cambridge Univ. Press, Cambridge, 1996. G. Zhang, Centered bodies and dual mixed volumes, Trans. Amer. Math. Soc. 345 (1994), 777–801. , Intersection bodies and the Busemann-Petty inequalities in R4 , Ann. of Math. (2) 140 (1994), 331–346. , A positive solution to the Busemann-Petty problem in R4 , Ann. of Math. (2) 149 (1999), 535–543.
Department of Mathematics, Polytechnic University, Six MetroTech Center, Brooklyn, New York 11201-3840, USA
Vol. 104, No. 3
DUKE MATHEMATICAL JOURNAL
© 2000
TRACES OF INTERTWINERS FOR QUANTUM GROUPS AND DIFFERENCE EQUATIONS, I PAVEL ETINGOF and ALEXANDER VARCHENKO
0. Introduction. This paper begins a series of papers whose goal is to establish a representation-theoretic interpretation of the quantum Knizhnik-ZamolodchikovBernard (qKZB) equations and to use this interpretation to study solutions of these equations. It was motivated by the recent work on the qKZB equations in [Fe], [FeTV1], [FeTV2], [MuV], [FeV2]–[FeV5] and by the theory of “quantum conformal blocks” that began with the classical paper [FR]. 0.1. The qKZB equations in [Fe] are difference equations with respect to an unknown function f (z1 , . . . , zN , λ, τ, µ, p) with values in V1 ⊗ · · · ⊗ VN ⊗ VN∗ ⊗ · · · ⊗ V1∗ , where Vi are suitable finite-dimensional representations of the quantum group Uq (g) (g is a simple Lie algebra), zi , p, τ ∈ C, and λ, µ are weights for g. The qKZB equations are a q-deformation of the Knizhnik-Zamolodchikov-Bernard (KZB) differential equations and an elliptic analogue of the quantum KnizhnikZamolodchikov (qKZ) difference equations, which are, in turn, generalizations of the usual (trigonometric) Knizhnik-Zamolodchikov (KZ) equations. It is proved in [FeTV2] (using an integral representation of solutions) that for g = sl2 the monodromy of the qKZB equations is given by the dual qKZB equations, which are obtained from the qKZB equations by interchanging (λ, τ ) with (µ, p). This fact generalizes the monodromy theorems for the KZB and qKZ equations: the monodromy of the KZB differential equations is the trigonometric degeneration of the qKZB equations (which involves dynamical R-matrices without spectral parameter) (see, e.g., [K3]), and the monodromy of the qKZ equations is given by elliptic dynamical R-matrices (but there is no difference equation) (see [TV1], [TV2]; see also [FR]). The self-duality of the qKZB equations leads one to expect that they should have symmetric solutions uV1 ,...,VN (z, λ, τ, µ, p), that is, such that uV1 ,...,VN (z, λ, τ, µ, p) = u∗V ∗ ,...,V ∗ (z, µ, p, λ, τ ), where u∗ is the dual of u (considered as an endomorN 1 phism of V1 ⊗ · · · ⊗ VN ). Such a solution u (for g = sl2 ) was constructed in [FeV3] and [FeTV2] by an explicit integral formula. It is called the universal hypergeometric function. This function has many interesting properties, in particular the SL(3, Z)-symmetry (see [FeV2], [FeV4], and [FeV5]), where the group SL(3, Z) acts on the lattice Z3 generated by the periods 1, τ, p. A consequence of this symmetry Received 31 August 1999. Revision received 29 November 1999. 2000 Mathematics Subject Classification. Primary 20G42; Secondary 33D52, 37K10. Etingof’s work partially supported by National Science Foundation grant number DMS-9700477. Varchenko’s work partially supported by National Science Foundation grant number DMS-9801582. 391
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is the qKZB heat equation in [FeV3] for the function u, which is a q-deformation of the KZB heat equation in [Ber]. 0.2. A central fact about the KZB and qKZ equations (and one of the main reasons why they are interesting) is that they are satisfied by conformal blocks. More precisely, the KZB equations are satisfied by conformal blocks of the Wess-Zumino-Witten conformal field theory on an elliptic curve (see [Ber]), and the qKZ equations are satisfied by quantum conformal blocks on the cylinder (see [FR]). In representation-theoretic terms, conformal blocks on an elliptic curve are traces of products of intertwining operators for affine Lie algebras (weighted by an element from the maximal torus) (see [Ber]), and quantum conformal blocks on the cylinder are highest matrix elements of products of intertwining operators for quantum affine algebras (see [FR]). This representation-theoretic interpretation of the KZB and qKZ equations is not only interesting by itself, but it also allows us to prove nontrivial properties of solutions, for example, monodromy theorems (see, e.g., [K3], [FR]). The goal of this series is to give a similar interpretation of the qKZB equations. In light of the above, the main idea is obvious: one should consider quantum conformal blocks on an elliptic curve or, representation theoretically, one should consider traces of products of intertwining operators for quantum affine algebras, weighted by an element of the maximal torus. It is natural to expect that such traces satisfy a pair of dual qKZB equations. This is actually true, and we plan to give a proof of it in a subsequent part of the series. However, the details of the proof are relatively complicated, and we would like to start with a simpler (“trigonometric”) limiting case, when τ, p → ∞. This limiting case is the main subject of this paper. 0.3. The structure of this paper is as follows. In Section 1, we introduce the main object of the paper—the renormalized universal trace function FV1 ,...,VN (λ, µ) ∈ (V1 ⊗ · · · ⊗ VN )[0] ⊗ (VN∗ ⊗ · · · ⊗ V1∗ )[0], where λ, µ are weights for g. It is obtained from traces of products of intertwining operators for Uq (g) weighted by an element of the maximal torus. At the end of the section, we formulate the main results of the paper, Theorems 1.1–1.5. Theorems 1.1 and 1.2 state that the function FV1 ,...,VN satisfies two systems of difference equations, one with shifts of λ and the other with shifts of µ, which go to each other under the transformation λ → µ, µ → λ. In the special case g = sln , N = 1, V1 = S mn Cn , these systems (as was shown in [EK1]) reduce to the trigonometric Macdonald-Ruijsenaars (MR) systems, so we call them the MR system and the dual MR system. Theorem 1.5 states that the function FV1 ,...,VN (λ, µ) is symmetric: FV1 ,...,VN (λ, µ) = FV∗ ∗ ,...,V ∗ (µ, λ), where ∗ is the permutation of components. It follows from TheoN 1 rems 1.1 and 1.2. Theorems 1.3 and 1.4 state that the function FV1 ,...,Vn satisfies two additional systems of difference equations—the trigonometric degenerations of the qKZB and the dual qKZB equations, respectively. Theorems 1.1–1.5 are proved in Sections 2–5.
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In Section 6, we study the symmetry of trace functions under q → q −1 , and we define a modified trace function uV (λ, µ) by renormalizing FV (λ, µ). (This function is introduced to connect our paper with the papers [FeV2]–[FeV5], and here we define it only for N = 1; we plan to define it in general in another paper.) Using the q → q −1 transformation properties and Theorem 1.5, we show that the function uV is symmetric. In Section 7, we compute the function FV (λ, µ), uV (λ, µ) explicitly in the case g = sl2 . In Section 8, we compute explicitly the trigonometric degeneration of the function u from [FeV3] in the case N = 1. We show that this function is the same as uV (λ, µ) up to normalization. In Section 9, we explain that Macdonald’s theory for root systems of type An−1 is a special case of the theory developed in this paper for g = sln , N = 1, V1 = S mn Cn . In Section 10, we consider limiting (degenerate) cases of the theory developed in this paper. 0.4. In subsequent papers of the series, we plan the following. (1) We will give a representation-theoretic proof of the qKZB heat equation and the orthogonality relations for the trigonometric degeneration of the function u (see [FeV3]), using the ideas of [EK1], [EK3], and [EK4]. Cherednik’s theory of difference Fourier transform and Macdonald-Mehta identities for root systems of type A is a special case of this theory, corresponding to the situation g = sln , N = 1, V1 = S mn Cn . (2) We will give a representation-theoretic derivation of the resonance relations from [FeV2] in the trigonometric case, using the ideas of [ESt2]. (3) We will generalize all the results to the case of quantum affine algebras. This involves a representation-theoretic definition of the function u from [FeV3] for generic values of parameters for any simple Lie algebra and representations, and a representation-theoretic proof of its main properties, such as qKZB and MR equations, orthogonality, and modular transformations (e.g., the qKZB heat equation). As a special case, this theory should contain Macdonald’s theory for affine root systems of type Aˆ n−1 , which was originated in [EK2] but has not been developed from an analytic standpoint. In particular, the classical limit (q → 1) of the modular transformation of the function u should yield the result of Kirillov [K1], [K2], which says that the modular transformation S of affine Jack polynomials (which are essentially the 1-point functions of the WZW model in genus 1; see [EK2]) is given by a matrix of special values of Macdonald polynomials at roots of unity. (4) Specializing this theory to the critical level, we will prove that radial parts of the n ) at the critical level corresponding to the representation central elements of Uq (sl S mn Cn are elliptic Ruijsenaars operators (as far as we know, this is known only in the trigonometric degeneration). Acknowledgments. The work of the first author was partly done while he was employed by the Clay Mathematics Institute (CMI) as a CMI Prize Fellow. He thanks the Department of Mathematics at the University of North Carolina at Chapel Hill for hospitality. The second author is grateful to the Department of Mathematics at
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Harvard University for hospitality. Both authors are grateful to A. A. Kirillov Jr. for useful suggestions on how to improve the paper. 1. Trace functions for Uq (g) 1.1. The trace functions. Let g be a simple Lie algebra over C. Let h be a Cartan subalgebra of g, and let αi be simple roots of g, i = 1, . . . , r. Let (aij ) be the Cartan matrix of g. Let di be relatively prime positive integers such that (di aij ) is a symmetric matrix. Let ei , fi , and hi be the Chevalley generators of g. Let t be a complex number that is not purely imaginary, and let q = et . For any operator A, we denote etA by q A . Let Uq (g) be the Drinfeld-Jimbo quantum group corresponding to g. We use the same definition of Uq (g) as in [EFK2]; namely, Uq (g) is a Hopf algebra with generators Ei , Fi , i = 1, . . . , r, q h , h ∈ h, (q 0 = 1), with relations q x+y = q x q y ,
q h Ej q −h = q αj (h) Ei ,
x, y ∈ h,
Ei Fj − Fj Ei = δij 1−aij
(−1)k
k=0 1−aij
k=0
(−1)k
1 − aij k
1 − aij k
qi
qi
q h Fj q −h = q −αj (h) Fi ,
q di hi − q −di hi , q di − q −di
1−aij −k
Ej Eik = 0,
i = j,
1−aij −k
Fj Fik = 0,
i = j,
Ei Fi
where qi = q di and we used notation [n]q ! n , [n]q ! = [1]q × [2]q × · · · × [n]q , = k q [k]q ![n − k]q !
[n]q =
q n − q −n . q − q −1
Comultiplication $, antipode S, and counit % in Uq (g) are given by $(Fi ) = Fi ⊗ 1 + q −di hi ⊗ Fi , $ q h = q h ⊗ q h , S(Fi ) = −q di hi Fi , S q h = q −h , S(Ei ) = −Ei q −di hi ,
$(Ei ) = Ei ⊗ q di hi + 1 ⊗ Ei ,
%(Ei ) = %(Fi ) = 0,
%(q h ) = 1.
Let Mµ be the Verma module over Uq (g) with highest weight µ, and let vµ be its highest weight vector. Let V be a finite-dimensional representation of Uq (g), and let v ∈ V be a vector of weight µv . It is well known that for a generic µ there exists a unique intertwining operator (vµ : Mµ → Mµ−µv ⊗ V such that (vµ vµ = vµ−µv ⊗ v + l. o. t. (here l. o. t. denotes lower order terms). It is useful to consider the
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generating function of such operators (Vµ ∈ HomC (Mµ , ⊕Mν ⊗ V ⊗ V ∗ ) defined by (Vµ = v∈B (vµ ⊗ v ∗ , where the summation is taken over a homogeneous basis B of V . Let V1 , . . . , VN be finite-dimensional representations of Uq (g), and let vi ∈ Vi be vectors of weights µvi , such that µvi = 0. Define the formal series in V1 ⊗ · · · ⊗ VN [0] ⊗ q 2(λ,µ) C[[q −2(λ,α1 ) , . . . , q −2(λ,αr ) ]] as + v1 ,...,vN (λ, µ) = Tr |Mµ (v1 N ⊗ 1N−1 · · · (vµN q 2λ . (1.1) µ−
i=2 µvi
It follows from [ESt2, Corollary 3.4] that this series converges (in a suitable region of values of the parameters) to a function of the form q 2(λ,µ) f (λ, µ), where f is a rational function in q 2(λ,αi ) and q 2(µ,αi ) , which is a finite sum of products of functions of λ and functions of µ. The function (1.1) is called a trace function. Define also the universal trace function, with values in V1 ⊗ · · · ⊗ VN ⊗ VN∗ ⊗ · · · ⊗ V1∗ : ∗ +V1 ,...,VN (λ, µ) = (1.2) + v1 ,...,vN (λ, µ) ⊗ vN ⊗ · · · ⊗ v1∗ , vi ∈Bi
where Bi are homogeneous bases of Vi . It is easy to see that this function takes values in (V1 ⊗ · · · ⊗ VN )[0] ⊗ (VN∗ ⊗ · · · ⊗ V1∗ )[0]. Using the generating functions (Vµ , one can express the universal trace function as (1.3) +V1 ,...,VN (λ, µ) = Tr (V1 N (∗i) ⊗ 1N−1 · · · (VµN q 2λ , µ+
i=2 h
where we label the components Vi by i and Vi∗ by ∗i, and the notation h(k) for a label k was defined in [Fe]: when acting on a homogeneous multivector, h(k) has to be replaced with the weight in the kth component. Example 1 (See Section 7). Let g = sl2 . In this case, let us represent weights by complex numbers, so that the unique fundamental weight corresponds to 1. If N = 1, and V = V1 is the 3-dimensional representation, then
2 q λµ q −2λ −2 . +V (λ, µ) = 1+ q −q 1 − q −2λ 1 − q 2µ 1 − q −2(λ−1) (Since V [0] is 1-dimensional, we view the function +V as a scalar function.) This example is also computed in [ESt1]. 1.2. The main results. It turns out that the trace function +V1 ,...,VN (λ, µ) satisfies some remarkable difference equations. These equations are written in terms of socalled dynamical R-matrices. Below we give a brief introduction to the theory of dynamical R-matrices, referring the reader to the expository paper [ESch] for a more detailed discussion of them.
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Let V , W be finite-dimensional representations of Uq (g). Recall from [EV] the definitions of the fusion matrix and the exchange matrix. The fusion matrix is the operator JW V (µ) : W ⊗ V → W ⊗ V defined by the formula w JW V (µ)(w⊗v) (1.4) . (µ−µv ⊗ 1 (vµ = (µ The exchange matrix RV W (µ) ∈ End(V ⊗ W ) is defined by 21 21 (1.5) RV W (µ) = JV−1 W (µ) V ⊗W JW V (µ), where is the universal R-matrix of Uq (g). We also use the universal fusion matrix J (λ) and the universal exchange matrix R(λ). They take values in a completion of Uq (g) ⊗ Uq (g) and are defined by the condition that they give JV W (λ), RV W (λ) when evaluated in representations V , W (cf. also [ABRR], [JKOS]). Remark. The fusion matrix describes how to “fuse” together two intertwining operators. The exchange matrix describes how to exchange the order of intertwining operators. The fusion matrix satisfies the 2-cocycle identity (see below) and is sometimes referred to as a “quasi-Hopf twist.” The exchange matrix satisfies the quantum dynamical Yang-Baxter equation and is referred to as a “dynamical R-matrix.” Let J(λ) := J (−λ − ρ), where ρ is the half-sum of positive roots, and let R(λ) = R(−λ − ρ). Let Q(λ) = m21 (1 ⊗ S −1 )(J(λ)), where m21 (a ⊗ b) := ba and where S is the antipode. It is easy to show that Q(λ) is invertible for generic λ. Define (1.6)
J1,...,N (λ) = J1,2,...,N (λ)J2,3,...,N (λ) · · · JN−1,N (λ),
where, for example, J1,2,...,N stands for (1 ⊗ $n−1 )(J), where $n−1 : Uq (g) → Uq (g)n−1 is the iterated coproduct. We agree that J1 (λ) = 1. Thus, J1,...,N (λ) describes how to “fuse” N intertwining operators. Let
δq (λ) = q 2(λ,ρ) (1.7) 1 − q −2(λ,α) α>0
be the Weyl denominator. Set (1.8)
ϕV1 ,...,VN (λ, µ) = J1,...,N (λ)−1 +V1 ,...,VN (λ, µ)δq (λ).
Finally, introduce the renormalized trace function (1.9)
FV1 ,...,Vn (λ, µ) (∗1) ϕV1 ,...,VN (λ, −µ − ρ). = Q−1 (µ)(∗N) ⊗ · · · ⊗ Q−1 µ − h(∗2,...,∗N)
It is convenient to formulate properties of trace functions using this renormalization.
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Example 2 (See Section 7). If g = sl2 , N = 1, and V = V1 is the 3-dimensional representation, then FV (λ, µ) = q −λµ
q 2(λ+µ) − q 2λ−2 − q 2µ−2 + 1 . 1 − q 2λ−2 1 − q 2µ−2
(This formula is obtained after simplifications from formula (7.20) below when m = 1.) Note that it is seen from this formula that FV is symmetric in λ and µ. The following theorems describe the properties of FV1 ,...,VN (λ, µ). For any finite-dimensional Uq (g)-module W , define the difference operator ᏰW acting on functions of λ ∈ h∗ with values in (V1 ⊗ · · · ⊗ VN )[0] given by the formula (2,...,N) 0N (1.10) ᏰW = Tr |W [ν] R01 (λ) Tν , λ + h · · · R W V1 W VN ν
where Tν f (λ) = f (λ + ν) and the component W is labeled by zero. Theorem 1.1 (Macdonald-Ruijsenaars equations). We have that ᏰλW FV1 ,...,VN (λ, µ) = χW q −2µ FV1 ,...,VN (λ, µ), (1.11) dim W [ν]x ν is the character of W , and by ᏰλW we mean the where χW (x) = operator ᏰW acting on F as a function of λ, in components V1 , . . . , VN . Theorem 1.1 is proved in Section 2. Example 3. If g = sl2 , N = 1, V = V1 is the 3-dimensional representation, and W is the 2-dimensional representation, then 1 − q 2λ−4 1 − q 2λ+2 −1 T , ᏰW = T + 1 − q 2λ−2 1 − q 2λ where Tf (λ) = f (λ+1). To prove this, it is enough to check that this operator is the unique operator of the form a(λ)T + b(λ)T −1 such that (1.11) holds for the function FV given above (note that in our case χW (q −2µ ) = q µ + q −µ ). We introduce also the dual Macdonald-Ruijsenaars operators Ᏸ∨ W , acting on functions of µ ∈ h∗ with values in (VN∗ ⊗ · · · ⊗ V1∗ )[0], by the formula 01 (∗1,...,∗N−1) 0N (1.12) Ᏸ∨ = Tr | (µ) Tν , µ + h · · · R R ∗ ∗ W [ν] W WV WV ν
N
1
where Vj∗ is considered as a module over Uq (g) via the antipode. Theorem 1.2 (Dual Macdonald-Ruijsenaars equations). We have that ∨,µ ᏰW FV1 ,...,VN (λ, µ) = χW q −2λ FV1 ,...,VN (λ, µ), (1.13) ∨,µ
∗ ∗ where ᏰW is Ᏸ∨ W acting on F as a function of µ, in components VN , . . . , V1 .
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Theorem 1.2 is proved in Section 3. To formulate the next two results, we need to define some operators acting on functions of λ and µ with values in (V1 ⊗ · · · ⊗ VN )[0] ⊗ (VN∗ ⊗ · · · ⊗ V1∗ )[0]. For j = 1, . . . , N , define the operators 2 Dj = q −2µ− xi (1.14) q −2 xi ⊗xi ∗j,∗1,...,∗j −1 , ∗j
where xi is an orthonormal basis of h. Also, define the operators (1.15)
−1 Kj = Rj +1,j λ + h(j +2,...,N) · · · RNj (λ)−1 7j × Rj 1 λ + h(2,...,j −1) + h(j +1,...,N) · · · Rjj −1 λ + h(j +1,...,N) ,
where 7j f (λ) := f (λ + h(j ) ), and hj,...,k acting on a homogeneous multivector has to be replaced with the sum of weights of components j, . . . , k of this multivector. It is easy to check that Dj commute with each other, and it is known that so do Kj (see [Fe]). Remark. As we have mentioned before, these operators are the trigonometric limits of the qKZB operators with spectral parameters, introduced by Felder. Analogously, define the operators 2 q −2 xi ⊗xi j,j +1,...,N , (1.16) Dj∨ = q −2λ− xi j
and (1.17)
−1 Kj∨ =R∗j −1,∗j µ + h(∗1,...,∗j −2) · · · R∗1,∗j (µ)−1 7∗j × R∗j,∗N µ + h(∗j +1,...,∗N−1) + h(∗1,...,∗j −1) · · · R∗j,∗j +1 µ + h(∗1,...,∗j −1) ,
where 7∗j f (µ) = f (µ + h∗j ). Like Dj , Kj , the operators Dj∨ , Kj∨ commute. Theorem 1.3 (The qKZB equations). The function FV1 ,...,VN satisfies the qKZB equations (1.18) FV1 ,...,VN (λ, µ) = Kj ⊗ Dj FV1 ,...,VN (λ, µ). Theorem 1.4 (The dual qKZB equations). We have (1.19) FV1 ,...,VN (λ, µ) = Dj∨ ⊗ Kj∨ FV1 ,...,VN (λ, µ). Example 4. Let g = sl2 , N = 2, V1 = V2 = C2 with standard basis v+ , v− . In this case, V1 ⊗ V2 [0] is 2-dimensional with basis v+ ⊗ v− , v− ⊗ v+ , and the action of the
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399
dynamical R-matrix in this basis is 1 R(λ) = −1 q −q q 2λ − 1
q −1/2 q −1 − q q −2λ − 1 −2λ −2λ . 2 −2 q −q q −q 2 q −2λ − 1
Therefore, if FV1 ,V2 (λ, µ) is represented by a 2-by-2 matrix with respect to the above basis, then the qKZB equation corresponding to j = 2 has the form −1 −2λ q −q q − q 2 q −2λ − q −2 2 q −2λ − 1 q −2λ − 1 −1 q −q 1 q 2λ − 1
F11 (λ + 1, µ) F12 (λ − 1, µ) F11 (λ, µ) F12 (λ, µ) q µ 0 . = × 0 q −µ F21 (λ, µ) F22 (λ, µ) F21 (λ + 1, µ) F22 (λ − 1, µ) Here for convenience we took the shift operator 72 from the left side of the equation to the right side. Remark 1. We should warn the reader that the term “qKZB equations” is normally used for equations that contain elliptic dynamical R-matrices with spectral parameters and are difference equations with respect to these parameters z1 , . . . , zN (see, e.g., [FeTV1]). The equations we consider are a limiting case of the “genuine” qKZB equations, when the modular parameter τ goes to infinity and the ratios of the spectral parameters zj /zj +1 go to zero, with e−2π Im τ 0 n=1
where for a positive root α we define kαV := max{n : V [nα] = 0}. Define the function (6.7)
uV (λ, µ) := δV ∗ (−λ − ρ)δV (−µ − ρ)uˆ V (λ, µ).
Proposition 6.3. The function uV is symmetric: uV (λ, µ) = u∗V ∗ (µ, λ), and it is a product of q −2(λ,µ) and a trigonometric polynomial of λ and µ. In particular, it is holomorphic in λ, µ. Proof. The symmetry of uV follows from Corollary 6.2. The fact that uV is holomorphic in µ is a consequence of Lemma 5.1. The fact that uV is holomorphic in λ follows from the symmetry. Using (6.5), one gets the following expression of uV in terms of FV : (6.8) uV (λ, µ) = q −(ν,ν+2ρ) δV ∗ (−λ − ρ)Q(q, λ) ⊗ δV (−µ − ρ)Q(q, µ) FV (λ, µ). In Section 8 we show that the function uV for Uq (sl2 ) coincides (up to a constant factor) with the trigonometric limit of the universal hypergeometric function u introduced in [FeV3].
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7. Calculation of the functions uV and FV for sl2 7.1. Calculation of the trace function. Recall that Uq (sl2 ) is generated by E, F, with relations
q h,
(7.1)
q h Eq −h = q 2 E,
q h F q −h = q −2 F,
EF − F E =
q h − q −h , q − q −1
and the coproduct is defined by (7.2)
$(E) = E ⊗ q h + 1 ⊗ E,
$(F ) = F ⊗ 1 + q −h ⊗ F.
Weights for Uq (sl2 ) can be identified with complex numbers: we say that a vector v in a Uq (sl2 )-module has weight µ if q h v = q µ v. In this case we have (µ, µ ) = µµ /2. We write q λ for q (λ,α) . Thus, the meaning of q λ in this section is different from the previous sections. Recall also that for any number a the q-number [a]q is defined by [a]q = (q a − q −a )/(q − q −1 ). Consider the function +V (λ, µ), where V is the representation of Uq (sl2 ) with highest weight 2m, m ∈ Z+ . Since the weight space V [0] is 1-dimensional, this function can be considered as a scalar function. Theorem 7.1. The function +V (λ, µ) is given by the formula +V (λ, µ) = q λµ
m
l q l(l−1)/2 q − q −1
l=0
(7.3) ×
[m + l]q ! q −2lλ l , l−1 −2(λ−j ) [l]q ![m − l]q ! j =0 1 − q 2(µ−j ) j =0 1 − q
where [n]q ! = [1]q · · · [n]q . Proof. We fix a generator w0 of V [0]. Let us compute the intertwining operator ( µ : Mµ → M µ ⊗ V . Let wβ , β = m, m − 1, . . . , −m, be the basis of V defined by the condition F wβ = wβ−1 if β = −m. (This basis is unique up to a common scalar.) The image of the highest-weight vector under the operator (µ has the form (7.4)
( µ vµ =
m
cj (µ)F j vµ ⊗ wj ,
j =0
where c0 = 1. Let us compute the coefficients cj (µ). They are computed from the condition that $(E) = E ⊗ q h + 1 ⊗ E annihilates the right-hand side of (7.4). Using the formula (7.5)
ewβ = [m + β + 1]q [m − β]q wβ+1 ,
β = m,
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we can rewrite this condition in the form (7.6)
q 2i [i]q [µ − i + 1]q ci (µ) + [m + i]q [m − i + 1]q ci−1 (µ) = 0,
i ≥ 1.
Solving this recurrence relation, we obtain (7.7)
ci (µ) = (−1)i q −i(i+1)
i
[m + i]q ! [µ − i + 1]−1 q . [i]q ![m − i]q ! j =1
F kv
Now we need to compute (µ the q-binomial theorem, we obtain (7.8)
µ.
For this, we need to compute $(F k ). Using
k k k q −lh F k−l ⊗ F l , $ F k = F ⊗ 1 + q −h ⊗ F = l q −2 l=0
where
k i k i=k−l+1 1 − p := l . i l p i=1 1 − p
(7.9)
Now, using the intertwining property of (µ , we obtain ( µ F k vµ = $ F k ( µ v µ
k m k (7.10) −lh k−l l = q F ⊗F cj (µ)F j vµ ⊗ wj , l q −2 l=0
j =0
This double sum reduces to a single sum if we use the following version of the q-binomial theorem: l
k −1 (7.11) x k−l = 1 − pi x . l p k≥l
i=0
Substituting (7.7) into (7.10) and using (7.11), we obtain (7.3). The theorem is proved. Corollary 7.2. We have uV (λ, µ) = q −λµ
m l=0
(7.12) ×
l q −l(l−1)/2 q − q −1
[m + l]q ! q −l(λ+µ) [l]q ![m − l]q !
m
q λ+j − q −λ−j q µ+j − q −µ−j .
j =l+1
Proof. The statement is obtained from Theorem 7.1 and formulas (6.4) and (6.7). The function uV is manifestly symmetric in λ and µ, as predicted by Proposition 6.3.
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7.2. A formula for Q(µ). Now we would like to compute the function FV . So it remains to compute the value of Q(µ) on the zero-weight subspace of V . This value is given by the following lemma. Lemma 7.3. The element Q(µ) acts on the zero-weight subspace of V by the formula Q(µ)|V [0] = q −2m
(7.13)
m
q −2µ−2j +2 − q −2m j =1
q −2µ−2j − 1
.
Proof. Denote by Qr,l (µ) the eigenvalue of Q(µ) on the subspace of weight l of the representation of Uq (sl2 ) with highest weight r. It is clear that Q0,0 = 1. The values of Q1,1 and Q1,−1 are easily computed from the definition. Namely, from the ABRR equation (see Lemma 2.4) we have (µ) = 1 +
q −1 − q F ⊗E +··· , q −2µ − q −2 q −h ⊗ q h
and, therefore, Q1,−1 = 1,
(7.14)
Q1,1 =
q −2µ − q −2 . q −2µ − 1
Now consider the subspace of weight l in the tensor product C2 ⊗ W , where W has highest weight r. Take the determinant of both sides of (2.38) restricted to this subspace. Using the strict triangularity of J, we can ignore the J terms and obtain (7.15)
Qr+1,l (µ)Qr−1,l (µ) = Qr,l−1 (µ − 1)Qr,l+1 (µ + 1)
q −2µ − q −2 . q −2µ − 1
This implies Qr+1,l (µ) Qr,l−1 (µ − 1) q −2µ − q −2 , = Qr,l+1 (µ + 1) Qr−1,l (µ) q −2µ − 1
(7.16) from which we get (7.17)
Qr+1,l (µ) = Qr,l+1 (µ + 1)
(r+l−1)/2
j =0
q −2µ − q −r−l−1 q −2µ+2j − q −2 = . q −2µ+2j − 1 q −2µ − 1
This yields (7.18)
Qr+1,l (µ) =
(r−l+1)/2
j =0
q −2µ−2j − q −r−l−1 . q −2µ−2j − 1
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In particular, Q2m,0 (µ) =
(7.19)
m
q −2µ−2j − q −2m j =0
q −2µ−2j − 1
= q −2m
m
q −2µ−2j +2 − q −2m j =1
q −2µ−2j − 1
,
as desired. 7.3. Calculation of FV (λ, µ) Proposition 7.4. The function FV (λ, µ) is given by (7.20) FV (λ, µ) = q
−λµ
m
m
j =1
l [m + l]q ! q −2µ−2j − 1 2m q q l(l−1)/2 q − q −1 −2µ−2j +2 −2m [l]q ![m − l]q ! q −q
× l
j =1
l=0
q −2lλ 1 − q −2(µ+j )
l
j =1
1 − q −2(λ−j )
.
Proof. Using the definition of FV , Theorem 7.1, and Lemma 7.3, we get (7.20). Remark. Note that expression (7.20) is not manifestly symmetric. In fact, the separate terms in the sum (7.20) are not symmetric, and it is only the whole sum that has the symmetry λ → µ. 8. Integral representation of the trace function uV for g = sl2 . In [FeV3], G. Felder and the second author, studying the qKZB difference equations, defined the universal hypergeometric function um (λ, τ, µ, p) (depending on a parameter q) with a number of interesting properties. In this section we consider the trigonometric limit of this function and show that it coincides, up to a constant factor, with the function uV (m) defined by (6.8), where V (m) is the irreducible representation of Uq (sl2 ) with highest weight 2m. By definition, the trigonometric limit of um (λ, τ, µ, p) is the leading coefficient of the asymptotic expansion of um (λ, τ, µ, p) as the modular parameters τ, p tend to i∞. We denote this leading coefficient by um (λ, µ). Sending τ, p to i∞ in the definition of [FeV3], one obtains the following definition of the function um (λ, µ). Let q = et with Re(t) > 0, and let 0 < |A| < 1. Define a function (8.1) Im (λ, µ, A) =q
−λµ
m
Tj A−2 q λ+m − q −λ−m Tj A−2 q µ+m − q −µ−m 1 − Tj A2 1 − Tj A−2 |T1 |=···=|Tm |=1 j =1
2 q −2 1 − Ti Tj−1 dTj × . ∧m √ −1 −2 −1 2 j =1 2π −1Tj 1 − T i Tj q 1 − T i Tj q 1≤i<j ≤m
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It is obvious that this function analytically continues to a rational function in A. We denote this analytic continuation also by Im . Definition. We put (8.2)
um (λ, µ) = Im λ, µ, q m .
Remark. Note that A = q m does not satisfy the condition |A| < 1, which is why we needed to talk about the analytic continuation. The main result of this section is the following theorem. Theorem 8.1. Let V (m) be the irreducible representation of Uq (sl2 ) with highest weight 2m. Then (8.3)
uV (m) (λ, µ) = q (3m−1)m
m [2m]q ! q − q −1 um (λ, µ), m!
where uV (m) is as in Section 6. Proof. The rest of the section is the proof of Theorem 8.1. Lemma 8.2. We have
(8.4)
2 1 − Ti Tj−1 dTj ∧m √ −1 −2 −1 2 j =1 2π −1Tj 1 − T i Tj q |T1 |=···=|Tm |=1 1≤i<j ≤m 1 − Ti Tj q = q −m(m−1)/2
m! . [m]q !
Proof. Let 0 ≤ |p| < 1. Consider the Macdonald denominator of type Am−1 (see [M2]): −1 xi xj , p ∞ $p,t x1 , . . . , xm = (8.5) , −1 i =j txi xj , p ∞
j where (a, p)∞ := ∞ j =0 (1 − ap ). The Macdonald constant term identity (see [M2, pp. 20–21]) says that the constant term of the Laurent series (8.5) (with respect to xi ) is given by j −i
t p, p ∞ t j −i , p ∞ . c. t.($p,t ) = m! (8.6) t j −i+1 , p ∞ t j −i−1 p, p ∞ i<j Setting in this identity p = 0, we get (8.7)
c. t.($0,t ) = m!
(1 − t)m . (1 − t) · · · (1 − t m )
Substituting t = q −2 , we obtain the lemma.
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421
Define the expression (8.8)
Ik,m = q −k(λ+µ+2m)−k(k−1)/2
k! , [k]q !
0 ≤ k ≤ m,
and the differential form (8.9)
k
Tj A−2 q λ+m − q −λ−m Tj A−2 q µ+m − q −µ−m 1 − Tj A−2 q 2m−2k−2 Ok,m = 1 − Tj A2 1 − Tj A−2 q 2m−2k 1 − Tj A−2 q −2 j =1 2
1 − Ti Tj−1 dTj × . ∧k √ −1 −2 −1 2 j =1 2π −1T T q T q 1 − T 1 − T j i i j j 1≤i<j ≤k Lemma 8.3. We have
(8.10)
|Tj |=A2 q −2(m−k) (1+%)
Ok,m = −kq
−2(m−k)
×
q λ+k − q −λ−k q µ+k − q −µ−k 1 − q −2 1 − A4 q −2(m−k) 1 − q −2(m−k+1)
|Tj |=A2 q −2(m−k+1) (1+%)
Ok−1,m + Ik,m .
Proof. Let us perform the integration with respect to Tk for fixed T1 , . . . , Tk−1 . It is obvious that the differential form Fk,m , as a function of Tk , has two simple poles inside the circle of integration: Tk = A2 q 2k−2m and Tk = 0. Therefore, the integral with respect to Tk is equal to the sum of residues at these two poles. The residue at the first pole can be found by a direct computation and equals the first term on the right-hand side of (8.10). The residue at zero equals to Ik,m by Lemma 8.2. Lemma 8.3 is proved. Now let us prove Theorem 8.1. Let us move the contour of integration in the definition of Im from |Tj | = 1 to |Tj | = A2 (1+%), via contours |Tj | = B, A2 (1+%) ≤ B ≤ 1. On the way, we do not run into any poles; therefore, we have −λµ−m(m−1) (8.11) Om,m |A=q m um (λ, µ) = q |Tj |=A2 (1+%)
(in the sense of analytic continuation). So, to prove Theorem 8.1, it is enough to compute |Tj |=A2 (1+%) Om,m |A=q m . We do it by using the recursive relation given in Lemma 8.3. Lemma 8.4. We have (8.12)
|Tj |=A2 q −2(m−k) (1+%)
Ok,m |A=q m =
k j =0
ckj Ij,m ,
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where (8.13) ckj = (−1)
k−j
1−q
−2 k−j
k k! j!
i=j +1
q λ+i − q −λ−i q µ+i − q −µ−i 2(i−m) q . 1 − q 2(i+m) 1 − q 2(i−m−1)
Proof. The proof is a straightforward induction in k using Lemma 8.3. Substituting k = m in Lemma 8.4, and using the definition of Ik,m , we find the following expression for um : (8.14) um (λ, µ) = q −m(3m−1)
m
−m −λµ −l(λ+µ)−l(l−1)/2 l m! q − q −1 q q q − q −1 [2m]q ! l=0
×
[m + l]q ! [l]q ![m − l]q !
m
q λ+i − q −λ−i q µ+i − q −µ−i .
i=j +1
Comparing (8.14) with (7.12), we get Theorem 8.1. 9. Trace functions and Macdonald theory. In this section, following [EK1] and [FeV1], we connect the results of this paper with the Macdonald-Ruijsenaars theory. We restrict ourselves to the case of g = sln , N = 1, and we let V be the q-analogue of the representation S mn Cn . The zero-weight subspace of this representation is 1dimensional, so the function +V can be regarded as a scalar function. We denote this scalar function by +m (q, λ, µ). Recall the definition of Macdonald operators (see [M1] and [EK1]). They are operators on the space of functions f (λ1 , . . . , λn ) that are invariant under simultaneous shifting of the variables λi → λi + c, and they have the form
tq 2λi − t −1 q 2λj TI , (9.1) Mr = 2λi − q 2λj q / I ⊂{1,...,n}:|I |=r i∈I,j ∈I / I and TI λj = λj + 1 if j ∈ I . Here q, t are parameters. We where TI λj = λj if j ∈ assume that t = q m+1 , where m is a nonnegative integer. It is known from [M1] that the operators Mr commute. From this it can be deduced that for a generic µ = (µ1 , . . . , µn ), µi = 0, there exists a unique power series fm0 (q, λ, µ) ∈ C[[q λ2 −λ1 , . . . , q λn −λn−1 ]] such that the series fm (q, λ, µ) := q 2(λ,µ−mρ) fm0 (q, λ, µ) satisfies difference equations (9.2) q 2 i∈I (µ+ρ)i fm (q, λ, µ). Mr fm (q, λ, µ) = I ⊂{1,...,n}:|I |=r
TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS
423
Remark. The series fm0 is convergent to an analytic (in fact, a trigonometric) function. The following theorem is contained in [EK1]. Theorem 9.1 [EK1, Theorem 5]. We have (9.3)
fm (q, λ, µ) = γm (q, λ)−1 +m q −1 , −λ, µ ,
where (9.4)
γm (q, λ) :=
m
q λl −λj − q 2i q λj −λl .
i=1 l<j
Remark. The exact statement of [EK1, Theorem 5], in our conventions, is that the function fm (q, λ, µ)γm (q, λ) is equal to Tr |Mµ ((Vµ (q −1 )q 2λ ), which is equivalent to Theorem 9.1. Let ᏰW (q −1 , −λ) denote the difference operator, obtained from the operator ᏰW defined in Section 1 by the transformation q → q −1 and by the change of coordinates λ → −λ. Let ;r Cn denote the q-analogue of the rth fundamental representation of sln . Corollary 9.2. We have Ᏸ;r Cn q −1 , −λ = δq (λ)γm (q, λ) ◦ Mr ◦ γm (q, λ)−1 δq (λ)−1 . Proof. This follows from Theorems 9.1 and 1.1. In conclusion of this section, we would like to make several important remarks. Remark 1. Corollary 9.2 is a degenerate (trigonometric) case of [FeV1, Theorem 5.2], which says that the elliptic Ruijsenaars operators are transfer matrices of the elliptic quantum sln acting in V [0]. Thus, Theorems 1.1 and 9.1 immediately imply the trigonometric case of [FeV1, Theorem 5.2] (i.e., the case without spectral parameter). Remark 2. Conversely, the trigonometric case of [FeV1, Theorem 5.2] together with Theorem 1.1 immediately implies Theorem 9.1 (and many other results of [EK1]). This is a “direct” proof of Theorem 9.1, in the sense that it involves (unlike the original proof of [EK1]) a direct computation of the radial parts of the central elements of Uq (g). (Another direct proof of Theorem 9.1 is given in [Mi], where the radial part of the central element corresponding to the vector representation is computed.) Remark 3. The line of argument discussed in Remark 2 can be extended to the elliptic case. Namely, combining an elliptic analogue of Theorem 1.1 (for affine Lie algebras at the critical level) and [FeV1, Theorem 5.2], one can prove an elliptic analogue of Theorem 9.1, which says that the radial parts of the central elements
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n ) at the critical level corresponding to evaluation modules ;r Cn (z), acting of Uq (sl on functions with values in V [0], are elliptic Ruisjsenaars operators. This has been a conjecture for a number of years (see, e.g., [Mi, p. 415]). We plan to do this in a subsequent paper of this series. Remark 4. In many arguments of this paper, Verma modules Mµ can be replaced with finite-dimensional irreducible modules Lµ with sufficiently large highest weight, and one can prove analogues of Theorems 1.1–1.5 in this situation (in the same way). ˆ Vµ q 2λ ), where ( ˆ Vµ : Lµ → Lµ ⊗ V ⊗ ˆ m (q, λ, µ) = Tr(( In particular, one may set + ∗ V [0] is the intertwiner with highest coefficient 1. (Such an operator exists if and only if µ − mρ ≥ 0; see [EK1].) Then one can show analogously to Theorem 9.1 ˆ m (q −1 , −λ, µ + mρ) is (see [EK1]) that the function fˆm (q, λ, µ) := γm (q, λ)−1 + 2λ the Macdonald polynomial Pµ (q, t, q ) with highest weight µ (µ is a dominant integral weight). In this case, Theorem 1.1 says that Macdonald’s polynomials are eigenfunctions of Macdonald’s operators, Theorem 1.2 gives recursive relations for Macdonald’s polynomials with respect to the weight (for sl(2), the usual 3-term relation for orthogonal polynomials), and Theorem 1.3 is the Macdonald symmetry identity (see [M1]). (This representation-theoretic derivation of the symmetry identity is somewhat different from the one in [EK3], where a pictorial argument is used.) 10. Limiting cases. In this section we discuss various degenerations of the function FV1 ,...,VN (q, λ, µ) and the corresponding degenerate versions of Theorems 1.1– 1.5. The main limiting cases we are interested in are the classical limit and the rational limit. The classical limit corresponds to passing from Uq (g) to g in the trace construction; in this limit the function F depends rationally on µ but trigonometrically of λ. This limit corresponds to the theory of spherical functions on the Lie group G associated with g, which is discussed in [EFK1]. In the rational limit, which corresponds to the theory of spherical functions on g rather than G, the function F becomes rational in both λ and µ, restoring the symmetry. In this limit, the function F is the Baker-Akhiezer function for a multivariable bispectral problem (see [Be]). 10.1. The classical (KZB) limit. Let (10.1)
λ FVc1 ,...,VN (λ, µ) = lim FV1 ,...,VN q = et , , µ . t→0 2t
We call this limit the classical limit. The existence of this limit follows from Proposition 10.1. Remark. Here and below we write the dependence of functions on q explicitly, since in this section q is allowed to vary. Example 1. If g = sl(2), N = 1, and V = V1 is the 3-dimensional representation, we have 1 1 + eλ µ FVc (λ, µ) = e−λµ/2 1− (10.2) . µ−1 µ 1 − eλ
TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS
425
The classical limit is obtained when, as in the situation of Section 1, we take the ordinary enveloping algebra U (g) instead of the quantized one Uq (g). More precisely, let (Vµ be intertwining operators for U (g) defined as in Section 1, and set +Vc 1 ,...,VN (λ, µ) = Tr (V1 N (∗i) ⊗ 1N−1 · · · (VµN eλ . (10.3) µ+
i=2 h
Also, set (10.4)
δ(λ) = e(λ,ρ)
1 − e−(λ,α) ,
α>0
and let Qc (µ) be the limit of Q(µ) as q → 1 (i.e., it is defined as in Section 1 from representation theory of U (g)). Then we have the following proposition. Proposition 10.1. We have (10.5) FVc1 ,...,VN (λ, µ) (∗1) c = δ(λ) Qc−1 (µ)(∗N) ⊗ · · · ⊗ Qc−1 µ − h(∗2,...,∗N) +V1 ,...,VN (λ, −µ − ρ). Proof. The proof is straightforward. Let us now look at the degenerations of the properties of FV1 ,...,VN in the classical limit. We start with the analogue of Theorem 1.1. First of all, we have the following analogue of Proposition 2.1, which is proved analogously to Proposition 2.1. Proposition 10.2. (i) For any element X of U (g), there exists a unique differential operator DX acting on V [0]-valued functions, such that Tr (Vµ Xeλ = DX Tr (Vµ eλ . (10.6) (ii) If X is central, then DXY = DY DX for all Y ∈ U (g). In particular, if X, Y are central, then DX DY = DY DX . The operator DX can be computed explicitly for any element X, but in general the answer is complicated. However, if X is the quadratic Casimir C, the answer is easy to write down. Namely, define D˜ X = δ(λ)DX δ(λ)−1 . Then we have (see [E], [ESt1]): (10.7)
D˜ C = $h −
f α eα
α>0
2 sinh (1/2)(λ, α) 2
− (ρ, ρ),
where fα , eα are root generators such that (eα , fα ) = 1, and $h is the Laplacian on the Cartan subalgebra associated with the standard invariant form.
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ETINGOF AND VARCHENKO
Thus, we have the following classical analogue of Theorem 1.1. Theorem 10.3. For any X in the center of U (g), let pX be the symmetric polynomial on h∗ such that X|Mµ = pX (µ + ρ). Then we have λ c D˜ X FV1 ,...,VN (λ, µ) = pX (−µ)FVc1 ,...,VN (λ, µ).
(10.8) In particular,
(10.9)
$h −
f α eα
2 sinh2 (1/2)(λ, α) α>0
FVc1 ,...,VN (λ, µ) = (µ, µ)FVc1 ,...,VN (λ, µ).
Formula (10.9) was obtained in [E] and [ESt1], but it can also be derived by taking the classical limit in Theorem 1.1. Now let us consider the classical analogue of Theorem 1.2. Let Ᏸ∨,c W denote the difference operators defined by formula (1.12) for q = 1 (i.e., R(µ) are the exchange matrices for U (g) with µ replaced by −µ − ρ). Then we have the following result, obtained by passing to the limit in Theorem 1.2. Theorem 10.4. We have ∨,c,µ
ᏰW
(10.10)
FVc1 ,...,VN (λ, µ) = χW e−λ FVc1 ,...,VN (λ, µ).
Example 2. In the case of Example 1, Theorems 10.3 and 10.4 have the form 2 1 ∂ µ2 c c − F (λ, µ) (λ, µ) = F V 4 V ∂λ2 2 sinh2 (λ/2) and
(µ − 2)(µ + 1) −1 c T+ T FV (λ, µ) = eλ/2 + e−λ/2 FVc (λ, µ) µ(µ − 1)
(where T is the shift by 1 in µ), which is easily checked from (10.2). Now consider the classical limit of Theorem 1.3. For this purpose introduce the classical dynamical r-matrix r(λ), which is the classical limit of the exchange matrix R(q, λ). This matrix is defined by the formula λ R q = et , = 1 − 2r(λ)t + O t 2 (10.11) 2t and is equal to
1 1 1 cotanh (λ, α)eα ∧ fα , r(λ) = − O + 2 2 2 α>0
where O is the Casimir tensor (see [EV]). Taking the quasi-classical limit in Theorem 1.3 and using that r(−λ) = r 21 (λ), we obtain the following result.
TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS
427
Theorem 10.5. For any j = 1, . . . , N , we have ∂ − rlj (λ) − rj l (λ) FVc1 ,...,VN (λ, µ) ∂h(j ) l<j l>j (10.12) (∗j ) 1 2 (∗l) = µ+ FVc1 ,...,VN (λ, µ), xi + xi ⊗ xi 2 ∗j l<j
where (∂/∂h(j ) )X(λ) = (∂/∂ν)X(λ) if X is a tensor-valued function whose j th component has weight ν. The last equation is the trigonometric limit of the KZB equation, which is why the classical limit is called “the KZB limit.” Let us now consider the classical limit of Theorem 1.4. Let Kj∨,c be the difference operators defined by formula (1.17) for q = 1 (i.e., R(µ) are exchange matrices for U (g) with µ replaced by −µ − ρ). Then we have the following result, obtained by passing to the limit in Theorem 1.4. Theorem 10.6. We have (10.13)
Kj∨,c FVc1 ,...,VN (λ, µ) = eλ j FVc1 ,...,VN (λ, µ).
Finally, Theorem 1.5 does not have an analogue in the classical limit. In this limit, the symmetry between λ and µ is destroyed, since F c is a product of e−(λ,µ) with a function that is trigonometric in λ but rational in µ. 10.2. The rational limit. The rational limit is a further degeneration of the classical limit. Namely, let µ (10.14) . FVr1 ,...,VN (λ, µ) = lim FVc1 ,...,VN λγ , γ →0 γ We call this limit the rational limit. The existence of this limit and the fact that det(F r ) = 0 can be deduced from [ESt1, Corollary 3.3]. Example 3. If g = sl(2), N = 1, and V = V1 is the 3-dimensional representation, we have 2 r −λµ/2 1+ (10.15) . FV (λ, µ) = e λµ r The degeneration of Theorem 1.1 in this limit is the following theorem. Let D˜ X r be the rational limit of D˜ X ; that is, D˜ X (λ) is the leading coefficient of D˜ X (γ λ) as γ → 0. For instance, 2fα eα (10.16) . D˜ Cr = $h − (λ, α)2 α>0
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Theorem 10.7. For any X in the center of U (g), let pX be the symmetric polyr be the top degree component of nomial on h∗ such that X|Mµ = pX (µ + ρ). Let pX pX . Then we have r,λ r r D˜ X FV1 ,...,VN (λ, µ) = pX (−µ)FVr1 ,...,VN (λ, µ).
(10.17) In particular,
2fα eα $h − FVr1 ,...,VN (λ, µ) = (µ, µ)FVr1 ,...,VN (λ, µ). (λ, α)2
(10.18)
α>0
The degeneration of Theorem 1.2 looks as follows. Theorem 10.8. Equations (10.17) and (10.18) are satisfied for the function FVr,∗∗ ,...,V ∗ . N
1
Example 4. In the situation of Example 3, Theorems 10.7 and 10.8 have the form 2 ∂ 2 µ2 r r F (λ, µ), F − (λ, µ) = V 4 V ∂λ2 λ2 2 2 λ2 r ∂ r F (λ, µ), − (λ, µ) = F V 4 V ∂µ2 µ2 which is easily checked from (10.15). Using the asymptotics of FVr1 ,...,VN (λ, µ) at infinity, similarly to arguments of Section 5, one can deduce from Theorems 10.7 and 10.8 the following analogue of Theorem 1.5 (the symmetry theorem). Theorem 10.9. The function FVr1 ,...,VN is symmetric: (10.19)
FVr1 ,...,VN (λ, µ) = FVr,∗∗ ,...,V ∗ (µ, λ). N
1
Thus, the symmetry, lost in the first limit, is restored after taking the second limit. Remark 1. Another proof of Theorem 10.8 is based on representation of the above sequence of two limits as a single limiting procedure, which is symmetric in λ and µ. Namely, one can show that λ µ FVr1 ,...,VN (λ, µ) = lim FV1 ,...,VN q = est/2 , , (10.20) , s,t→0 t s after which Theorem 10.9 follows from Theorem 1.5. Remark 2. Theorems 10.7 and 10.8 show that the function FVr1 ,...,VN (λ, µ) is a solution of the matrix bispectral problem in several variables (on the bispectral problem, see, e.g., [DuG], [G]). The Baker-Akhiezer function of the rational Calogero
429
TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS
system of type A, which is a known solution of the multidimensional bispectral problem (see [VeStC]; see also [Be]), is a special case of FVr1 ,...,VN (λ, µ) (N = 1, V = V1 = S mn Cn ). Finally, let us consider the rational limit of Theorems 1.3 and 1.4. To formulate the analogue of Theorem 1.3, introduce the rational limit of the classical dynamical r-matrix, r 0 (λ) = limγ →0 γ r(γ λ). It has the form r 0 (λ) =
(10.21)
eα ∧ f α α>0
(λ, α)
.
Taking the rational limit in Theorem 10.4, we get the following. Theorem 10.10. For any j = 1, . . . , N , we have (10.22)
∂ − rlj0 (λ) − rj0l (λ) FVr1 ,...,VN (λ, µ) = µ∗j FVr1 ,...,VN (λ, µ). ∂h(j ) l<j
l>j
The analogue of Theorem 1.4 is the following. Theorem 10.11. Equation (10.22) is satisfied for the function FVr,∗∗ ,...,V ∗ . N
1
10.3. The qKZ and KZ limits. Assume that |q| < 1. The qKZ limit is defined by (10.23)
qKZ
FV1 ,...,VN (λ, µ) =
lim
(λ,αi )→−∞
q 2(λ,µ) FV1 ,...,VN (λ, µ).
It is easy to check using [EV, Theorem 50] that qKZ
(10.24)
FV1 ,...,VN (λ, µ) (∗1) = Q−1 (µ)(∗N) ⊗ · · · ⊗ Q−1 µ − h(∗2,...,∗N) ! " × (V1 N (∗i) ⊗ 1N−1 · · · (VµN µ+
i=2 h
(∗1) 1,...,N = Q−1 (µ)(∗N) ⊗ · · · ⊗ Q−1 µ − h(∗2,...,∗N) J (µ)∗ , where , denotes the highest-matrix element. (The last expression is an endomorphism of (VN∗ ⊗· · ·⊗V1∗ )[0], which is regarded as an element of (V1 ⊗· · ·⊗VN )[0]⊗ (VN∗ ⊗ · · · ⊗ V1∗ )[0].) In particular, this function is independent on λ. Let us now consider the behavior of the equations given by Theorems 1.1–1.5 in the qKZ limit. The MR equations given by Theorem 1.1 become trivial. Namely, when (λ, αi ) → −∞, we have J(λ) → 1, and hence R(λ) → 21 . The matrix 21 is triangular, so
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only its diagonal part contributes to the trace. Inspection of this diagonal part shows that lim(λ,αi )→−∞ ᏰW = ν dimW [ν] Tν , and the limiting equation is
dimW [ν] q −2(ν,µ) Tν F qKZ = χW q −2µ F qKZ
ν
(where Tν is the shift of λ), which is a trivial consequence of the fact that F qKZ is independent on λ. The dual MR equations given by Theorem 1.2 have a slightly more interesting limit. It is easy to see that the only term on each side of (1.11) which survives in the limit is the term corresponding to the lowest weight νW of W . Therefore, the limiting equation has the form (10.25) qKZ ∗ (∗1,...,∗N−1) vW ⊗ 1, R01 · · · R0N WV ∗ µ+h W V ∗ (µ)(vW ⊗ 1) FV1 ,...,VN (µ + νW ) N
1
qKZ
= FV1 ,...,VN (µ). The qKZB equations become the trigonometric limit of the qKZ equations. Namely, for all j = 1, . . . , N , we have (10.26)
qKZ
qKZ
−1 −1 −2µ j,j · · · j −1,j ⊗ Dj FV1 ,...,VN (µ) = FV1 ,...,VN (µ). +1 · · · j N q j 1j
If N = 2, these equations are closely related to the ABRR equation (see Lemma 2.4). In general, they are essentially the N-component version of the ABRR equation. Remark. The dual qKZB equations do not seem to have a reasonable qKZ limit. Also, the symmetry relation (Theorem 1.5) does not hold in the qKZ limit since the function depends on µ and not on λ. The KZ limit is obtained from the qKZ limit as q → 1, in which case the qKZ equations degenerate into the trigonometric KZ equations (the quasi-classical limit of the ABRR equation). We leave it to the reader to derive the limiting equations in this case. We plan to consider these limits in more detail in another paper in the more interesting case of affine Lie algebras and quantum affine algebras. References [ABRR] [Be]
D. Arnaudon, E. Buffenoir, E. Ragoucy, and Ph. Roche, Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys. 44 (1998), 201–214. Yu. Berest, “Huygens’ principle and the bispectral problem” in The Bispectral Problem (Montreal, 1997), CRM Proc. Lecture Notes 14, Amer. Math. Soc., Providence, 1998, 11–30.
TRACES OF INTERTWINERS AND DIFFERENCE EQUATIONS [Ber] [C] [D] [DuG] [E] [EFK1] [EFK2] [EK1] [EK2] [EK3] [EK4] [ESch] [ESt1] [ESt2] [EV] [Fe]
[FeTV1]
[FeTV2] [FeV1] [FeV2]
[FeV3] [FeV4] [FeV5] [FR]
431
D. Bernard, On the Wess-Zumino-Witten models on the torus, Nuclear Phys. B 303 (1988), 77–93. I. Cherednik, Macdonald’s evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), 119–145. V. G. Drinfeld, On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321–342. J. J. Duistermaat and F. A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177–240. P. I. Etingof, Quantum integrable systems and representations of Lie algebras, J. Math. Phys. 36 (1995), 2636–2651. P. Etingof, I. Frenkel, and A. A. Kirillov Jr., Spherical functions on affine Lie groups, Duke Math. J. 80 (1995), 59–90. , Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Math. Surveys Monogr. 58, Amer. Math. Soc., Providence, 1998. P. I. Etingof and A. A. Kirillov Jr., Macdonald’s polynomials and representations of quantum groups, Math. Res. Lett. 1 (1994), 279–296. , On the affine analogue of Jack and Macdonald polynomials, Duke Math. J. 78 (1995), 229–256. , Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials, Compositio Math. 102 (1996), 179–202. , On Cherednik-Macdonald-Mehta identities, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 43–47, available from http://www.ams.org/era/. P. Etingof and O. Schiffmann, Lectures on the dynamical Yang-Baxter equations, preprint, http://xxx.arXiv.org/abs/math.QA/9908064. P. Etingof and K. Styrkas, Algebraic integrability of Schrödinger operators and representations of Lie algebras, Compositio Math. 98 (1995), 91–112. , Algebraic integrability of Macdonald operators and representations of quantum groups, Compositio Math. 114 (1998), 125–152. P. Etingof and A. Varchenko, Exchange dynamical quantum groups, Comm. Math. Phys. 205 (1999), 19–52. G. Felder, “Conformal field theory and integrable systems associated to elliptic curves” in Proceedings of the International Congress of Mathematicians (Zürich, 1994), Birkhäuser, Basel, 1995, 1247–1255. G. Felder, V. Tarasov, and A. Varchenko, “Solutions of the elliptic QKZB equations and Bethe ansatz, I” in Topics in Singularity Theory, Adv. Math. Sci. 34, Amer. Math. Soc. Transl. Ser. 2 180 Amer. Math. Soc., Providence, 1997, 45–76. , Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, preprint, http://xxx.arXiv.org/abs/q-alg/9705017. G. Felder and A. Varchenko, Elliptic quantum groups and Ruijsenaars models, J. Statist. Phys. 89 (1997), 963–980. , Resonance relations for solutions of the elliptic QKZB equations, fusion rules, and eigenvectors of transfer matrices of restricted interaction-round-a-face models, Commun. Contemp. Math. 1 (1999), 335–403. , Quantum KZB heat equation, modular transformations and GL(3, Z), I, preprint, http://xxx.arXiv.org/abs/math.QA/9809139. , The elliptic gamma function and SL(3, Z)×Z3 , preprint, http://xxx.arXiv.org/abs/ math.QA/9907061. , Quantum KZB heat equation, modular transformations, and GL(3, Z), II, preprint, 1999. I. Frenkel and N. Reshetikhin, Quantum affine algebras and holonomic difference
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Etingof: Department of Mathematics, Room 2-165, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA; [email protected] Varchenko: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA; [email protected]
Vol. 104, No. 3
DUKE MATHEMATICAL JOURNAL
© 2000
ON THE FINITE-GAP ANSATZ IN THE CONTINUUM LIMIT OF THE TODA LATTICE A. B. J. KUIJLAARS
1. Introduction. The finite, nonperiodic Toda lattice is a dynamical system given by the Lax pair dL/dt = BL − LB for tridiagonal n × n matrices
a1
b 1 L= 0
b1
0
a2
b2
b2
a3 .. .
.. ..
. .
bn−1
, bn−1
0
−b 1 B =
an
b1
0
0
b2
−b2
0 .. .
0
.. ..
. .
−bn−1
. bn−1 0
This corresponds to the system of equations dak 2 = 2 bk2 − bk−1 , k = 1, . . . , n, dt dbk = bk (ak+1 − ak ), k = 1, . . . , n − 1, dt
(1.1) (1.2)
with b0 = bn = 0. The Toda lattice is an integrable system that is solved explicitly by the inverse spectral method (see [M]). Deift and McLaughlin [DM] studied the continuum limit of the Toda lattice. Here the size n tends to infinity, and we write ak (t; n) and bk (t; n) to indicate the dependence on n. For given continuous functions a0 (y) and b0 (y) > 0 for y ∈ (0, 1), Deift and McLaughlin [DM] take initial values
k k ak (0; n) = a0 , bk (0; n) = b0 (1.3) n n and study the limiting behavior of a[xn] (tn; n),
b[xn] (tn; n)
as n → ∞ with fixed x ∈ (0, 1) and t > 0. Received 7 June 1999. 2000 Mathematics Subject Classification. Primary 37J35, 37K10; Secondary 31A15, 35Q53. Author’s work supported in part by research project number G.0278.97 and by a research grant from the Fonds voor Wetenschappelijk Onderzoek–Vlaanderen (FWO). 433
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This continuum limit has many similarities to the zero dispersion limit of the Korteweg–de Vries (KdV) equation ut − 6uux + 2 uxxx = 0,
u(x, 0) = u0 (x).
The limit 0 was first studied by Lax and Levermore [LL]. Working with singlewell initial data u0 (x) < 0 and using the inverse scattering transform method for the KdV equation, they obtained for each (x, t) a quadratic, constrained minimization problem over density functions in the spectral parameter. Their analysis revealed the significance of the set where the constraints are not effective. Under the assumption that this set is a finite number of intervals, Lax and Levermore described the weak limit of u(x, t; ) as 0 in terms of the endpoints of the intervals. The number N + 1 of intervals depends on (x, t). In a region of space-time where N is constant, it was moreover shown that the endpoints satisfy a hyperbolic system of equations, identical to the multiphase modulation equations derived earlier by Flaschka, Forest, and McLaughlin [FFM]. The nature of rapid, small-scale oscillations that lead to the weak limit if N ≥ 1 was later examined by Venakides [V]. These higher-order asymptotics are constructed out of the theta function associated with the hyperelliptic Riemann surface of genus N based on the 2N + 2 endpoints. See [DVZ] and [ELZ] for recent advances on the zero dispersion limit of the KdV equation. Also in the continuum limit of the Toda lattice, a quadratic minimization problem with constraints plays a prominent role. The nature of the support of the minimizer is again basic in the description of the singular limit. Based on a finite-gap ansatz (i.e., under the assumption that the set where the constraints are not valid is a finite union of intervals), weak limits are described in terms of the endpoints of these intervals (see [DM]). A similar approach was developed to describe the semiclassical limit of the defocusing nonlinear Schrödinger equation (see [JLM]). The zero-gap ansatz (i.e., N = 0) was established in all of the above cases at small times t < tb , where tb is the shock time for the formal limit, which is, for example, ut − 6uux = 0,
u(x, 0) = u0 (x)
in the case of the KdV equation, and the system at = 4bbx ,
bt = bax ,
a(x, 0) = a0 (x),
b(x, 0) = b0 (x)
(1.4)
in the case of the Toda lattice. In a number of cases a global zero or one-gap ansatz was established (see [DM], [T]). However, a general result was lacking until the work of Deift, Kriecherbauer, and McLaughlin [DKM] for the continuum limit of the Toda lattice. They proved that the finite-gap ansatz holds for real analytic spectral data provided that only one constraint is active. In general there are two constraints for the minimizer ψ, namely, ψ ≥ 0 and ψ ≤ φ, where φ is determined by the initial data. The result of [DKM] applies to the case in which the upper constraint φ is not active. It is the purpose of this paper to extend this result to the case in which both the nonnegativity constraint and the upper constraint are active. It turns out that the
ON THE FINITE-GAP ANSATZ
435
methods of [DKM] cannot be easily generalized to this case—at least this author was unable to do it. Instead a number of new ideas are used, coming from potential theory and finding their origin in approximation theory and the theory of orthogonal polynomials (see, e.g., [DK], [DS], [KM], [ST], and [To]). We expect that a combination of the ideas of [DKM] and this paper may be used to obtain the finite-gap ansatz for the zero dispersion limit of the KdV equation with real-analytic initial data. The next section contains a more detailed description of the extremal problem as it arises in the continuum limit of the Toda lattice. The main result of the paper is stated in Section 3, and its proof is given in Section 4. 2. The extremal problem. We describe the extremal problem as it appears in the continuum limit of the Toda lattice (see [DM]). For general information on this kind of extremal problem, we refer the reader to [D] and [ST]. The problem associated with the Toda lattice involves two “spectral functions” V and φ that are obtained from the initial data a0 and b0 of (1.3). We write α(y) = a0 (y) − 2b0 (y),
β(y) = a0 (y) + 2b0 (y).
(2.1)
Note that α(y) < β(y) for y ∈ (0, 1), since b0 (y) > 0. The following assumptions are made on the functions α(y) and β(y) (see [DM]): • α(y) has at most one critical point that, if it exists, is a minimum; • β(y) has at most one critical point that, if it exists, is a maximum; • α(0) = β(0) and α(1) = β(1). In [DM] it is assumed that α is strictly decreasing, but the same analysis goes through under the above assumptions (see also [KVA1]). Let A := min α(y),
B := max β(y).
y∈[0,1]
y∈[0,1]
It follows from the assumptions that for each x ∈ [A, B], the set {y ∈ [0, 1] : α(y) ≤ x ≤ β(y)} is a closed interval, which is denoted by [y− (x), y+ (x)]. Then V0 and φ are defined on [A, B] by the formulas
y− (x)
V0 (x) :=
cosh 0
1 φ(x) := π
y+ (x)
y− (x)
x − a0 (y) dy, 2b0 (y)
(2.2)
1 dy β(y) − x x − α(y)
(2.3)
−1
(see [DM]). Note that V0 was denoted by θ in [DM]. We are also using a different normalization for φ. We refer to the function V0 as the external field. The function φ is nonnegative and is called the constraint.
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For an arbitrary, continuous V : [A, B] → R and for a finite, nonnegative Borel measure µ on [A, B], the energy in the presence of V is 1 IV (µ) := dµ(x) dµ(y) + 2 V (x) dµ(x). (2.4) log |x − y|
For a positive Borel measure σ on [A, B] and a constant c ∈ (0, dσ ), we let σ,c ᏹ := µ : 0 ≤ µ ≤ σ, dµ = c (2.5) be the collection of positive Borel measures µ on [a, b] which are dominated by σ and have total mass c. We refer to c as the normalization constant. Associated with V , σ , and c is the energy minimization problem µ,c EV = inf IV (µ) : µ ∈ ᏹσ,c . (2.6) σ,c such It is known (see [DM], [DS]) that there exists a unique measure µ = µσ,c V in ᏹ σ,c σ,c that EV = IV (µV ). The minimizer is called the equilibrium measure (or extremal measure) associated with V , σ , and c. We introduce an operator L acting on measures by Lµ(x) := log |x − y| dµ(y).
If µ has a density ψ, we also write Lψ instead of Lµ. Then the equilibrium measure is characterized by the variational inequalities Lµ(x) − V (x) ≤ l
if x ∈ supp(σ − µ),
(2.7)
Lµ(x) − V (x) ≥ l
if x ∈ supp(µ),
(2.8)
which are satisfied for some constant l, which depends on V , σ , and c (see [DM], [DS]). Here and in what follows, supp denotes the closed support of a measure. In the description of the continuum limit of the Toda lattice presented in [DM], the external field V depends on time V (x) = Vt (x) = V0 (x) − tx,
x ∈ [A, B],
with V0 given by (2.2), and dσ (x) = φ(x) dx is the measure supported on [A, B] with
B density φ given by (2.3). The constraint φ is such that A φ(x) dx = 1. Therefore, the equilibrium measures µσ,c Vt exist for 0 < c < 1 and t ≥ 0. Of interest are the sets I 0 (c, t) := x ∈ [A, B] : Lµσ,c Vt − Vt (x) = l , I + (c, t) := x ∈ [A, B] : Lµσ,c Vt − Vt (x) > l , I − (c, t) := x ∈ [A, B] : Lµσ,c Vt − Vt (x) < l , where l is the constant in the variational inequalities (2.7) and (2.8) which also depends
437
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on c and t. Every interval in I 0 is called a band. The bands in I 0 are separated by gaps. It is known that at time t = 0, we have I 0 (c, 0) = [α(c), β(c)] for every c ∈ (0, 1); this means that the zero-gap ansatz holds at time t = 0. The zero-gap ansatz continues to hold until the shock time for the system (1.4) (see [DM]). To analyze the continuum limit beyond the time of shock formation, it is important that the set I 0 (c, t) is a finite union of intervals. The endpoints of these intervals then give important information on the behavior of the Toda lattice in the continuum limit, as already indicated in the introduction. The finite-gap ansatz says that there are only finitely many gaps. Knowing the gaps, the extremal problem can be turned into a Riemann-Hilbert problem, which can be analyzed explicitly. For various cases, it was verified in [DM] that the finite-gap ansatz holds. Deift, Kriecherbauer, and McLaughlin [DKM] obtained the finite-gap ansatz for real-analytic V in the absence of a constraint. More precisely, if we denote by µcV the minimizer for the problem to minimize IV (µ) among all measures in the class c ᏹ := µ ≥ 0 : supp(µ) ⊂ [A, B], dµ = c , (2.9) then it was shown in [DKM] that for C 2 external fields V , the minimizer µcV has a density ψ given by ψ(x) = where
1 − q (x), π
V (x) − V (y) c dµV (y) x −y 1 c2 − 2 V (y)(x + y) dµcV (y) − (B − x)(x − A)
2 q(x) = V (x) − 2
(2.10)
(2.11) if x ∈ [A, B],
and q − denotes the negative part of q. See [J] for related results. Note that the V used in [DKM] differs from ours by a factor of −2. If V is real-analytic in a neighborhood of [A, B], then q is real-analytic; it follows that ψ is supported on a finite number of intervals. This result also establishes the finite-gap ansatz if V is real-analytic and the constraint σ is not effective, that is, if I + (c, t) is empty. Remark 2.1. The extremal problem (2.6) with constraint was also introduced recently in the theory of orthogonal polynomials to describe asymptotics for polynomials that satisfy a discrete orthogonality relation (see [DS], [KVA2], [R], and [KR] for a survey). Some of the techniques used in this paper are based on ideas originating in this area. 3. Statement of main result. We state and prove our main theorem for external fields and constraints that are defined on the full real line.
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For a continuous external field V : R → R, we impose the growth condition lim
|x|→∞
V (x) = +∞. log |x|
(3.1)
Given such V and a positive Borel measure σ on R with a positive continuous density φ, the extremal problem is defined in a way analogous to (2.6). Thus, µ,c EV = inf IV (µ) : µ ∈ ᏹσ,c , where now the measures µ in ᏹσ,c may have support anywhere on the real line. Because of the growth condition on V , the minimizer µσ,c V exists and is unique. It is called the equilibrium measure associated with V , σ , and c. The equilibrium measure has a compact support (see [DS]). Since it is dominated by σ , it has a density, which we denote by ψVσ,c , or simply ψ, and 0 ≤ ψ ≤ φ. The inequalities (2.7) and (2.8) characterize the equilibrium measure. We refer to the constant l in (2.7) and (2.8) as the equilibrium constant. Following [JLM] and [LL], we partition R into three sets: (3.2) I 0 = x ∈ R : Lψ(x) − V (x) = l , + (3.3) I = x ∈ R : Lψ(x) − V (x) > l , − (3.4) I = x ∈ R : Lψ(x) − V (x) < l . These sets depend on V , σ , and c. Since φ is continuous and 0 ≤ ψ ≤ φ, it is known that Lψ is a continuous function (see, e.g., [DS], [R]). Since V is also assumed to be continuous, it is clear that I + and I − are open sets that are separated by the closed set I 0 . From (2.7) and (2.8), it follows that I + is the set where the constraint φ is active, and I − is the set where the nonnegativity constraint ψ ≥ 0 is active (i.e., ψ = φ on I + and ψ = 0 on I − ). Also, since ψ has compact support, we have that I 0 and I + are bounded, and I − is unbounded. It is the aim of this paper to establish the finite-gap ansatz in all cases where both V and φ = dσ/dx are real-analytic on R. The following is our main result. Theorem 3.1. Let V be a real-analytic function on R such that V (x) = ∞, x→±∞ log |x| lim
and let σ be a measure on R with positive, real-analytic density φ. Then for each c ∈ (0, dσ ), the following hold. (a) The constrained equilibrium measure µσ,c V has a continuous density ψ. − (b) The sets I = {x : Lψ(x) − V (x) < l} and I + = {x : Lψ(x) − V (x) > l} are both finite unions of open intervals. (c) The density ψ is real-analytic on the open set {x : 0 < ψ(x) < φ(x)}.
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439
(d) The density ψ has the representation ψ(x) =
1 π
q1− (x),
x ∈ I 0 ∪ I −,
where q1− is the negative part of a function q1 defined on I 0 ∪ I − , which is realanalytic in the interior of I 0 ∪ I − . The function q1 is positive on I − , so that ψ ≡ 0 on I − . (e) The density ψ has the representation ψ(x) = φ(x) −
1 π
q2− (x),
x ∈ I 0 ∪ I +,
where q2− is the negative part of a function q2 defined on I 0 ∪ I + , which is realanalytic in the interior of I 0 ∪ I + . The function q2 is positive on I + , so that ψ ≡ φ on I + . If the conclusions of the theorem hold, we say that the finite-gap ansatz holds for the triple (V , σ, c). The representations (d) and (e) are the extensions of (2.10) to the constrained case. It is clear from (d) and (e) that for x ∈ I 0 , 1 1 − q1 (x) + q2− (x) = φ(x), π π which shows that there exists a strong connection between the functions q1 and q2 . From the continuity of ψ, as claimed in Theorem 3.1(a), it follows that the sets {x : ψ(x) = 0} and {x : ψ(x) = φ(x)} are closed. Since they are disjoint and the latter is compact, they have a positive distance. Thus the density ψ cannot change abruptly from zero to φ. There is always a continuous (in fact, real-analytic) transition between zero and φ. The main problem in the proof of Theorem 3.1 is to exclude the possibility of such a sudden transition from zero to φ and also to exclude the possibility of an infinite number of transitions. There is no problem if over some interval (a, b) the constraint φ is not effective, since then we may use arguments as in [DKM], based on formulas such as (2.10) and (2.11) to conclude that ψ is real-analytic on those parts of (a, b) where it is positive. Similarly, it can be shown that there is no real problem on an interval where the nonnegativity condition is not active. The proof of Theorem 3.1 is in Section 4. Remark 3.2. The reader may have noticed that Theorem 3.1 does not correspond exactly to the extremal problem associated with the continuum limit of the Toda lattice, as described in Section 2. Indeed, the extremal problem (2.6) is restricted to the finite interval [A, B], while Theorem 3.1 covers problems defined on the whole real line.
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The problem on [A, B] is somewhat more difficult because of the endpoint effects. If the support of the equilibrium measure µσ,c V stays away from the endpoints A and B, then the analysis leading to Theorem 3.1 goes through; thus the finite-gap ansatz holds, provided of course that V and φ are real-analytic. It is possible to obtain the same conclusion if one (or both) of the endpoints is in the support and if both V and φ are real-analytic in a neighborhood of [A, B]. The proof of this result would involve a number of additional technicalities. For the sake of clarity of presentation, it is not included here. Remark 3.3. Based on the previous remark, we conclude that if V0 given by (2.2) and if φ given by (2.3) are real-analytic in a neighborhood of [A, B], then the finitegap ansatz holds for Vt , φ, and c ∈ (0, 1), for all t ≥ 0. Recall that Vt (x) = V0 (x)−tx. It is possible to construct initial data a0 (y) and b0 (y) giving rise to real-analytic V and φ. However, the fact that a0 and b0 themselves are real-analytic in a neighborhood of [0, 1] is not enough for the real-analyticity of V0 and φ. In fact, real-analyticity of a0 and b0 only implies that the spectral functions are real-analytic on (A, B) \ {α(0), α(1)}, where α is defined by (2.1). From this, it is possible to prove that there are only finitely many gaps in each compact subinterval of (A, B)\{α(0), α(1)}, but these gaps could possibly accumulate near one of the points A, B, α(0), or α(1). In specific cases it seems likely that one might be able to use special properties to exclude such accumulation of gaps. Then the true finite-gap ansatz would hold. Remark 3.4. Let V , σ , and c satisfy the assumptions of Theorem 3.1. We say that the triple (V , σ, c) is regular (otherwise, singular) if the following hold. • The function q1 from part (d) of Theorem 3.1 does not vanish on the interior of I 0 , and it has a simple zero at each of the endpoints of I − . • The function q2 from part (e) of Theorem 3.1 does not vanish on the interior of I 0 , and it has a simple zero at each of the endpoints of I + . • The set I 0 has no isolated points. This notion was introduced for the situation without constraint σ in [DKMVZ], where it was used in the asymptotic analysis of orthogonal polynomials, which is different for the regular and singular cases. Generically the regular case holds. This was shown by McLaughlin and the author (see [KM]), who proved that for a fixed V , the regular case holds for all c, with at most a countable number of exceptions. We expect a similar result for the case with constraint. 4. The proof of Theorem 3.1 4.1. Outline. Since the proof of Theorem 3.1 is rather long, we first present an outline. In the proof, the external field V and constraint σ are fixed, and the problem is considered depending on the normalization constant c. To simplify notation, we write µc instead of µσ,c V , and we denote its density by ψc . The equilibrium constant from (2.7) and (2.8) is denoted by lc .
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As a function of c, the measure µσ,c V increases with c. This is a very useful result. For extremal problems without constraint, it is well known (see, e.g., [ST], [To]). We also need certain continuity properties of µσ,c V in the parameter c. These results are included in Proposition 4.1 and are stated for more general V and σ , as in Theorem 3.1. We use σ |K to denote the restriction of a measure σ to the set K. Proposition 4.1. Let V be a continuous function on R such that limx→±∞ V (x)/ log |x| = +∞, and let σ be a nonnegative Borel measure on R such that L(σ |K ) is continuous for every compact set K ⊂ R. Then the following hold. σ,c for c ∈ (0, dσ ). That is, if 0 < c1 < c2
dµ1 ), L(µ2 − µ1 ) ≥ l2 − l1
on C.
We also conclude from (2.7) and (2.8) that L(µ2 − µ1 ) ≤ l2 − l1
on supp(µ1 ) ∩ supp(σ − µ2 ),
(4.6)
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443
so that by (4.6), L(µ2 − µ1 ) = l2 − l1
on supp(µ1 ) ∩ supp(σ − µ2 ).
(4.7)
The relations (4.6) and (4.7) imply that µ2 ≥ µ1 on supp(µ1 ) ∩ supp(σ − µ2 ) (see [ST, Theorem IV.4.5]). On the complement of supp(µ1 ) ∩ supp(σ − µ2 ), we have either µ1 = 0 or µ2 = σ ; in both cases, it is clear that µ2 ≥ µ1 . Thus µ2 ≥ µ1 everywhere, which completes the proof of part (a). (b) The limit (4.1) is an immediate consequence of part (a). ∗ For the proof of (4.2) and (4.3), we write µc instead of µσ,c V . Let x ∈ R, and let xc be real numbers such that lim xc = x ∗ .
(4.8)
c→c0
From (4.1) and the principle of descent for logarithmic potentials (see [ST, Theorem I.6.8]), it follows that Lµc0 (x ∗ ) ≥ lim sup Lµc (xc ). c→c0
(4.9)
Let K be a compact set such that supp(µc ) ⊂ K for some c > c0 . Then µc ≤ σ |K for all c sufficiently close to c0 , and we have lim σ |K − µc = σ |K − µc0 c→c0
in the sense of weak∗ convergence for positive measures. The principle of descent combined with the continuity of L(σ |K ) (see the assumptions of the proposition) then yields Lµc0 (x ∗ ) ≤ lim inf Lµc (xc ). c→c0
(4.10)
Hence by (4.9) and (4.10) we have Lµc0 (x ∗ ) = lim Lµc (xc ), c→c0
and this holds for every x ∗ and every xc satisfying (4.8). This implies that the limit (4.2) holds uniformly on compact subsets of R. For (4.3) we choose a point x ∗ from supp(µc0 ) ∩ supp(σ − µc0 ). Note that this is possible, since supp(µc0 ) and supp(σ − µc0 ) are nonempty closed sets whose union is R. Their intersection is then nonempty since R is connected. We have, by (2.7) and (2.8), Lµc0 (x ∗ ) − V (x ∗ ) = lc0 . Since the measures µc increase to µc0 as c increases to c0 by part (a), it follows that x ∗ ∈ supp(σ − µc ) for all c < c0 ; so by (2.7), Lµc (x ∗ ) − V (x ∗ ) ≤ lc ,
if c < c0 .
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Taking the limit c c0 , we then find because of (4.2) that lc0 ≤ lim inf lc .
(4.11)
c c0
Next, we note that it follows from part (a) and (4.1) that supp µc0 = supp(µc ).
(4.12)
cc0
instead of (4.12), we prove limc c0 lc = lc0 and the limit (4.3) follows. The following more precise version of Proposition 4.1(a) is used in the proof of Lemma 4.10. 1 2 and µ2 = µσ,c Lemma 4.3. Let V and σ be as in Proposition 4.1. Let µ1 = µσ,c V V with 0 < c1 < c2 < dσ . Let ω be the equilibrium measure of the set supp(µ2 ). Then (4.14) µ1 + (c2 − c1 )ω E ≤ µ2 ,
where E = supp(µ1 ) ∩ supp(σ − µ2 ). Proof. As in the proof of Proposition 4.1(a) (see (4.6) and (4.7)), we have Lµ1 ≤ Lµ2 + l1 − l2
on C
with equality on E. The potential of ω is constant on supp(µ2 ), say, Lω = C on supp(µ2 ). Then (4.15) L µ1 + (c2 − c1 )ω ≤ Lµ2 + l1 − l2 + (c2 − c1 )C on supp(µ2 ) with equality on E. Since µ1 + (c2 − c1 )ω and µ2 are measures with the same total mass and both are supported on supp(µ2 ), the principle of domination for logarithmic potentials (see [ST, Theorem II.3.2]) yields that (4.15) holds on C. Then, by a theorem of de la Vallée Poussin (see [ST, Theorem IV.4.5]), the inequality (4.14) holds.
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4.3. Proof that Ꮿ is nonempty. For an external field V and normalization c > 0, we use µcV to denote the equilibrium measure (without constraint). That is, µcV minimizes IV (µ) among all positive measures on R with total mass c. To prove that Ꮿ, defined in (4.4), is nonempty, we show that for c small enough, the measure µcV is dominated by σ . We show this for the more general class of C 1+ external fields, where an external field is said to be of class C 1+ if it is differentiable with a derivative satisfying a Hölder inequality with exponent > 0. In this more general version, the lemma is used in the proof of Lemma 4.10. The lemma also contains some additional facts that we need later. Lemma 4.4. Let W be a C 1+ function for some > 0 with lim|x|→∞ W (x)/ log |x| = ∞. Let µcW be the equilibrium measure with external field W and normalization constant c > 0. Then the following hold. c with respect to Lebesgue (a) For every c > 0, the measure µcW has a density ψW c measure, and ψW is a bounded function. (b) For every m > 0, there is c0 > 0 such that for every c ∈ (0, c0 ], c ψW (x) ≤ m
for all x ∈ R.
(c) Let * = {x ∈ R : W (x) = minW }. Then supp µcW = *. c>0
(d) Assume that W is convex in a neighborhood of *. Then for every c > 0, the support supp(µcW ) contains * in its interior. Proof. Take a bounded interval [a, b] containing the support of µcW in its interior. The integral equation Lv(x) − W (x) = l with supp(v) ⊂ [a, b],
b a
if x ∈ [a, b]
v(t) dt = c, is equivalent to the singular integral equation PV
b a
v(t) dt = W (x), x −t
where PV denotes the Cauchy principal value. It is well known (see, e.g., [G, Section 42.3] that for Hölder-continuous W , there is a unique solution b 1 W (s) 1 v(x) = √ (b − s)(s − a) ds if x ∈ [a, b], c + PV π π (b − x)(x − a) a s −x and v is Hölder-continuous in the open interval (a, b). From [KD, Lemma 3] we c and ψ c ≤ v + . Here know that dµcW (x) ≤ v + (x) dx, so that µcW has a density ψW W c + v denotes the positive part of v. The support of ψW is strictly contained in [a, b] by
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A. B. J. KUIJLAARS
c . It follows the choice of [a, b], so that v is finite and continuous on the support of ψW c that ψW is bounded from above. This proves part (a). c is bounded, we have that ψ c is in Lp for every p > 1, and this is enough Since ψW W for the analysis of [DKM, Theorem 1.34]. It follows that 1 c (x) = qc− (x) (4.16) ψW π
with qc− the negative part of the function 2 W (x) − W (y) c ψW (y) dy qc (x) = W (x) − 2 x −y
(4.17)
(see also [DKMVZ]). We have by the Hölder continuity of W , W (x) − W (y) ≤ M|x − y| if x, y ∈ [a, b], for some M > 0. Thus from (4.17), − qc (x) ≤ 2M
c (y) ψW dy |x − y|1−
if x ∈ R.
c (y) → 0 for almost every y. Then it is easy to see that for c As c 0, we have ψW c (x) ≤ m by (4.16), sufficiently small, we have qc− (x) ≤ π 2 m2 for all x ∈ R. Then ψW and this proves part (b). Part (c) is essentially contained in [BR]. Indeed, from the proof of [BR, Lemma 4] we have *⊂ supp µcW . c>0
Conversely, if
x0 ∈ supp(µcW )
for every c > 0, then
LµcW (x0 ) − W (x0 ) = lc
for every c > 0.
Since limc→0 lc = − min W (see [BR, remark after Theorem 2]) and since the measures µcW decrease to zero as c → 0, it follows that W (x0 ) = minW , that is, x0 ∈ *. This proves part (c). To prove part (d), we assume that W is convex in a neighborhood of *. Let c x0 ∈ *, c0 > 0, and suppose that x0 is not an interior point of the support of µW0 . Since W is convex in a neighborhood of x0 , say, in [x0 −, x0 +], it follows by [ST, Theorem IV.1.6] that /c := supp µcW ∩ x0 − , x0 + is an interval for every c. Since x0 ∈ supp(µcW ) by part (c), we have x0 ∈ /c . Since the supports are increasing with the parameter c, we see that x0 is not an interior
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447
point of any /c with c < c0 . It follows that x0 is either a right endpoint of all /c with c ≤ c0 or a left endpoint. Suppose for definiteness that x0 is a right endpoint of all /c with c ≤ c0 . Then there is δ > 0 such that for every c ∈ [c0 /2, c0 ], x0 − δ, x0 ⊂ /c ⊂ supp µcW
and
x0 , x0 + δ ∩ supp µcW = ∅. (4.18)
A formula of Buyarov and Rakhmanov [BR, Theorem 2] says that c0 c ωsupp(µcW ) dc, µW0 =
(4.19)
0
where ωS denotes the equilibrium measure (without external field) of the set S. Let c [a, b] be an interval containing the support of µW0 . Then by (4.18) we have for c ∈ [c0 /2, c0 ], supp µcW ⊂ [a, x0 ] ∪ x0 + δ, b ; this implies that
ωsupp(µcW ) |supp(µcW ) ≥ ω[a,x0 ]∪[x0 +δ,b] |supp(µcW ) .
By (4.18) we then have for c ∈ [c0 /2, c0 ], dωsupp(µcW ) dx
(x) ≥
dω[a,x0 ]∪[x0 +δ,b] (x), dx
x ∈ [x0 − δ, x0 ].
Since the density of the equilibrium measure of the union of the two intervals [a, x0 ]∪ [x0 + δ, b] is infinite at x0 , we see that dωsupp(µcW ) c0 (x0 ) = ∞, for c ∈ , c0 . dx 2 c
Because of (4.19) we then get ψW0 (x0 ) = ∞. This is a contradiction with part (a), c since ψW0 is bounded. This contradiction proves part (d). Proposition 4.5. The collection Ꮿ is nonempty. Proof. Since φ is positive and continuous, and supp(ψ1 ) is compact, there exists a constant m > 0 such that φ > m on supp(ψ1 ). By Lemma 4.4(b), there is c0 such that ψVc ≤ m for all c ∈ (0, c0 ]. Here ψVc is the density of the minimizer without upper constraint. We may assume that c0 ≤ 1. Let c ∈ (0, c0 ]. Then supp(ψVc ) ⊂ supp(ψc ) by [DS, Theorem 2.6], and supp(ψc ) ⊂ supp(ψ1 ) by Proposition 4.1(a). Therefore, ψVc ≤ m < φ on supp(ψVc ). It follows that the constraint φ is not effective if c ≤ c0 . Then (V , σ, c) satisfies the finite-gap ansatz by [DKM, Theorem 1.38], and so c0 ∈ Ꮿ.
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4.4. Auxiliary lemmas. Before we continue with the proof of Theorem 3.1, we state and prove a number of lemmas that are used later in the paper. The first is a straightforward extension of the result of Deift, Kriecherbauer, and McLaughlin [DKM]. Lemma 4.6. Let V be continuous on [a, b] and real-analytic on (a, b). Let µcV be the equilibrium measure with external field V and normalization c > 0; that is, µcV minimizes IV (µ) among all positive Borel measures on [a, b] with total mass c. Then µcV is absolutely continuous with respect to Lebesgue measure dµcV (x) = ψ(x) dx,
(4.20)
and there exists a real-analytic function q on (a, b) such that ψ(x) =
1 − q (x) for x ∈ [a, b], π
where q − is the negative part of q. The function q satisfies 2 V (x) − V (y) c dµV (y) q(x) = V (x) − 2 x −y
1 2 c + c − 2 V (y)(x + y) dµV (y) . (x − a)(x − b)
(4.21)
(4.22)
Proof. This follows by considering V as an external field on [a + , b − ], where we can use [DKM, Theorem 1.36] since V is real-analytic in a neighborhood of [a + , b − ], and by letting → 0. Our second lemma gives estimates on the density ψ of the equilibrium measure. It is important that the estimates do not depend on the normalization c. Lemma 4.7. Let V be a real-analytic function on an interval I . Let σ be a measure with real-analytic density φ. Let ψ be the density of the minimizer with external field V , constraint σ , and a certain normalization. Suppose a < b in I are such that [a, b] ⊂ supp(ψ) ∩ supp(φ − ψ) and ψ vanishes at a and b. (a) If ψ ≡ 0 outside (a, b), then ψ(x) ≤
b−a max V (y) if x ∈ (a, b). 2π y∈[a,b]
(b) If ψ ≡ 0 on (b, ∞) and on (A, a) with A < a and A ∈ supp(ψ), then ψ(x) ≤
b−a max V (y) π y∈[A,b]
if x ∈ (a, b).
(4.23)
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(c) If ψ ≡ 0 on (−∞, a) and on (b, B) with B > b and B ∈ supp(ψ), then ψ(x) ≤
b−a max V (y) π y∈[a,B]
if x ∈ (a, b).
(d) If ψ ≡ 0 on (A, a) and on (b, B) with A < a, B > b, and A, B ∈ supp(ψ), then ψ(x) ≤
b−a max V (y) π y∈[A,B]
if x ∈ (a, b).
Proof. (a) If ψ vanishes outside (a, b), then by (4.23) ψ is the density of the minimizer of the extremal problem without constraint. Since the support of ψ is equal to the interval [a, b] and ψ vanishes at a and b, we have for x ∈ (a, b) (cf. [G, Section 42.3]), b 1 ds V (x) − V (s) ψ(x) = (b − x)(x − a) , (4.24) √ π x − s π (b − s)(s − a) a and (a) immediately follows. (b) Suppose ψ vanishes to the right of b, and let A < a be as in part (b). Because of (4.23) and the choice of A, we have for some constant l that Lψ −V ≤ l on [A, b] with equality on [a, b] and also at A. Let ψ0 denote the restriction of ψ to [a, b], and define A V0 (x) := V (x) − log |x − t|ψ(t) dt for x ∈ [A, b]. (4.25) −∞
Then Lψ0 − V0 ≤ l
on [A, b],
(4.26)
with equality on the support of ψ0 and at A. It follows that ψ0 is the density of the minimizer with external field V0 and certain normalization. Thus for x ∈ (a, b), b V0 (x) − V0 (s) 1 ds ψ(x) = ψ0 (x) = (b − x)(x − a) √ π x − s π (b − s)(s − a) a (4.27) b−a max V (y). ≤ 2π y∈[a,b] 0 Let M = maxy∈[A,b] |V (y)|, and suppose, to get a contradiction, that ψ(x) > (b − a)/π ·M for some x ∈ (a, b). Then it follows from (4.27) that there is y ∈ [a, b] such that 2M < V0 (y). Since V0 (x) = V (x) +
A
ψ(t) dt 2 −∞ (x − t)
if x ∈ [A, b]
(4.28)
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A. B. J. KUIJLAARS
and |V (y)| ≤ M, it follows that
A
∞
Since y ≥ a and
ψ(t) dt > M. (y − t)2
(4.29)
A
ψ(t) dt 2 ∞ (x − t) is decreasing for x > A, it follows from (4.29) that A ψ(t) dt > M if x ∈ [A, a]. (x − t)2 ∞ Using (4.28) and the definition of M, we then have that V0 (x) > 0 for x ∈ [A, a]. Thus, V0 is convex on the interval [A, a]. Since ψ0 has no support on (A, a), we get that Lψ0 is strictly concave on [A, a]. Now we have a contradiction, since according to (4.26) the inequality Lψ0 − V0 ≤ l holds on [A, a] with equality at a and A. This contradiction completes the proof of part (b). (c) Part (c) is proved similarly. (d) Let A < a and B > b be as in part (d). Let ψ0 be the restriction of ψ to [a, b], and let V0 = V −L(ψ −ψ0 ). As in the proof of part (b), we have that ψ0 is the density of the minimizer with external field V0 on [A, B] and with a suitable normalization; as in (4.27) we obtain the estimate b−a max V (y) if x ∈ (a, b). 2π y∈[a,b] 0
ψ(x) = ψ0 (x) ≤
Let M = maxy∈[A,B] |V (y)|, and suppose that ψ > (b − a)/π · M somewhere in (a, b). Then we have 2M < V0 (y) for some y ∈ [a, b], and in the same way we obtained (4.29) in the proof of part (b), we now find ψ(t) dt > M. (4.30) 2 R\(A,B) (y − t) The function
ψ(t) dt 2 R\(A,B) (x − t)
is convex for x ∈ (A, B). Because of (4.30) with y ∈ [a, b], this leads to the fact that ψ(t) dt > M (4.31) (x − t)2 R\(A,B) holds, either for every x in the interval [A, a] or for every x in [b, B]. Suppose (4.31) holds for x ∈ [A, a]. Then we have as in the proof of part (b), V0 (x) > 0
if x ∈ [A, a],
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and we see that V0 is convex on [A, a]. This leads to a contradiction in the same way as before. Similarly, if the inequality (4.31) holds on [b, B], then V0 is convex on [b, B] and we get a contradiction as well. This proves part (d). The third lemma shows that it is enough for Theorem 3.1 to be able to cover the real line with a finite number of closed intervals, such that on each of these intervals, either the constraint σ or the lower bound zero is not active. Lemma 4.8. Let V , σ , and φ be as in Theorem 3.1. Let µ = µσ,c V be the equilibrium measure corresponding to some c ∈ (0, dσ ). Suppose there exists a finite sequence of points (ak )2n k=1 with a1 < · · · < a2n , such that (a) (−∞, a1 ] ⊂ supp(σ − µ), (b) [a2j −1 , a2j ] ⊂ supp(µ) for j = 1, . . . , n, (c) [a2j , a2j +1 ] ⊂ supp(σ − µ) for j = 1, . . . , n − 1, and (d) [a2n , ∞) ⊂ supp(σ − µ). Then the finite-gap ansatz holds for (V , σ, c). Proof. Let ψ be the density of µ, and let l be the equilibrium constant in the relations (2.7) and (2.8). Write a0 := min supp(µ) and a2n+1 := max supp(µ). For j = 0, 1, . . . , n, define V2j (x) := V (x) −
R\[a2j ,a2j +1 ]
log |x − t|ψ(t) dt
if x ∈ a2j , a2j +1 .
From the assumptions of the lemma, (2.7), and (2.8), it follows that Lψ − V ≤ l on [a2j , a2j +1 ], with equality on supp(ψ) ∩ [a2j , a2j +1 ]. Thus, if ψ2j denotes the restriction of ψ to [a2j , a2j +1 ], we have Lψ2j − V2j ≤ l
on a2j , a2j +1
with inequality on the support of ψ2j . This implies that ψ2j is the density of the minimizer of the unconstrained problem with external field V2j and normalization [a2j ,a2j +1 ]
ψ(t) dt.
Since V2j is continuous on [a2j , a2j +1 ] and real-analytic on the interior, Lemma 4.6 yields ψ(x) = ψ2j (x) =
1 − q2j (x) if x ∈ a2j , a2j +1 π
for a real-analytic function q2j on (a2j , a2j +1 ).
(4.32)
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A. B. J. KUIJLAARS
Next, we define for j = 1, . . . , n, a2j V2j −1 (x) := − V (x) + log |x − t|φ(t) dt −
a2j −1
R\[a2j −1 ,a2j ]
log |x − t|ψ(t) dt
if x ∈ a2j −1 , a2j .
On the interval [a2j −1 , a2j ], by (2.7), (2.8), and the assumption (b) of the lemma, we have the inequality Lψ − V ≤ l with equality on supp(φ − ψ) ∩ [a2j −1 , a2j ]. If we let φ2j −1 and ψ2j −1 be the restrictions to [a2j −1 , a2j ] of φ and ψ, respectively, then it follows that L φ2j −1 − ψ2j −1 − V2j −1 ≤ l on a2j −1 , a2j with equality on the support of φ2j −1 − ψ2j −1 . Thus φ2j −1 − ψ2j −1 (which is nonnegative) is the density of the minimizer with external field V2j −1 and normalization a2j φ(t) − ψ(t) dt. a2j −1
Since φ is real-analytic, it is easy to show that log |x −t|φ2j −1 (t) dt is real-analytic on (a2j −1 , a2j ), so that V2j −1 is real-analytic there. It is also continuous on the closure. Hence, Lemma 4.6 gives 1 − q2j −1 (x) (4.33) φ(x) − ψ(x) = π with q2j −1 a real-analytic function on (a2j −1 , a2j ). From (4.32) it follows that supp(ψ)∩[a2j , a2j +1 ] consists of an at most countable number of intervals, which, if infinite, can accumulate only at a2j and a2j +1 . Similarly, from (4.33) it follows that supp(φ −ψ)∩[a2j −1 , a2j ] consists of an at most countable number of intervals, which, if infinite, can accumulate only at a2j −1 and a2j . Let us now assume that for some j = 1, . . . , n, a2j − , a2j +1 ⊂ supp(σ − µ) for some > 0. Then it is clear that the representation (4.32) extends to the bigger interval [a2j − , a2j +1 + ] and that q2j is real-analytic on (a2j − , a2j +1 ). Since [a2j −, a2j ] ⊂ supp(ψ), it then follows that there is no accumulation of components of supp(ψ) near the point a2j . Similarly, there is no accumulation of components of supp(φ − ψ) near a2j . The same conclusion would hold if we would assume that a2j −1 , a2j + ⊂ supp(µ). Similar considerations apply to the points a2j +1 .
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453
Thus, to obtain the finite-gap ansatz, it is enough to show that for each j = 1, . . . , 2n, there exists an > 0 such that we have either aj , aj + ⊂ supp(µ) ∩ supp(σ − µ) or
aj − , aj ⊂ supp(µ) ∩ supp(σ − µ).
To establish this, we consider without loss of generality an odd-numbered point a2j +1 , which, for convenience, we assume is zero, and we assume that a2j < −1 and a2j +2 > 1, which can be achieved by a rescaling. We then have for some η > 0, − 1 − η, 0 ⊂ supp(σ − µ), 0, 1 + η ⊂ supp(µ), (4.34) and we want to show that for some > 0, either [0, ] ⊂ supp(σ − µ)
(4.35)
[−, 0] ⊂ supp(µ).
(4.36)
or
We assume that (4.35) does not hold. We claim that in such a case we have 1 ψ(t) dt = ∞. (4.37) t 0 The relation (4.37) certainly holds if for some > 0, (0, ] ∩ supp(σ − µ) = ∅,
(4.38)
since then ψ(x) = φ(x) for x ∈ (0, ]. If both (4.35) and (4.38) do not hold, then supp(σ − µ) ∩ (0, 1] is the disjoint union of a sequence of closed intervals [αj , βj ] accumulating at zero. Choose a compact interval I containing the support of µ in its interior. On I , we consider the external field W = −V + L(σ |I ), where σ |I denotes the restriction of σ to I . Since σ has a real-analytic density, W is real-analytic on the interior of I . From the variational inequalities (2.7) and (2.8), it follows that L σ |I − µ (x) − W (x) ≤ −l if x ∈ supp(µ), L σ |I − µ (x) − W (x) ≥ −l if x ∈ supp(σ − µ).
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This implies that σ |I − µ is the minimizer with external field W on I , constraint σ , and normalization d(σ − µ). I
For this extremal problem, the constraint σ is not effective on the support of µ, and so by (4.34) it is not effective on the interval (0, 1). The minimizer σ |I − µ has a density that vanishes at αj and βj for every j . Then we can apply Lemma 4.7 to each of the intervals [αj , βj ], and it follows that β j − αj M if x ∈ αj , βj , π where M is the maximum of the absolute value of the second derivative of W on [0, 1]. Since the length of the intervals [αj , βj ] tends to zero as j → ∞, it then easily follows that ψ(x) is bounded away from zero in some neighborhood of zero. Hence the claim (4.37) follows also in case (4.38) does not hold. Thus, assuming that (4.35) does not hold, we have found that (4.37) holds. We now want to establish (4.36). Consider V (x) − log |x − t|ψ(t) dt (4.39) φ(x) − ψ(x) ≤
R\[−1,0]
as an external field on [−1, 0]. With a suitable normalization, the equilibrium measure for (4.39) is the restriction of µ to [−1, 0]. The second derivative is ψ(t) V (x) + dt, (x − t)2 R\[−1,0] which tends to +∞ as x → 0− because of (4.37). Thus the external field is convex in an interval [−δ, 0] with δ > 0, and therefore the intersection of supp(µ) with [−δ, 0] is an interval (see [ST, Theorem IV.1.10]). Since there is equality at zero in the variational inequalities associated with the extremal problem on [−1, 0], it follows that supp(µ) ∩ [−δ, 0] = [−, 0] for some ≥ 0. If = 0, then ψ = 0 on [−δ, 0). Then ψ(t) lim (Lψ) (x) = lim dt = −∞ x→0− x→0− x −t by (4.37). Thus Lψ −V is strictly decreasing in some left neighborhood of zero. This is incompatible with the relation Lψ − V ≤ l since
on [−δ, 0]
Lψ − V (0) = l.
Thus = 0 cannot hold. Therefore > 0, and it follows that [−, 0] ⊂ supp(µ) with > 0. This proves that (4.36) holds. This completes the proof of Lemma 4.8.
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4.5. Proof that Ꮿ is open Proposition 4.9. The collection Ꮿ is open. Proof. Because of the definition (4.4) of Ꮿ, it is enough to prove that for every c0 ∈ Ꮿ, there is c1 > c0 also belonging to Ꮿ. Thus, let c0 ∈ Ꮿ. Then the support of ψc0 consists of a finite number of closed nondegenerate intervals, say, n supp ψc0 = a2j −1 , a2j j =1
with a1 < a2 < · · · < a2n . It follows from Proposition 4.1(a) that a2j −1 , a2j ⊂ supp(µc ) for j = 1, . . . , n,
(4.40)
for every c > c0 .
Now we want to show that there exists c1 ∈ (c0 , dσ ) such that for every c ∈ (c0 , c1 ), (−∞, a1 ] ⊂ supp(σ − µc ),
(4.41)
a2j , a2j +1 ⊂ supp(σ − µc ) for j = 1, . . . , n − 1,
(4.42)
[a2n , ∞) ⊂ supp(σ − µc ).
(4.43)
and
If we could show (4.41)–(4.43), then the conditions of Lemma 4.8 would be satisfied, and it would follow that the finite-gap ansatz holds for (V , σ, c). Then c1 ∈ Ꮿ, and Ꮿ would be open. Hence what remains to be done is to prove (4.41)–(4.43). Since ψc0 = 0 < φ on each of the intervals (−∞, a1 ], [a2j , a2j +1 ], and [a2n , ∞), the inclusions (4.41)–(4.43) follow from Lemma 4.10, which we state as a separate lemma since it is also used in the proof of Proposition 4.11. Lemma 4.10. Let c0 ∈ Ꮿ and a < b. Suppose that ψc0 < φ on [a, b]. Then there exists c1 > c0 such that ψc < φ on [a, b] for every c ∈ (c0 , c1 ). Proof. On [a, b] we have Lψc0 − V ≤ lc0 with equality on [a, b] ∩ supp(ψc0 ). If Lψc0 − V < lc0 on the full interval [a, b], then by compactness and continuity in the parameter c (see Proposition 4.1(b)), there is c1 > c0 such that Lψc −V < lc on [a, b] for every c ∈ (c0 , c1 ). Then for such c, ψc = 0 on [a, b] and the lemma follows. Therefore, in the rest of the proof, we assume * := x ∈ [a, b] : Lψc0 (x) − V (x) = lc0 = ∅. (4.44) Observe that V − Lψc0 is differentiable and its derivative is Hölder-continuous with
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A. B. J. KUIJLAARS
exponent 1/2. It is possible to find an auxiliary external field W such that W (x) = V (x) − Lψc0 (x) if x ∈ [a, b], W (x) > max − lc0 , V (x) − Lψc0 (x) if x ∈ R \ [a, b],
(4.45) (4.46)
W (x) = +∞, |x|→∞ log |x| lim
and such that W is differentiable with a Hölder-continuous derivative. In addition, we may assume that W is convex in a neighborhood of *. We let µδW be the minimizer with external field W and normalization δ, and we let δ be its density (see Lemma 4.4(a)). We let l ψW W,δ be the equilibrium constant in the δ variational conditions for µW . Let > 0 be such that ψc0 < φ on [a − , b + ]. It is possible to find such an since ψc0 and φ are continuous. Define φ(x) − ψc0 (x) , (4.47) m := min x∈[a−,b+]
which is a positive number. Since W assumes its minimum −lc0 on * ⊂ [a, b] only, we have by Lemma 4.4(c), supp µδW = * ⊂ [a, b]. δ>0
It follows that there is δ0 > 0 such that supp µδW ⊂ a − , b +
(4.48)
δ of the for all δ < δ0 . By Lemma 4.4(b), there is δ1 ≤ δ0 such that the density ψW δ measure µW satisfies δ ψW <m
if δ < δ1 .
(4.49)
Let c = c0 + δ with δ ∈ (0, δ1 ). We are going to apply Lemma 4.2 with σ,c
0 µ1 := µσ,c V − µV ,
µ2 := µδW ,
σ,c
0
and upper constraint σ˜ = σ − µV . Note that µ1 and µ2 are positive measures with dµ1 = dµ2 , µ1 ≤ σ˜ , and also µ2 ≤ σ˜ because of (4.47)–(4.49). For x ∈ supp(µ2 ), we have Lµ2 (x) = W (x) + lW,δ ≥ V (x) − Lψc0 (x) + lW,δ ,
where we used (4.46), and for x ∈ supp(σ˜ − µ1 ) = supp(σ − µc ), we have Lµ1 (x) = Lψc (x) − Lψc0 (x) ≤ V (x) + lc − Lψc0 (x).
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Thus, min L(µ2 − µ1 )(x) : x ∈ supp(µ2 ) ∩ supp σ˜ − µ1 ≥ lW,δ − lc .
(4.50)
Next, if x0 ∈ [a, b] ∩ supp(µ2 ) ∩ supp(ψc ), then Lµ2 (x0 ) = V (x0 ) − Lψc0 (x) + lW,δ and Lµ1 (x0 ) ≥ V (x0 ) + lc − Lψc0 (x0 ), so that L(µ2 − µ1 )(x0 ) ≤ lW,δ − lc .
(4.51)
Then by (4.50), (4.51), and Lemma 4.2, we have [a, b] ∩ supp(µ2 ) ∩ supp(ψc ) ⊂ supp σ˜ − µ1 = supp(σ − µc ). Since clearly R \ supp(ψc ) ⊂ supp(σ − µc ), it follows that [a, b] ∩ supp(µ2 ) ⊂ supp(σ − µc ).
(4.52)
Since W is convex in a neighborhood of *, we see from Lemma 4.4(d) that the support of µ2 contains * in its interior. Then we have by the definition (4.44) of *, Lψc0 − V > lc0
on [a, b] \ supp(µ2 ).
By continuity in c (see Proposition 4.1(b)), there is c1 ∈ (c0 , c0 + δ1 ] such that Lψc − V > lc
on [a, b] \ supp(µ2 )
for every c ∈ (c0 , c1 ], which implies [a, b] \ supp(µ2 ) ⊂ supp(σ − µc ).
(4.53)
Then if c ∈ (c0 , c1 ], both (4.52) and (4.53) hold and it follows that [a, b] ⊂ supp(σ − µc ).
(4.54)
Note that (4.54) does not imply that ψc < φ on [a, b], since there may be isolated points in [a, b] where equality holds. However, we claim that equality can happen only for c = c1 and not for c < c1 . To see this, we recall from Lemma 4.3 that for c < c1 , we have ψc (x) + (c1 − c)
dω (x) ≤ ψc1 (x) if x ∈ supp(µc ) ∩ supp σ − µc1 , dx
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where ω is the equilibrium measure of the support of µc1 . This measure has a positive density on supp(µc1 ). Since supp(µc ) ∩ supp σ − µc1 ⊂ supp µc1 , we see that ψc (x) < ψc1 (x) if x ∈ supp(µc ) ∩ supp σ − µc1 .
(4.55)
Now, if x ∈ [a, b], then we have either ψc (x) = 0, in which case it is clear that ψc (x) < φ(x), or ψc (x) > 0. In the latter case, we have x ∈ supp(µc ) and also x ∈ supp(σ − µc1 ) because of (4.54). Hence (4.55) implies that ψc (x) < ψc1 (x) ≤ φ(x). This completes the proof of Lemma 4.10.
4.6. Proof that Ꮿ is closed in (0, dσ )
Proposition 4.11. Let c0 < dσ be such that c ∈ Ꮿ for every c < c0 . Then c0 ∈ Ꮿ. Proof. For c < c0 , the finite-gap ansatz holds. Thus, the support supp(µc ) consists of a finite number of disjoint closed intervals. We let Nc be the number of those components of supp(µc ) on which ψc meets the constraint φ in at least one point, and we denote these intervals by [a2j −1 (c), a2j (c)], j = 1, . . . , Nc , with a1 (c) < a2 (c) < · · · < a2Nc (c). We also put *c :=
Nc
a2j −1 (c), a2j (c) for c < c0 .
(4.56)
j =1
By Proposition 4.1(a), the sets *c grow as c increases. First, we show that c → Nc is continuous from the right. To this end, let c < c0 . In each of the intervals a2j (c), a2j +1 (c) , a2Nc (c), ∞ , (4.57) − ∞, a1 (c) , the density ψc satisfies ψc < φ. Then Lemma 4.10 implies that ψc+δ < φ on the intervals (4.57) for every δ > 0 sufficiently small. In addition, we have that ψc vanishes identically on some subinterval of each of the intervals (4.57). Thus, Lψc − V < lc somewhere in each of the intervals. By continuity, then also Lψc+δ − V < lc+δ somewhere in each of the intervals (4.57), for δ > 0 small enough. This prevents the intervals in *c from growing together in going from c to c + δ. Thus, Nc+δ = Nc for δ > 0 small enough. Next, we want to show that Nc remains bounded as c increases to c0 . If the number Nc would be unbounded, then it follows from the above that there would be infinitely
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459
many c < c0 such that Nc−δ < Nc for all δ > 0 small enough. We show that there can be at most finitely many such c. Thus, let c < c0 be such that Nc−δ < Nc for δ small enough. Then there is an interval in *c in (4.56) which has no intersection with *c−δ if δ > 0. Thus, there is a j , such that [a2j −1 (c), a2j (c)] is disjoint from *c−δ for all δ > 0. Then ψc vanishes in a left neighborhood of a2j −1 (c), in a right neighborhood of a2j (c), and is equal to φ at a finite number of points of (a2j −1 (c), a2j (c)) only. Indeed, because of realanalyticity, the density cannot be equal to the constraint on a set with an accumulation point. We have a2j −1 (c), a2j (c) ⊂ supp(ψc ) ∩ supp(φ − ψc ). From Lemma 4.7, it follows that ψ(x) ≤
a2j (c) − a2j −1 (c) max V (y) y∈I π
if x ∈ a2j −1 (c), a2j (c) ,
(4.58)
where I is the convex hull of supp(µc0 ). Let m := min φ, I
M := max V , I
and note that these numbers do not depend on c and j . Since ψ(x) = φ(x) ≥ m for some x ∈ (a2j −1 (c), a2j (c)), we get from (4.58), a2j (c) − a2j −1 (c) ≥
πm =: /, M
(4.59)
and / > 0 is independent of c and j . Since (a2j −1 (c), a2j (c)) is disjoint from *c−δ for every δ > 0, we get from (4.59) that |*c | ≥ |*c−δ | + /, where |*c | denotes the Lebesgue measure of *c . Thus, each time we have Nc > Nc−δ , the Lebesgue measure of *c increases by an amount of at least /. Since all the sets *c are contained in *c0 and are increasing with c and since *c0 has finite Lebesgue measure, such an increase can happen only a finite number of times. It follows that Nc > Nc−δ for every δ > 0 small enough can happen for only finitely many c < c0 . Thus, the numbers Nc remain bounded as c increases to c0 . It also follows that Nc is nonincreasing for c close to c0 , and since these are nonnegative integers, it follows that Nc is in fact constant for c sufficiently close to c0 . Say, Nc = N if c ∈ (c0 − δ, c0 ) with δ > 0. Then the aj (c) are monotone functions of c ∈ (c0 − δ, c0 ), decreasing if j is odd, and increasing if j is even. Thus, the limits aj := lim aj (c) for j = 1, . . . , 2N c→c0 −
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exist. Since the measures µc increase to µc0 if c increases to c0 and since [a2j −1 (c), a2j (c)] ⊂ supp(µc ) for c ∈ (c0 − δ, c0 ), it follows that a2j −1 , a2j ⊂ supp µc0 . (4.60) Next, let j be such that
a2j , a2j +1 ⊂ supp µc0 ,
(4.61)
which implies in particular that a2j < a2j +1 . We have a2j , a2j +1 ⊂ a2j (c), a2j +1 (c) ⊂ supp(σ − µc ). Thus, Lψc − V ≤ lc on [a2j , a2j +1 ] for every c < c0 , and so, by continuity, Lψc0 − V ≤ lc0 on [a2j , a2j +1 ]. It follows that the restriction of ψc0 to [a2j , a2j +1 ] is the density of the extremal measure for the external field V (x) − log |x − t|ψc0 (t) dt, x ∈ a2j , a2j +1 . R\[a2j ,a2j +1 ]
This external field is real-analytic on the interior. Then we get from Lemma 4.6 that ψc0 has the form ψc0 (x) =
1 − q (x) if x ∈ a2j , a2j +1 π
with q real-analytic on (a2j , a2j +1 ). Thus ψc0 is real-analytic on {x ∈ (a2j , a2j +1 ) : ψc0 (x) > 0}. Also ψc0 ≤ φ. Since φ is positive and real-analytic, it follows that ψc0 hits the constraint φ only in a number of isolated points in (a2j , a2j +1 ) (if at all). Therefore, (4.62) a2j , a2j +1 ⊂ supp σ − µc0 if (4.61) holds. In a similar way, we show that (−∞, a1 ] ⊂ supp σ − µc0 , [a2N , ∞) ⊂ supp σ − µc0 .
(4.63) (4.64)
Now we delete from the sequence a1 < a2 < · · · < a2N the points a2j and a2j +1 if j is such that [a2j , a2j +1 ] ⊂ supp(µc0 ). We renumber the remaining points as a1 < a2 < · · · < a2M with M ≤ N. Then (4.60) holds for j = 1, . . . , M and (4.62) holds for j = 1, . . . , M −1. Also (4.63) holds, and (4.64) holds with N replaced by M. By Lemma 4.8, the finite-gap ansatz then holds for (V , σ, c0 ). Thus c0 ∈ Ꮿ. 4.7. Conclusion of the proof of Theorem 3.1. Combining Propositions
4.5, 4.9, and 4.11, we see that Ꮿ is a nonempty, open and closed subset of (0, dσ ). Thus Ꮿ = (0, dσ ), and (V , σ, c) satisfies the finite-gap ansatz for every c < dσ . This completes the proof of Theorem 3.1.
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Acknowledgments. I wish to thank Ken McLaughlin and Walter Van Assche for their interest in this work and for many stimulating discussions. I am grateful to the anonymous referee for the detailed remarks and suggestions which helped to improve the manuscript. References [BR]
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[KVA1]
V. S. Buyarov and E. A. Rakhmanov, Families of equilibrium measures in an external field on the real axis (in Russian), Mat. Sb. 190 (1999), 11–22; English transl. in Russian Acad. Sci. Sb. Math. 190 (1999), 791–802. S. B. Damelin and A. B. J. Kuijlaars, The support of the equilibrium measure in the presence of a monomial external field on [−1, 1], Trans. Amer. Math. Soc. 351 (1999), 4561–4584. P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lec. Notes Math. 3, Courant Institute, New York, 1999. P. Deift, T. Kriecherbauer, and K. T.-R. McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), 388–475. P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335–1425. P. Deift and K. T.-R. McLaughlin, A continuum limit of the Toda lattice, Mem. Amer. Math. Soc. 131 (1998), no. 624. P. Deift, S. Venakides, and X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Internat. Math. Res. Notices 1997, 286–299. P. D. Dragnev and E. B. Saff, Constrained energy problems with applications to orthogonal polynomials of a discrete variable, J. Anal. Math. 72 (1997), 223– 259. N. M. Ercolani, C. D. Levermore, and T. Zhang, The behavior of the Weyl function in the zero-dispersion KdV limit, Comm. Math. Phys. 183 (1997), 119–143. H. Flaschka, M. G. Forest, and D. W. McLaughlin, Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equation, Comm. Pure Appl. Math. 33 (1980), 739–784. F. Gakhov, Boundary Value Problems, Pergamon Press, Oxford, 1966. S. Jin, C. D. Levermore, and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math. 52 (1999), 613–654. K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), 151–204. A. B. J. Kuijlaars and P. D. Dragnev, Equilibrium problems associated with fast decreasing polynomials, Proc. Amer. Math. Soc. 127 (1999), 1065–1074. A. B. J. Kuijlaars and K. T.-R. McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Comm. Pure Appl. Math. 53 (2000), 736–785. A. B. J. Kuijlaars and E. A. Rakhmanov, Zero distributions for discrete orthogonal polynomials, J. Comput. Appl. Math. 99 (1998), 255–274; Corrigendum, J. Comput. Appl. Math. 104 (1999), 213. A. B. J. Kuijlaars and W. Van Assche, The asymptotic zero distribution of orthogonal
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A. B. J. KUIJLAARS polynomials with varying recurrence coefficients, J. Approx. Theory 99 (1999), 167–197. , Extremal polynomials on discrete sets, Proc. London Math. Soc. (3) 79 (1999), 191–221. P. Lax and C. D. Levermore, The small dispersion limit of the Korteweg–de Vries equation, I, Comm. Pure Appl. Math. 36 (1983), 253–290; II, 571–593; III, 809–829. J. Moser, “Finitely many mass points on the line under the influence of an exponential potential—an integrable system” in Dynamical Systems: Theory and Applications (Seattle, 1974), ed. J. Moser, Lecture Notes in Phys. 38, Springer, Berlin, 1975, 467–497. E. A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials of a discrete variable (in Russian), Mat. Sb. 187 (1996), 109–124; English transl. in Russian Acad. Sci. Sb. Math. 187 (1996), 1213–1228. T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Stud. Texts 28, Cambridge Univ. Press, Cambridge, 1995. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren Math. Wiss. 316, Springer, Berlin, 1997. F. R. Tian, Oscillations of the zero dispersion limit of the Korteweg–de Vries equation, Comm. Pure Appl. Math. 46 (1993), 1093–1129. V. Totik, Weighted Approximation with Varying Weight, Lecture Notes in Math. 1569, Springer, Berlin, 1994. S. Venakides, The Korteweg–de Vries equation with small dispersion: Higher order Lax-Levermore theory, Comm. Pure Appl. Math. 43 (1990), 335–361.
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, Leuven, Belgium; [email protected]
Vol. 104, No. 3
DUKE MATHEMATICAL JOURNAL
© 2000
HOLOMORPHIC DIFFERENTIAL INVARIANTS FOR AN ELLIPSOIDAL REAL HYPERSURFACE S. M. WEBSTER 0. Introduction. A Levi nondegenerate real hypersurface M 2n−1 in a complex nmanifold has a complete system of local differential invariants, under biholomorphic mappings. These invariants may be described either by the coefficients in a normal form or by the curvature of a connection. For a smooth, bounded, strictly pseudoconvex domain D in complex space Cn , the invariants of M = ∂D are intimately related to the global complex analytic invariants of D. Despite their importance, until now these invariants have been fully computed, to our knowledge, only in the case of the unit ball D = B n , where they all vanish! This is partially explained by the fact that if D is not holomorphically equivalent to B n , then the automorphism group of M is greatly reduced. Our aim is to carry out the computation of invariants in a significant case where there is virtually no symmetry. The simplest and most useful of the boundary invariants is the fourth-order pseudoconformal curvature tensor S of Chern and Moser [6]. For n ≥ 3 it vanishes identically if and only if M is locally biholomorphically equivalent to the sphere ∂B n , in the strictly pseudoconvex case. For n = 2 it vanishes identically by default—its role being taken by Cartan’s sixth-order invariant (see [4]). For further information on these invariants and their application, we refer to [1], [2], and [10], for example. In this paper we compute the fourth-order curvature tensor S for an ellipsoidal real hypersurface M (see [12] and [14]), M = z ∈ Cn | r(z, z) = 0 ,
r(z, ζ ) =
n j =1
zj ζj + Aj zj2 + ζj2 − 1,
(0.1)
which we call generic, if 1 0 < A1 < · · · < A n < . 2
(0.2)
A generic ellipsoid (and the domain that it bounds) has only a finite group of holomorphic symmetries (see [14]), which are of little use in computing differential invariants. Previously in [13], we were able to compute S only at points of intersection of M with the coordinate axes, but the methods of [13] were too ungainly for the more complete computation. Here we proceed by the method of complexification, which Received 23 June 1999. Revision received 6 January 2000. 2000 Mathematics Subject Classification. Primary 32V40; Secondary 32V05. Author’s work partially supported by the National Science Foundation. 463
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reveals the essential geometric structure, as it does in a number of other problems involving real submanifolds. In particular, we prove the following result. Theorem 0.1. Let M ⊂ Cn , n ≥ 3, be a generic ellipsoidal real hypersurface. Then the fourth-order pseudoconformal tensor S does not vanish at any point of M. Thus, in the terminology of [6], M is nowhere umbilic. This result was rather unexpected. It raises the general question of the behavior of the umbilic set under perturbation. The higher-order invariants of an ellipsoid, especially for n = 2, should also be accessible by our methods, but we do not go into their computation here. These methods, based on the developments of [15], [16], and [17], are at least as interesting as the above result. The Chern-Moser tensor S (= Sαβρσ ) can be interpreted as a relatively invariant (2, 2)-tensor on the holomorphic tangent bundle H (M). Its norm S θ , relative to the Levi form of the contact form θ = −i∂r, is multiplied by u−1 , if θ is multiplied by a positive function u on M. Thus, the theorem gives an everywhere defined, holomorphically invariant contact form , the principal contact form, for which S = 1. The principal characteristic vector field V on M is defined by ιV = 1,
ιV d = 0.
(0.3)
It is a holomorphically invariant infinitesimal contact transformation of the structure (M, ). Theorem 0.1 and the method of proof lead to the following result. Theorem 0.2. Let M ⊂ Cn , n ≥ 3, be a generic ellipsoidal real hypersurface. Then the flow of the principal characteristic vector field is completely integrable. The term “completely integrable” is necessarily somewhat vague, as it sometimes is even in the more familiar Hamiltonian case. Roughly speaking, it means that, off a proper subvariety, M is foliated by n-dimensional, V -invariant submanifolds on which the flow is susceptible to a simple, explicit description. The details of this are given in Section 4. The proofs of Theorems 0.1 and 0.2 make essential use of the complexification ᏹ of M, ᏹ = (z, ζ ) ∈ C2n | r(z, ζ ) = 0 , (0.4) M = FP(ρ), ρ(z, ζ ) = ζ , z , where FP means “fixed point set.” The proofs also make use of the Segre polar varieties Qζ = z ∈ Cn | r(z, ζ ) = 0 . (0.5) By Levi nondegeneracy there is locally a unique such complex hypersurface through a given point and tangent to a given hyperplane. The map (z, ζ ) → (z, Tz Qζ ) locally
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identifies ᏹ with the space of holomorphic contact elements. This map also identifies the complexification of the Chern-Moser invariants of M with the invariants of Tresse, Cartan, and Hachtroudi (see [3], [8], [5], and [7]) associated with the family of complex hypersurfaces {Qζ }. For the generic ellipsoid (see [15] and [16]), the family of complex quadrics ᏽ = {Qζ } sits in a linear family Pn+1 as a nondegenerate n-dimensional quadric. For each point z ∈ Cn , the linear condition z ∈ Qζ determines a point in the dual space P∗n+1 . The set of all such points determines another nondegenerate n-dimensional quadric in P∗n+1 , which we identify with its projective dual ⊂ Pn+1 . Under the birational polar transform (z, ζ ) −→ (z, Qζ ) ≡ (ξ, η) −→ (η, l),
l = [ξ η],
(0.6)
ᏹ corresponds to the set ᏸ0 of pointed lines (η, l) in Pn+1 with η ∈ l ∩ ᏽ and l tangent to (at ξ ). As in [15] we study ᏹ ∼ = ᏸ0 by means of moving ᏽ-frames adapted to . We
use the Maurer-Cartan forms of the complex orthogonal group determined by the complex quadric ᏽ to construct the Cartan-Hachtroudi connection. This allows us to compute the tensor S in terms of the invariants of the quadric ⊥ᏽ ∩ l ⊥ᏽ , relative to the quadric ᏽ ∩ l ⊥ᏽ , where ⊥ ᏽ means dual with respect to the nonsingular quadric ᏽ. (This brings to mind the computation of the intrinsic curvature of a surface in real Euclidean space in terms of the invariants of the second fundamental form relative to the first fundamental form.) The vanishing of the tensor S at a point translates into the condition l ⊥ᏽ ⊂ ⊥ᏽ , which contradicts nondegeneracy of ⊥ᏽ , if n ≥ 3. This summarizes the proof of Theorem 0.1. Again following [15], we next use the confocal family of n-quadrics ᏽλ ⊂ Pn+1 , of ᏽ relative to , to construct special “confocal” coordinates on ᏹ, via ᏸ0 . These coordinates are used in the proof of Theorem 0.2. This is achieved by showing that the n-dimensional variety of pointed lines (η, l), with l tangent to n − 1 additional fixed confocal quadrics ᏽλ , is invariant by the flow of the meromorphic continuation of V to ᏹ ∼ = ᏸ0 . A generalized hyperelliptic Abel-Jacobi map takes this variety to the quotient of Cn by a lattice of rank 2n − 1, with the flow of V going to a linear flow. This work provides another example of a general fact about ellipsoids. Certain problems tend to be explicitly solvable for ellipsoids, regardless of the underlying geometry, by means of the appropriate confocal theory. Thus, the main result of [15] on the dynamics of double-valued reflection is the analogue of the complete integrability of billiards in an ellipsoidal domain in real Euclidean space. Theorem 0.2 now appears to be the proper complex analogue of Jacobi’s theorem [9] on the complete integrability of the geodesic flow on an ellipsoid in Rn . The analogy is perhaps clearer if, in Jacobi’s case, we pass to the unit cotangent bundle and take the restriction of the canonical 1-form, since its characteristic vector field then generates the geodesic flow.
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1. Local structure and complexification. As in [6, Section 4] we describe the local structure of a nondegenerate real hypersurface M by means of local coframes. The 1-form θ = −i∂r is a real contact form on M and is determined up to a nonzero real factor. We choose local complex forms θ α , 1 ≤ α ≤ n − 1, so that {θ, θ α } and α {θ, θ } span the (1,0)- and (0,1)-forms restricted to M, respectively. Then the Levi form (relative to θ ) is representated by the hermitian matrix gαβ defined by β
dθ = i∂∂r = igαβ θ α ∧ θ + θ ∧ ϕ.
(1.1)
Here, and in what follows, repeated Greek indices are summed from 1 to n − 1. In preparation for complexification, it is convenient to pass to a “bar-free” notation, β so we set θα = igαβ θ . Then, as in [6], the integrability conditions for the two systems α {θ, θ }, {θ, θα } give dθ = θ γ ∧ θγ + θ ∧ ϕ, dθα = ϕαβ − δαβ ϕ ∧ θβ + ψα ∧ θ,
dθ α = θ β ∧ ϕβα + θ ∧ ψ α , dϕ = ψγ ∧ θ γ + ψ γ ∧ θγ + θ ∧ ψ,
(1.2)
for certain auxiliary 1-forms, ϕβα , ψ α , ψα , ψ. The first equation is just (1.1). Taking its exterior derivative gives the form of the last two (see [6]). For fixed forms {θ, θ α , θα , ϕ}, the auxiliary forms are determined, by Cartan’s lemma, up to changes of the form ϕαβ − ϕα0β = Bαβ θ, ψα − ψα0 = Bαβ θβ + Bα θ,
ψ α − ψ 0α = Bβα θ β + B α θ, ψ − ψ 0 = −Bα θ α − B α θα + Bθ.
(1.3)
Here the forms with “0” are any initial choices satisfying (1.2), and the coefficients B are to be chosen so that certain further normalizations hold. If M and the coframe are real analytic, the above continues locally to the complexification ᏹ in (0.4). Then we have θ = −i∂z r(z, ζ ), up to a factor, and the two systems are spanned by the dzj and the dζ j , respectively. The (complexified) Levi-degeneracy locus is 0 rζ Ꮾ = (z, ζ ) ∈ ᏹ | det (z, ζ ) = 0 . (1.4) rz rzζ It is disjoint from M in the present (nondegenerate) case. On the set ᏹ − Ꮾ, the map (z, ζ ) → (z, Tz Qζ ) ∈ Cn × P∗n−1 is locally invertible. Thus, the varieties {Tz Qζ | z ∈ Qζ } give locally a foliation of the space of holomorphic contact elements. Another foliation is given by the fibers z = const. Associated to such a structure is the invariant connection of Cartan [3] and Hachtroudi [8]. We briefly explain this in the next section. 2. The Cartan-Hachtroudi connection. This may be motivated by first considering the “flat case,” which consists of the family of pointed hyperplanes in Pn , that is,
467
ELLIPSOIDAL INVARIANTS
the subset of Pn × P∗n satisfying the incidence relation. To such a configuration we attach a projective frame (basis of Cn+1 ), Z = (Z0 , Zα , Zn ), det Z = +1, so that [Z0 ] is the point, and [Z0 ], . . . , [Zn−1 ] span the hyperplane. If Z = Z(t) depends smoothly on some parameters t, then exterior differentiation gives j
dZ = π Z ⇐⇒ dZi = πi Zj , j
tr π = πii = 0, j
dπ = π ∧ π ⇐⇒ dπi = πik ∧ πk .
(2.1) (2.2)
In this section Latin indices run from 0 to n, Greek indices from 1 to n − 1, and the corresponding summation conventions are used. The geometric point [Z0 ] remains fixed if the forms {π0n , π0α } vanish, while the hyperplane [Z0 , . . . , Zn−1 ] is constant if {π0n , παn } vanish. This singles out two integrable systems. An SL(n + 1, Cn ) Cartan connection manifests itself locally by such a matrix π of 1-forms, tr π = 0. In place of (2.2), we have the structure equation dπ = π ∧ π + ,,
tr , = 0.
(2.3)
The connection is torsion free if the matrix of 2-forms , satisfies j
,0 = 0,
,ni = 0.
(2.4)
Then tr , = ,αα = 0.
(2.5)
Taking the exterior derivative of (2.3) gives the Bianchi identity 0 = , ∧ π − π ∧ , + d,,
(2.6)
which together with (2.4) gives, in particular, 0 = ,ki ∧ πkn ,
j
0 = π0k ∧ ,k .
(2.7)
Now consider the complexified real hypersurface ᏹ and local coframe of Section 1. To associate a connection matrix π to this data, we first set π0n = θ,
π0α = θ α ,
παn = θα .
(2.8)
Comparison of (1.2) with (2.3) and (2.4) suggests the relations ϕ = πnn − π00 ,
ϕαβ = παβ − δαβ π00 ,
ψ α = πnα ,
ψα = πα0 ,
ψ = πn0 . (2.9)
Applying Cartan’s lemma to (2.7) and using (2.8) and (2.5) gives βσ ρ ,βα ≡ Sαρ θ ∧ θσ mod θ,
(2.10)
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S. M. WEBSTER
where the coefficients satisfy the symmetry and trace conditions βσ σβ βσ = Sαρ = Sρα , Sαρ
ασ Sαρ = 0.
(2.11)
In fact, a matrix π satisfying all these conditions exists. Further conditions, namely, the full trace condition (2.5) and one further trace condition, are needed to determine π uniquely. However, this is not needed here (see [6] and [8]). We carry out the normalizations only to the point needed to determine S. This amounts to determining Bβα in (1.3). S is a relatively invariant tensor. If we change the contact form θ → uθ , change {θ α , θα } → {θ α , uθα }, and follow this through the structure equations, then we see β that ,α is unchanged. Hence, S → u−1 S. We define the contraction of S by αρ
βσ Sβσ , S, S = Sαρ
(2.12)
which transforms as S, S → u−2 S, S . Where (2.12) is not zero, we can choose the contact form locally to make it identically 1. On M the sign of θ is determined by requiring a positive definite Levi form, in the strictly pseudoconvex case. This is the principal contact form. 3. Polar transform and curvature tensor. We describe the map (0.6) and its image. This was first carried out explicitly in [15], to which we refer for full details. −1/2 By the change zj → Aj zj , we pass from (0.1) to the defining function (3.1) r(z, ζ ) = A−1 z · ζ + z · z + ζ · ζ − 1,
where the dot product is z · w = nj=1 zj wj and where A is the diagonal matrix with the eigenvalues in (0.2). Then the varieties Qζ are complex spheres. The family of all spheres is an (n + 1)-dimensional linear family denoted Pn+1 . In terms of homogeneous coordinates ξ = (ξ0 , ξ , ξ∗ ) ∈ Cn+2 (∗ = n + 1), equation (0.6) is given by ξ = F (z) = (1, z, z · z), 1 A−1 ζ, ζ · ζ − 1 . η = G(ζ ) = 1, − 2
(3.2) (3.3)
We have F (Cn ) ⊆ = {s(ξ, ξ ) = 0} and G(Cn ) ⊆ ᏽ = {q(ξ, ξ ) = 0}, where the two quadrics are defined by the symmetric bilinear forms 1 s(ξ, η) = ξ · η − ζ0 η∗ + ξ∗ η0 , (3.4) 2 1 ζ0 η∗ + ξ∗ η0 . (3.5) q(ξ, η) = 4A2 ξ · η − ξ0 η0 − 2 We write q(ξ, η) = s(Ꮽξ, η), where Ꮽ is a q-symmetric operator.
ELLIPSOIDAL INVARIANTS
469
Since r(z, ζ ) = −2s F (z), G(ζ ) = −2s(ξ, η), ᏹ corresponds to the set ᏸ0 of (ξ, η) ∈ × ᏽ such that the line l = [ξ η] is tangent to at ξ . The set Ꮾ (defined in (1.4)) corresponds to q(ξ, η) = 0. This just means that l is also tangent to ᏽ at η. For the contact form we may take θ = −2is(ξ, dη). The system θ = θ α = 0 corresponds to dξ ∧ ξ = 0, and θ = θα = 0 corresponds to dη ∧ η = 0. As in [15] we study ᏸ0 via moving q-frames e adapted to . By definition, a q-frame e = (e0 , eα , en , e∗ ) is a basis of the homogeneous coordinate space Cn+2 , 1 ≤ α ≤ n−1 as before, and ∗ = n+1, such that e0 , e∗ ∈ l ∩ ᏽ, and eα , en span l ⊥q . It is said to be adapted to , if l is also tangent to at Ꮽen and if l ⊥q is tangent to ⊥q at en . Algebraically, the conditions defining a q-frame may be specified by setting q(ei , ej ) = gij ,
(3.6)
where g0∗ = gαα = gnn = 1 and where all the other gij are zero. The condition that e be adapted to is equivalent to aij = aj i = q Ꮽei , ej , (3.7) aαn = ann = 0, where the second equation defines the coefficients aij . Adapted frames exist at any point off Ꮾ, where we can choose e0 = e∗ . By changing (3.2) and (3.3) by multiples, which is immaterial in homogeneous coordinates, we may take η = e0 ,
ξ = Ꮽ en .
(3.8)
For moving q-frames, exterior differentiation gives, in (n + 2) × (n + 2) -matrix notation, de = ωe,
dω = ω ∧ ω.
(3.9)
So, for example, de0 = ω00 e0 + ω0α eα + ω0n en + ω0∗ e∗ and dω0α = ω00 ∧ ω0α + ω0β ∧ ωβα + ω0n ∧ ωnα + ω0∗ ∧ ω∗α . Differentiating (3.6) and (3.7) gives 0 = ωg + gωt
(3.10)
da = ωa + aωt .
(3.11)
and
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S. M. WEBSTER
More explicitly, (3.10) is equivalent to ωα∗ = −ω0α , ω∗∗ = −ω00 ,
ωn∗ = −ω0n ,
ω∗0 = ω0∗ = 0,
ω∗α = −ωα0 ,
ωαβ = −ωβα ,
ω∗n = −ωn0 ,
ωαn = −ωnα ,
ωnn = 0. (3.12)
From the first two equations in (3.7) and (3.11), we get 0 = daj n = ωj k akn + aj k ωnk ,
1 ≤ j ≤ n,
where the index k is summed over the n + 2 values from 0 to ∗ = n + 1. This yields, using (3.12) and (3.7), an∗ ω0n , an0 aαβ an∗ aα0 an∗ − an0 aα∗ ω0α − ωnβ − ω0n . ωα0 = an0 an0 (an0 )2 ωn0 =
(3.13) (3.14)
We may further normalize the frame so that s(e0 , e0 ) = 1, off the set ∩ ᏽ. Differentiation of this condition gives −ω00 = s(e0 , eα )ω0α + s(e0 , en )ω0n .
(3.15)
From s(ξ, dη) = s(Ꮽen , de0 ) = q(en , de0 ) = ω0n and dω0n = ω0j ∧ ωj n = ω00 ∧ ω0n + ω0α ∧ ωαn ,
(3.16)
we see that we may take θ = −2iω0n ,
θ α = ωnα ,
θα = −2iω0α ,
ϕ = −ω00 .
(3.17)
From (3.16) we get the first equation in (1.2). Similarly computing dωnα , dω0α , and dω00 shows that we get the rest of the equations in (1.2), if we choose 1 an∗ ωα0 + ψ 0α = − ψ 0 = 0. ω0α , ψα0 = −ωnα , ϕα0β = ωαβ , 2i an0 (3.18) The next structure equation in (2.3) is, using (2.8) and (2.9), β
,βα = dπαβ − πα0 ∧ π0 − παγ ∧ πγβ − παn ∧ πnβ
= dϕαβ − ϕαγ ∧ ϕγβ − ψα ∧ θ β − θα ∧ ψ β + δαβ θ γ ∧ ψγ + θ ∧ ψ .
(3.19)
Computing this expression modulo the ideal of θ with the forms (3.18) gives 0βσ ρ ,0β α ≡ Sαρ θ ∧ θσ mod θ, −1 0βσ Sαρ = 2ia0n aαρ δ βσ .
(3.20) (3.21)
ELLIPSOIDAL INVARIANTS
471
Observe that a0n = q(ξ, η) is nonzero off the set Ꮾ. Making the substitution (1.3) into (3.19) gives (2.10), after a little algebra, with βσ 0βσ βσ Sαρ = Sαρ + Eαρ ,
(3.22)
βσ = Bαβ δρσ + Bασ δρβ + Bρβ δασ + Bρσ δαβ . Eαρ
(3.23)
β
We want to choose Bα to achieve the trace condition in (2.11); that is, ασ ασ 0 = Sαρ = Sρ0σ + Eαρ = Sρ0σ + Bαα δρσ + (n + 1)Bρσ ,
(3.24)
where we have defined −1 0ασ Sρ0σ = Sαρ = 2ia0n aρσ .
(3.25)
0 = Sγ0γ + 2nBγγ .
(3.26)
Contracting ρ and σ gives
Hence, we choose Bαβ =
−1 1 Sα0β − Sγ0γ δαβ . n+1 2n
(3.27)
To maintain the trace condition in (2.11), we must restrict to changes (1.3) with β Bα = 0. Such changes do not affect S. Hence, we have proved the following. Lemma 3.1. On the subset ᏹ − Ꮾ of the complexified ellipsoid, the fourth-order curvature tensor S, computed relative to the 1-form θ in (3.17), is given by (3.22) and (3.23) with (3.27), (3.21), and (3.25). Next we compute the contraction of S,
S, S = S 0 , S 0 + 2 S 0 , E + E, E . We readily find, using symmetry of the indices,
0 0 0γ 2 S , S = Sγ ,
0 2 0αρ 2 S , E = 8Sβσ Bαβ δρσ = −8(n + 1)−1 Sα0β Sβ0α − (2n)−1 Sγ0γ , 2 αρ E, E = 4Eβσ Bαβ δρσ = 4 (n + 1)Bαβ Bβα + Bγγ 2 = 4(n + 1)−1 Sα0β Sβ0α − (2n)−1 Sγ0γ .
(3.28)
(3.29) (3.30)
(3.31)
Hence, S, S =
−4 0β 0α n(n + 1) + 2 0γ 2 S S + Sγ . n+1 α β n(n + 1)
(3.32)
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S. M. WEBSTER
4. Proof of the theorems Proof of Theorem 0.1. Suppose that S vanishes at a point (z, ζ ) of ᏹ − Ꮾ. Then, near the image point (ξ, η) ∈ × ᏽ, we can choose an adapted frame e and apply the results of Section 3. Taking α = ρ = β = σ in (3.22) (which is possible, since n − 1 ≥ 2) gives ββ 0ββ 0 = Sαα = Sαα ,
(4.1)
β
so that aαα = 0, 1 ≤ α ≤ n − 1 by (3.21). Also, Bα = −(n + 1)−1 Sα = 0, if α = β. Now taking β = ρ = σ gives ββ
0 = Sαβ = Sα0β + 2Bαβ =
0β
n − 1 0β S . n+1 α
(4.2)
It follows that aαβ = 0, for 1 ≤ α, β ≤ n − 1. Since we already have aαn = ann = 0 by (3.7), it follows that the (n − 1)-dimensional space spanned by e1 , . . . , en lies on the nondegenerate, n-dimensional quadric given by q(Ꮽξ, ξ ) = 0. This contradicts a basic property of quadrics for n > 2, and it finishes the proof. Proof of Theorem 0.2. We must determine precisely the principal contact form . By (3.17) and the remarks after (2.12) it can be written as = ±i S, S s(ξ, dη), (4.3) since ω0n = q(en , de0 ) = s(ξ, dη). We express this form in terms of the confocal coordinates of [15], which we briefly recall. The family of quadrics in Pn+1 confocal to ᏽ relative to is defined by ᏽλ = η | s (λ − B)−1 η, η = 0 , (4.4) where λ ∈ C is not an eigenvalue of the operator B = Ꮽ−1 . The (nonhomogeneous) confocal coordinates λj , 0 ≤ j ≤ n, of a point η of Pn+1 are those values of λ for which η ∈ ᏽλ . The quantities −1 xj = xj (ξ, η) = s λj − B ξ, η ,
λj = λj (η)
(4.5)
are “conjugate coordinate pairs,” in that they give the Pfaffian canonical form n
s(ξ, dη) = −
1 xj dλj . 2
(4.6)
j =0
In general, the point η lies on (n + 1) of the quadrics ᏽl , while the line l = [ξ η] is tangent to n of them, and ᏽµ , µ = µ1 , . . . , µn . Since, for ᏹ ∼ = ᏸ0 , η ∈ ᏽ = ᏽ0
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and l is tangent to = ᏽ∞ , we set λ0 ≡ 0, µn ≡ ∞. Then (λ1 , . . . , λn , µ1 , . . . , µn−1 ) generically provide local “confocal” coordinates on ᏹ. By q-duality (see [15]), the plane l ⊥q is tangent to the duals of the quadrics ᏽ1 , . . . , ᏽn at points that we take to be e1 , . . . , en in our q-frame. Then relative to such a frame, we have, in addition to (3.7), as in [15], aαβ = 0,
α = β;
aαα = µ−1 α .
(4.7)
If we normalize so that s(e0 , e0 ) = 1 (away from e0 ∈ ᏽ ∩ ), then by [15, Prop. 7.2], the conjugate coordinate pairs satisfy p λ, µ " 2 λj , xj ∈ Cµ" ≡ (λ, x) | x + =0 , (4.8) f (λ) n−1 (λ−µα ). where f (λ) is the characteristic polynomial of B and where p(λ, µ) " = α=1 The hyperelliptic curve Cµ" generically has genus n − 1, since f has one double root λ = 1. Thus −p 0, µ " a0n = q(ξ, η) = x0 = ± , (4.9) f (0) so that, relative to confocal coordinates, (3.32) has the form n−1 2 n−1 −f (0) n(n + 1) + 2 , S, S = µ−2 µ−1 α − α 4n (n + 1)p 0, µ " α=1
(4.10)
α=1
which is symmetric in µ1 , . . . , µn−1 . Hence, we have the necessary form in [15] for , =
n
aj λ j , µ " dλj .
(4.11)
j =1
The (meromorphic) characteristic vector field V = satisfies ιV d = 0, or n
n−1
j =1
α=1
∂µα aj vj = 0,
j vj ∂/∂λj
wα ∂µα aj = 0.
+
α wα ∂/∂µα
(4.12)
We claim that the matrix (∂µα aj ) generically has rank n − 1. This forces wα = 0, so that V is tangent to the n-dimensional varieties ᏸ µ " = (η, l) | l tangent to ᏽµ1 , . . . , ᏽµn−1 , (4.13) on which the µα are constant.
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To prove the rank statement, put " xj , aj = ih µ
−1 ∂µα aj = −2−1 λj − µα + hα µ " aj
(4.14)
for functions h, hα of µ1 , . . . , µn−1 . For generic choices of these values of µ, and of λ1 , . . . , λn−1 , the determinant −1 det λj − µα + hα µ " is not zero, since the functions of one variable λ → (λ−µα )−1 +hα (µ) " are independent. As in [15] the mapping (ξ, η) → ((λj , xj ))nj=1 takes ᏸ(µ) " several-to-one to the (n)
symmetric product Cµ" . The equations for the integral curves of V are Abelian differential equations of the third kind via (4.12). The generalized Jacobi inversion theorem applied to the corresponding Abelian sums gives a basically one-to-one (n) correspondence of Cµ" with a quotient Cn /?µ" , where the lattice ?µ" has rank 2n−1. n The isogeny C /(2?µ" ) → Cn /?µ" factors through ᏸ(µ), " and the integral curves of V on ᏸ(µ) " are the images of straight lines as in [15] and [17]. Restricting back to the real locus M finishes the proof of Theorem 0.2. It should be possible to give explicit parametrizations of these principal curves by means of (generalized) theta functions. This would be the analogue of the results of Weierstrass [18] and Knörrer [11], for the geodesics of the ellipsoid in real Euclidean space. References [1] [2]
[3] [4] [5] [6] [7] [8] [9] [10]
M. Beals, C. Fefferman, and R. Grossman, Strictly pseudoconvex domains in Cn , Bull. Amer. Math. Soc. (N.S.) 8 (1983), 125–322. D. Burns and S. Shnider, “Real hypersurfaces in complex manifolds” in Several Complex Variables (Williamstown, Mass., 1975), Proc. Sympos. Pure Math. 30, Part 2, Amer. Math. Soc., Providence, 1977, 141–168. É. Cartan, Sur les variétés à connexion projective, Bull. Soc. Math. France 52 (1924), 205– 241. , Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes, Ann. Mat. Pura Appl. (4) 11 (1932), 17–90. S. S. Chern, On the projective structure of a real hypersurface in Cn+1 , Math. Scand. 36 (1975), 74–82. S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271; Erratum, Acta Math. 150 (1983), 297. J. J. Faran V, Segre families and real hypersurfaces, Invent. Math. 60 (1980), 135–172. M. Hachtroudi, Les espaces d’éléments à connexion projective normale, Actualités Sci. Indust. 565, Hermann, Paris, 1937. C. G. J. Jacobi, Gesammelte Werke: Vorlesungen über Dynamik, supplemental volume, Chelsea, New York, 1969. H. Jacobowitz, An Introduction to CR Structures, Math. Surveys Monogr. 32, Amer. Math. Soc., Providence, 1990.
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[17] [18]
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H. Knörrer, Geodesics on the ellipsoid, Invent. Math. 59 (1980), 119–143. S. M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977), 53–68. , Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25–41. , Some birational invariants for algebraic real hypersurfaces, Duke Math. J. 45 (1978), 39–46. , Real ellipsoids and double valued reflection in complex space, Amer. J. Math. 120 (1998), 757–809. , “Segre polar correspondence and double valued reflection for general ellipsoids” in Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends Math., Birkhäuser, Boston, 1999, 273–288. , Stationary curves and complete integrability in the complex domain, to appear in Proceedings of the 1997 Lelong Conference, Birkhäuser. K. Weierstrass, “Über die geodätischen Linien auf dem dreiaxigen Ellipsoid” in Mathematische Werke, Vol. 1: Abhandlungen, 1, Mayer & Müller, Berlin, 1894, 257–266.
Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA; [email protected]
Vol. 104, No. 3
DUKE MATHEMATICAL JOURNAL
© 2000
RATIONAL POINTS ON QUARTICS JOE HARRIS and YURI TSCHINKEL
Contents 1. 2. 3. 4.
5. 6. 7. 8.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Fano 3-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 A Chow ring calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 The argument via monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 4.1. Generalities about quartic surfaces with a line . . . . . . . . . . . . . . . . . . . . . . . 486 4.2. Analysis of the points of CH ∩ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 Rational points on quartic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 An example: The Fermat quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 Quartic 3-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Other elliptic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
1. Introduction. Of all the possible extensions to higher dimensions of Faltings’s theorem, probably the most fundamental is the following conjecture. Conjecture 1.1 (Weak Lang conjecture). Let X be a variety defined over a number field K. If X is of general type, then the set X(K) of K-rational points of X is not Zariski dense. (While the name “weak Lang conjecture” has become standard usage—in part to distinguish it from the “strong Lang conjecture” below—we should point out that as stated here it was first ventured by Bombieri for surfaces (see, e.g., [28]) and by Vojta in [32].) We ask now whether a converse to this statement might hold. As it stands, the converse to the weak Lang conjecture cannot possibly be true. For example, if we take the product X = P1 × C of a rational curve and a curve C of genus g ≥ 2, we get a surface that is not of general type; but by Faltings’s theorem the rational points of X must lie in a finite union of fibers of X over C. The point is that the Kodaira dimension of a variety is not a sufficiently sensitive measure of the positivity or negativity of its canonical bundle. One possible modification, if we hope to have a plausible converse to the weak Lang conjecture, is to Received 25 February 1999. Revision received 30 December 1999. 2000 Mathematics Subject Classification. Primary 14G05. Harris’s work partially supported by the National Science Foundation. Tschinkel’s work partially supported by the National Security Agency. 477
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replace the hypothesis “X is of general type” with “X admits a dominant rational map to a positive-dimensional variety of general type”; or, given the counterexamples to this found by Colliot-Thélène, Skorobogatov, and Swinnerton-Dyer [10], with “there exists a finite étale cover Y → X and a dominant rational map Y → Z to a positive-dimensional variety of general type.” In other words, we may make the following conjecture, which was suggested to us by Dan Abramovich and Jean-Louis Colliot-Thélène. Conjecture 1.2. Let X be a smooth, connected projective variety defined over a number field K. There exists a finite extension K of K such that the set X(K ) of K -rational points of X is Zariski dense if and only if no finite étale cover Y → X admits a dominant rational map Y → Z to a positive-dimensional variety of general type. Alternatively, we can give up trying to find an if-and-only-if statement and simply ask what sort of condition on the canonical bundle of X ensures that X has a Zariski dense collection of rational points over some finite extension of K: For example, we may make the following conjecture. Conjecture 1.3. Let X be a smooth, connected projective variety defined over a number field K. If the canonical bundle KX of X is negative (i.e., −KX is ample), then for some finite extension K of K the set X(K ) of K -rational points of X is Zariski dense. This conjecture is easily seen to be true for curves and surfaces, where the hypothesis ensures that X is rational. The first real test cases are thus Fano 3-folds. In this paper, we examine the available evidence for this conjecture and add to it by analyzing one further class of Fano 3-folds, the smooth quartic hypersurfaces in P4 . Specifically, we prove the following theorem. Theorem 1.4. Let S ⊂ Pn be a smooth quartic hypersurface defined over a number field K. If n ≥ 4, then for some finite extension K of K the set S(K ) of K rational points of S is Zariski dense. In Section 7 we show that Theorem 1.4 follows as a straightforward corollary of a result about quartic surfaces. Theorem 1.5. Let S ⊂ P3 be a smooth quartic surface defined over a number field K, and let L be a line in P3 contained in S, likewise defined over K. Then (a) for some finite extension K of K the set S(K ) of K -rational points of S is Zariski dense; and (b) if we assume further that L does not meet six or more other lines contained in S, then in fact the set S(K) of K-rational points of S is Zariski dense. Our proof of Theorem 1.5 is based on an analysis of the fibration of S over P1 given by projection from the line L and on an analysis of the trisection of S → P1
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given by the points of L itself. For any plane H ⊂ P3 containing L, let CH ⊂ H be the cubic residual to L in the intersection of H with S. The key point in our argument has to do with the relation (or lack thereof) between the points p of intersection of the curves CH with L and the hyperplane class in Pic(CH ). The basic result, which we establish subject to various hypotheses in the following sections, states that, for all but finitely many H and any point p ∈ CH ∩ L, the classes of p and the line bundle ᏻCH (1) are linearly independent in Pic(CH ); that is, no multiple of the point p is linearly equivalent to any multiple of the hyperplane class on CH . This implies the desired density of rational points—it suffices to pull back the elliptic fibration to L. The new fibration S → L has a section of infinite order. By a result of Néron, rational points are Zariski dense on S and consequently on S. It is worth mentioning that Conjecture 1.3 is not the strongest possible converse to the weak Lang conjecture. It may well be that we do not need the canonical bundle to be negative—as Theorem 1.5 shows—but only nonpositive in a suitable sense. Thus, for example, we could make the following stronger conjecture. Conjecture 1.6. Let X be a smooth projective variety defined over a number field K. If the anticanonical bundle −KX of X is nef (e.g., −KX has nonnegative degree on every curve C ⊂ X), then for some finite extension K of K the set X(K ) of K -rational points of X is Zariski dense. The first interesting test case for this conjecture is K3 surfaces. In fact, we prove it for a large class of K3 surfaces, but the question remains open for general K3 surfaces defined over a number field.1 We should also mention here the strong Lang conjecture. Conjecture 1.7 (Strong Lang conjecture). Let X be a variety defined over a number field K. If X is of general type, then there exists a proper subvariety X such that, for any finite extension K of K, #(X \ )(K ) < ∞; that is, the set of K -rational points of X lying outside of is finite. The converse to this statement seems plausible. For one thing, if φ : A → X is any nonconstant map from a rational or abelian variety to X, it is not hard to see that the image of φ has to be contained in the Langian exceptional subvariety . Moreover, as a consequence of the theorem of Kollár, Miyaoka, and Mori [21] and of Campana [7], we know that varieties that are not of general type admit a rational quotient, namely, a maximal fibration X → Y , where the generic fiber of X → Y is rationally connected. The variety Y may admit a further fibration with generic fiber 1 Building on ideas of the present paper, Bogomolov and the second author have recently proved Conjecture 1.6 for Enriques, elliptic K3 surfaces, and K3 surfaces with infinite automorphism groups (see [4] and [5]). In particular, part (a) of Theorem 1.5 is now a special case of [5].
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of Kodaira dimension zero (see [21], [7], and [11]). Thus the converse to the strong Lang conjecture hinges on whether a variety of Kodaira dimension zero possesses a dense collection of images of rational and/or abelian varieties. Since this is known for curves and surfaces, the converse to the strong Lang conjecture is likewise known for all curves and surfaces, and for all 3-folds except for Calabi-Yau 3-folds, which represent the first real test. In general, however, it remains very much open. It is also worth mentioning that there are conjectures describing asymptotics for the number of rational points of bounded height. For example, let us consider a smooth quartic hypersurface in P4 . Then it is expected that the number of rational points, contained in some appropriate Zariski open subset and defined over a sufficiently large finite extension of the ground field of bounded height (induced from a standard height on P4 ), grows linearly with the height (cf. [13]). Acknowledgment. We are very grateful to the referee for many helpful comments. 2. Fano 3-folds. In this section we give a brief survey of known classification and rationality results for Fano 3-folds over an algebraically closed field of characteristic zero (cf. [25], [26], [3], and [23]). More details and references can be found in the recent book [19]. For our purposes it suffices to consider minimal Fano 3-folds (not isomorphic to a blow-up of a Fano variety). The main invariants of Fano 3-folds are: r(X)—the index, defined as the maximal r ∈ Z such that −KX = rL for some L in the Picard group Pic(X); ρ(X)—the rank of Pic(X) and the normalized degree δ(X) = (−KX )3 /r(X)3 . Group I: r ≥ 2, ρ = 1. (1) We have P3 (r = 4). (2) Q3 is the nonsingular quadric hypersurface in P4 (r = 3). The remaining five families have r = 2. They are indexed by δ. Let H be a line bundle such that |2H | = | − KV |. (3) We have that φH : V1 −→ P2 is a rational map with one indeterminacy point and with irreducible elliptic fibers. V1 can be realized as a double cover of the Veronese cone in P6 whose branch locus is a smooth intersection of this cone and a cubic hypersurface not passing through the vertex of the cone. Another realization is as a hypersurface of degree 6 in the weighted projective space P(1, 1, 1, 2, 3). The general V1 is nonrational. Unirationality is unknown. (4) We have that φH : V2 −→ P3 is a double covering ramified along a smooth quartic surface. All are unirational, and the general V2 is nonrational.
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(5) We have that φH : V3 −→ P4 is a smooth cubic hypersurface. All are unirational, and all are nonrational. (6) We have that φH : V4 −→ P5 is a smooth intersection of two quadrics. All are rational. (7) V5 is birational to a smooth quadric Q3 . Group II: r = 1, ρ = 1. (1) We have that φ−KV : W2 −→ P3 is a double covering ramified along a smooth sextic surface. They are nonrational, and unirationality is unknown. (2) We have that φ−KV : W4 −→ P4 is a smooth quartic. All are nonrational, and some are unirational. In general, unirationality is unknown. (3) We have that φ−KV : W6 −→ P5 is a smooth complete intersection of a quadric and a cubic. All are unirational and nonrational. (4) We have that φ−KV : W8 −→ P6 is a smooth complete intersection of three quadrics. All are unirational and nonrational. (5) W10 are all unirational, and rationality is unknown: The general one is nonrational. Geometrically, it is a section of Gr(2, 5) in its Plücker embedding by a subspace of codimension 2 and a quadric. (6) W12 , W16 , W18 , and W22 are all rational. (7) W14 are birational to a cubic 3-fold. All are unirational and all are nonrational. Group III: ρ = 2, 3 Theorem 2.1 [26, p. 104]. If ρ(X) = 3, then X is a conic bundle over P1 × P1 and has either a horizontal divisor D P1 × P1 or another conic bundle structure over P1 × P1 . In particular, all varieties in this group are unirational. Nonrational varieties are pointed out in the following list. There are four types of minimal Fano 3-folds with ρ = 3: (1) double cover of P1 × P1 × P1 ramified in a (2, 2, 2)-divisor, all nonrational;
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(2) smooth member of |L⊗2 ⊗ᏻP1 ×P1 ᏻ(2, 3)| on PP1 ×P1 (ᏻ ⊕ ᏻ(−1, −1)⊕2 ) such that X ∩ Y is irreducible (here L is the tautological line bundle and Y ∈ |L|); (3) P1 × P1 × P1 ; (4) PP1 ×P1 (ᏻ ⊕ ᏻ(1, 1)). The remaining varieties in this group have Picard number ρ = 2, and all are conic bundles over P2 : (5) double cover of P3∗ (i.e., P3 blown up in one point), ramified in a divisor in | − KP3∗ |, all nonrational; (6) double cover of P1 × P2 ramified in a (2, 2)-divisor; (7) double cover of P1 × P2 ramified in a (2, 4)-divisor, all nonrational; (8) hypersurface of bidegree (2, 2) in P2 × P2 , all nonrational; (9) hypersurface of bidegree (1, 2) in P2 × P2 ; (10) hypersurface of bidegree (1, 1) in P2 × P2 ; (11) P1 × P2 ; (12) PP2 (ᏻ ⊕ ᏻ(2)); (13) PP2 (ᏻ ⊕ ᏻ(1)). There are many more forms of Fano varieties over nonclosed fields. The main result of this paper together with the above classification implies: If X is a Fano variety defined over some number field K such that X is not isomorphic over C to (a blow-up of) V1 or W2 , then there exists a finite extension K /K such that the set X(K ) is Zariski dense.2 One may ask for conditions that ensure unirationality. Already, for (minimal) Del Pezzo surfaces of degree 1 (i.e., smooth minimal over K surfaces S with ample anticanonical bundle and KS2 = 1), it is unknown whether or not rational points are Zariski dense. 3. A Chow ring calculation. In this section we work over C. We use the notation of the introduction: S ⊂ P3 is a smooth quartic surface containing a line L. We consider hyperplanes H ⊂ P3 passing through L, and we denote by CH the cubic curve residual to L in the intersection of H with S—that is, S · H = L + CH as divisors on H —and let DH = CH ∩L be the intersection of CH with the line L. Note that projection from the line L ⊂ P3 gives a regular map π from the surface S to the line M ∼ = P1 parametrizing planes through L, and note that the curves CH are simply the fibers of this map. Note also that for any point p ∈ L, the point p lies in CH if and only if H is the tangent plane to S at p. Thus the restriction of the map π : S → M to L ⊂ S is simply the restriction to L of the Gauss map on S, mapping L onto the line in (P3 )∗ dual to L. Since the general plane H containing L is tangent to S at the three points of DH , this map has degree 3. The divisors DH are the fibers of the restriction of this map, and so in particular the divisors DH form a linear system on L of degree 3. 2 The
case of V1 was treated in the recent paper [6].
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Similarly, for any point p ∈ L, let Tp S ⊂ P3 be the tangent plane to S at p, and let Cp = CTp S ⊂ Tp S be the cubic residual to L in the intersection of Tp S with S. Let Dp = DTp S = Cp ∩ L be the intersection with L; note that p ∈ Dp tautologously. We begin by establishing a weak form of our basic result. We show that if L meets no other line of S (in particular, all CH are irreducible), then for all but countably many H ⊃ L the points of CH ∩ L are not rationally related to ᏻCH (1) in Pic(CH ). The proof is a relatively elementary argument using a calculation in the Néron-Severi group of an associated surface. In the following section, we give a more refined analysis, which allows us to conclude the same statement subject only to the weaker hypothesis that L does not meet six or more lines of S; while the present argument is superseded by that one, the argument here is useful for its (relative) simplicity and its applications to similar situations. (We see some of these in Section 8.) Theorem 3.1. Assume that no other lines lying on S meet L. For every positive integer n, there are only finitely many points p ∈ L(C), such that the classes of p and the line bundle ᏻCp (n) satisfy 3n · p ∼ ᏻCp (n). Proof. We begin by introducing a basic surface associated to this configuration. The incidence correspondence T = (p, q) : q ∈ Cp ⊂ L × S. To see T more clearly, note first that projection from the line L gives a regular map φ : S → M of S to the line M ∼ = P1 parametrizing the pencil of planes containing L; the curves CH are the fibers of this map. Similarly, the divisors cut on L by the curves CH form a base-point-free pencil of degree 3 on L; the restriction φ = φ|L is the map associated to this pencil and correspondingly has degree 3. In these terms, the surface T is simply the fiber product T = L ×M S. In particular, T is a 3-sheeted cover of S, branched over the union of the fibers CH of S → M such that CH is tangent to L. Note that the surface T need not always be smooth. It can be singular when some curve CH is simultaneously singular and not transversal to L. At worst, however, it has isolated singularities, since by the hypothesis that L meets no other lines lying on S, no curve CH can have a multiple component. Note that T → L has a tautologous section = (p, p) : p ∈ L ⊂ T ; this is just the intersection of T = L ×M S ⊂ S ×M S with the diagonal ⊂ S ×M S. As a Weil divisor, the pullback ν ∗ (L) of the line L under the 3-sheeted covering ν : T → S is thus a sum ν ∗ (L) = + R
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with R ⊂ T flat of degree 2 over L. Note that since the divisors DH form a basepoint-free linear series, all but finitely many divisors DH are reduced; in particular, R does not contain . The curve R is reducible if and only if the covering L → M is cyclic. Now, suppose that the conclusion of Theorem 3.1 is false; that is, for some n, we have infinitely many p ∈ L(C) with Cp smooth and 3n · p ∼ ᏻCp (n). Fixing a plane # ⊂ P3 , there is thus for infinitely many p a rational function on Cp with a pole of order 3n at p and with zeroes of order n at the points of intersection # ∩ Cp , and this rational function on Cp is nonzero and regular everywhere else. It follows in turn that there is a rational function f on T with divisor (f ) = −3n · + n · ν ∗ # + D, where D is supported on a finite union of fibers of T → L. Since the hypothesis that L meets no other line of S ensures that all fibers of T → L are irreducible, D must consist of a sum of fibers Cp of T → L. Since all fibers of T → L are linearly equivalent, Theorem 3.1 thus follows from the next lemma. Lemma 3.2. The classes σ , γ , and φ ∈ A1 (T ) of the divisors , ν ∗ #, and C are independent in the group A1 (T ) of Weil divisors modulo linear equivalence on T . Proof. We need to begin with a basic fact (due to Mumford), whose proof is mapped out in Fulton [14, Examples 7.1.16 and 8.3.11]. Lemma 3.3. Let T be a reduced, irreducible, normal, and projective surface. We may define, for every point p ∈ T and Weil divisors D, E ∈ Z1 (T ) whose supports have no common component in a neighborhood of p, an intersection multiplicity j (p, D · E) ∈ Q, bilinear in D and E, and a bilinear intersection pairing (· , · ) : A1 (T ) × A1 (T ) −→ Q on the group A1 (T ) of Weil divisors on T modulo rational equivalence, with the following properties. (1) If D and E are effective and both contain p, then j (p, D · E) > 0. (2) If D is locally principal at p, that is, D = (f ) for some rational function f in a neighborhood of p, and E is effective and irreducible, then j (p, D · E) = ordp (f |E ). (3) If D and E have no common components, then ([D] · [E]) = j (p, D · E). p∈D∩E
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We may now establish Lemma 3.2 by calculating the matrix of intersection numbers of the classes σ , γ , and φ ∈ A1 (T ) and by showing that this matrix is nonsingular. All but one of these numbers are readily calculated. To begin with, φ is the class of a fiber of the map T → L, so of course φ 2 = 0; inasmuch as σ is the class of a section of that map, we have (φ · σ ) = 1. Next, γ is the pullback of the hyperplane class under the map ν : T → S *→ P3 ; since the map T → S has degree 3, we have γ 2 = 3 · deg(S) = 12. Since the curves Cp map forward to plane cubics under the map ν, moreover, we have (γ · φ) = 3, and similarly, since the curve maps one-to-one onto the line L ⊂ S, (γ · σ ) = 1. In sum, then, we have Table 1. Table 1 Intersection Numbers γ
φ
σ
γ
12
3
1
φ
3
0
1
σ
1
1
σ2
The only mystery is the self-intersection σ 2 of the curve on T . To find this, we use a relation of linear equivalence between and a curve not containing . As we saw above, ν ∗ L = + R, so that if ρ = [R] ∈ A1 (T ) is the class of R, we have σ 2 = (σ · [ν ∗ L] − ρ). Now, by the projection formula (which is easily checked on some desingularization of the surface), ( · ν ∗ L)T = (ν∗ · L)S = (L · L)S = −2, and so σ 2 = −2 − (σ · ρ).
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Alternatively, if we choose the plane # ⊂ P3 to contain L, we see that the inverse image in T consists of , R, and the three fibers of the map T → L over the points of intersection of C# with L. Thus γ = σ + ρ + 3φ and
σ 2 = σ · γ − 3φ − ρ = −2 − (σ · ρ).
At this point, we may readily complete the calculation for smooth T —the curves and R intersect transversely over the points p, where Cp is tangent to L at p. The curves CH cut out a pencil of degree 3 on L, which by Riemann and Hurwitz have four branch points; thus ( · R) = 4. For arbitrary S and L, however, T may be singular at the points of intersection of with R—it is so exactly when a curve CH has a singularity at a point of L—and we can no longer say precisely what the intersection multiplicity is. All we do know, in fact, is that must meet R somewhere, so that ( · R) > 0 and, correspondingly, σ 2 < −2. Now, we may calculate the determinant of the matrix of pairwise intersection of the classes σ , γ , and φ. It is −12 + 3 − 9σ 2 + 3 = −6 − 9σ 2 > 0. The matrix is thus nonsingular, the classes σ , γ , and φ are independent, and Lemma 3.2 is proved. Thus, Theorem 3.1 is also proved. 4. The argument via monodromy. We continue to work over C. A finer analysis of the fibration π : S → M ∼ = P1 , specifically of the monodromy of the family on the torsion points in the Jacobians of fibers, yields a stronger result. 4.1. Generalities about quartic surfaces with a line. To begin, we recall some basic facts about the fibration S → M. First, since S is smooth, the Gauss map Ᏻ : S → (P3 )∗ is regular and, hence, is finite. It follows that no plane H ⊂ P3 can be tangent to S along a curve; in other words, every hyperplane section of S is reduced. The same is thus true of the fibers CH of the fibration π : S → M. We may thus list the possible singular fibers of π. They are • a cubic with one, two, or three nodes, that is, an irreducible nodal curve, the union of a line and a conic meeting transversely, or the union of three nonconcurrent lines, called fibers of type Ib with b = 1, 2, or 3, respectively; • a cuspidal cubic, called a fiber of type II; • the union of a line and a tangent conic, called a fiber of type III; or
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• the union of three concurrent lines, called a fiber of type IV. Note that fibers of type Ib correspond to poles of order b of the j -function on M associated to the elliptic fibration S → M. By contrast, fibers of type II and IV correspond to zeroes of j —after base changes of order 6 and 3, respectively, we may replace the singular curve by an elliptic curve C˜ of j -invariant zero—and fibers of type III correspond to points where j = 1728. After a base change of order 4 we may replace the singular curve by an elliptic curve C˜ of j -invariant 1728. The monodromy action on the homology of the smooth fibers around a fiber of type Ib is thus given by the Picard-Lefschetz transformation, while in the case of fibers of type II, III, and IV the monodromy is just the action of the automorphism given by ˜ We list here these actions in Table 2; see the original paper of the base change on C. Kodaira or the discussion in Barth, Peters, and van de Ven for details (cf. [20], resp., [1, pp. 150–160]). Table 2 Type Ib II
III
Monodromy 1 b MIb = 0 1
1 1 MII = −1 0
0 1 MIII = −1 0
IV
0 1 MIV = −1 −1
There is one constraint on the number of singular fibers CH . By a standard Euler characteristic calculation, χ(S) = 24 = χ(CH ). H ∈M
The Euler characteristics of fibers of type Ib , II, III, and IV are b (with b = 1, 2, or 3 in our case), 2, 3, and 4, respectively, giving a linear relation on the numbers of fibers of each type. In particular, we see that there must be at least six singular fibers CH . A related issue is the count of curves CH that are not transverse to L. Given that the divisors DH cut out on L by the curves CH form a pencil of degree 3, it follows by Riemann and Hurwitz that there must be a total of four branch points, counting multiplicity; that is, either two curves CH having a point of intersection multiplicity
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3 with L, one such curve and two others having a double point of intersection with L, or four curves CH having a double point of intersection with L. If the points p of CH ∩ L differ from each other by torsion in Pic(CH ) for almost all H , then since sections of an elliptic fibration that differ by torsion in the generic fiber can intersect only at singular points of fibers, the multiple point p of intersection of CH0 with L must be a singular point of CH0 . On the other hand, since there can be at most four fibers CH having multiple points of intersection with L, and each can contribute at most 4 to the Euler characteristic, we may draw one conclusion in particular that turns out to be vital to the following analysis: There must be singular fibers CH that intersect L transversely. 4.2. Analysis of the points of CH ∩ L Theorem 4.1. Let S ⊂ P3 be a smooth quartic surface and L ⊂ S a line in P3 contained in S; assume that L does not meet six or more other lines contained in S. Let n be any positive integer. For all but finitely many p ∈ L(C), 3n · p ∼ ᏻCp (n). The rest of this section is devoted to the proof of this fact. Let H be a plane containing L, and let p1 , p2 , and p3 be the three points on intersection of CH with L. Assume that some multiple of pi is linearly equivalent to a multiple of the hyperplane section of CH , and let n be the smallest positive integer such that 3n · pi ∼ ᏻCH (n). Note that since the monodromy on the three points pi as H varies is at least transitive, this hypothesis holds for one pi if and only if it holds for all three, and the value of n is the same for all three. In this case, the pairwise differences αi,j = pi − pj ∈ Pic0 (CH ),
i = j
are torsion, we let m be their order (the curve CH can be singular). Again, since the monodromy is transitive on the three pairs {±αi,j } = {αi,j , αj,i }, the value of m is the same for all i and j . Note that the classes αi,j are all nonzero, but they need not be distinct. If m = 2, of course, we have αi,j = αj,i ; while if m = 3, we could have α1,2 = α2,3 = α3,1 . If m > 3, however, we can see from the transitivity of the monodromy and the fact that α1,2 + α2,3 + α3,1 = 0 that they must be distinct. Note finally that if m = 2, the monodromy on the points pi must be cyclic, rather than the symmetric group S3 . A transformation fixing p1 , for example, and exchanging p2 and p3 , would exchange α1,2 and α1,3 and send α2,3 to −α2,3 ; given that α1,2 − α1,3 + α2,3 = 0, this implies 2α2,3 = 0. It follows, in particular, that in case m = 2, there are exactly two fibers CH not transverse to L, and each has a triple point of intersection with L. To carry out the further analysis of the monodromy action on the points pi and the classes αi,j , we consider in turn three potential cases: m > 3, m = 3, and m = 2.
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Case a: m > 3 Lemma 4.2. Every fiber not transverse to L must be of type IV. Proof. For each such fiber, the subgroup G ⊂ Pic0 (CH ) spanned by (any one of) the αi,j is an eigenspace for the action of the monodromy on the points of order m in Pic0 (CH ). The monodromy action MIb associated to a singular fiber of type Ib has only one eigenspace, with eigenvalue 1. Since the monodromy on the points pi is necessarily nontrivial and cyclic, and since m = 3, it follows that the action on the classes αi,j is nontrivial. Thus a fiber of type Ib singular at a point of L cannot occur. Next, suppose that we have a singular fiber of type II. The monodromy MII has the characteristic polynomial p(λ) = λ2 − λ + 1. Suppose λ is a root of this polynomial modulo m. Then we have λ2 ≡ λ − 1 mod m and hence λ3 ≡ λ2 − λ ≡ −1
mod m.
Alternatively, we can just multiply out and see that MII3 = −1. Either way, we see that MII3 cannot fix any element of Pic0 (CH ) of order m = 2, and so no fiber of this type can occur. A similar analysis shows that no fiber of type III can occur (or we could just observe that the union of a line and a tangent conic cannot have a point of intersection multiplicity 3 with L). This concludes the proof of the lemma. By the discussion above, each fiber that is not transverse to L has a triple point of intersection with L and (since the classes αi,j are torsion) this triple point of intersection has to be the singular point of the fiber. Therefore, L intersects at least six other lines on S. We do not know of any examples of this case. Case b: m = 3. The same analysis shows that the fibers CH not transverse to L must both be of type IV. None of the other transformations MIb , MII , and MIII has a 3-cycle on the points of order 3. Now we turn to fibers that are transverse to L. Suppose that the classes αi,j do not lie in a cyclic subgroup (for almost all fibers). This could be the case if every singular fiber CH transverse to L is of type I3 . But if there were δ such fibers, the formula for the Euler characteristic gives 24 = 2 · 4 + δ · 3, which has no solution. Thus the classes αi,j lie in a cyclic subgroup, which means that we must have α1,2 = α2,3 = α3,1 . This implies that 3pi ∼ p1 + p2 + p3 ∼ ᏻCH (1); that is, n = 1, or, in other words, all three points pi are flexes of CH for almost all planes H . Again, we do not know if this is possible. In any event, we see again that this case can occur only when six or more other lines of S meet L.
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Case c: m = 2. This case gives us the least amount of control over the behavior of the singular fibers CH not transverse to L but happily the most over the behavior of those that are. Very simply, in this case the classes αi,j cannot lie in a cyclic subgroup. They must comprise all three classes of order 2 in Pic0 (CH ). It follows that the monodromy associated to each fiber CH transverse to L must be trivial on the points of order 2 in Pic0 (CH ), which means that every singular fiber CH transverse to L must be of type I2 ; that is, they must consist of the union of a line and a conic. Now, as far as we can tell, the fibers CH not transverse to L can be of type II, III, or IV, but in any event the total contribution of such fibers to the Euler characteristic can be at most eight (if we have either four fibers of type III or two of type IV). The remaining singular fibers must therefore contribute at least 16 to the Euler characteristic, which means we must have at least eight fibers of type I2 ; in particular, L must intersect at least eight other lines of S. 5. Rational points on quartic surfaces. In this section we work over a number field K (and we fix an embedding of Q into C). We assume that S and L are defined over K. To deduce part (b) of Theorem 1.5 from the analysis carried out so far, we need one more ingredient. Briefly, Theorems 3.1 and 4.1 assure us (subject to their hypotheses) that for a very general point—that is, for all but countably many points p ∈ L(C)—the point p ∈ Cp (C) is not rationally related to the hyperplane class in Pic(Cp ). If such a point p lies in L(K), the cubic curve Cp has positive rank over K and, hence, a dense set of rational points. It seems reasonable to expect that for “most” of the points p ∈ L(K) this would be true. But there are countably many points p ∈ L(Q) for which p is rationally related to ᏻCp (1)—for each n, there is a finite subset 2n ⊂ L(Q) such that 3n · p ∼ ᏻCp (n) in Pic0 (Cp )—and it is still a logical possibility, if not a plausible one, that all the points of L(K) lie in the union of these sets. There are two ways of eliminating this possibility. The first is to invoke an extremely powerful theorem due to Merel [24]. Theorem 5.1 (Merel). Let K be any number field. There is an integer n0 = n0 (K) such that no elliptic curve defined over K has a K-rational point of order n > n0 . This theorem assures us that for n > n0 , the subset 2n is disjoint from L(K); so that for all but finitely many p ∈ L(K), the point p is not rationally related to ᏻCp (1) in Pic(Cp ). A second way to arrive at this fact uses a theorem that is easier to prove, if less simple to state. Suppose that E → B is any family of elliptic curves and σ a section of E → B, both defined over a number field K. For each point t ∈ B, let hB (t) be the height of t (relative to any divisor of degree 1 on B), let h(σt ) be the canonical height of the value σt of the section in the fiber Et of the family over t, and let h(σ ) be the canonical height of the section σ in the fiber of E over the generic point
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of B. We have then the following theorem of Dem’janenko [12], Manin [22], and Silverman [31, Th. B, p. 197]. Theorem 5.2. We have lim
hB (t)→∞
h(σt ) = h(σ ). hB (t)
In particular, to say that σ is not a torsion point in the fiber of E over the generic point η of B is to say that h(σ ) > 0, and we can deduce the following corollary. Corollary 5.3. If ση is not a torsion point in the fiber Eη of E over the generic point η ∈ B, then there are only finitely many points t ∈ B(K) such that σt is a torsion point in Et . Applying this to the family T → L of elliptic curves introduced in Section 3 (with the origin given by the tautologous section and with the section σ given by the divisor class ᏻCp (1) in each fiber), we deduce again that (subject to the hypotheses of Theorem 4.1) for all but finitely many p ∈ L(K) the point p is not rationally related to ᏻCp (1) in Pic(Cp ). In fact, a weaker version, due to Néron, suffices—he proved that there are infinitely many points in B(K) with fiber containing infinitely many K-rational points (see Serre’s book [30, p. 153]). Now, S ⊂ P3 is a smooth quartic surface, and L ⊂ S is a line in P3 contained in S (both defined over K); assume that L does not meet six or more other lines contained in S (not necessarily defined over K). For each point p ∈ L(K) and for each integer n there are unique points qn and rn ∈ Cp such that qn + (3n − 1) · p ∼ ᏻCp (n) and −rn + (3n + 1) · p ∼ ᏻCp (n), also defined over K. Moreover, for all but finitely many p ∈ L(K) we have 3n · p ∼ ᏻCp (n) for every n, so that these points are all distinct. We have, accordingly, an infinite collection of K-rational points on Cp , so that Cp is contained in the Zariski closure of S(K); since this is true of infinitely many curves Cp , it follows that the Zariski closure of S(K) is all of S. As for the proof of part (a) of Theorem 1.5, given part (b) this requires only one further trick, and it is a relatively simple one. Lemma 5.4. Let S ⊂ P3 be a smooth quartic surface and L, L , and L ⊂ S three lines in P3 contained in S; assume that L does not meet either L or L , but that L and L do meet. For each plane H containing L, let qH = CH ∩L and rH = CH ∩L .
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For all but finitely many H (defined over C) containing L, the difference qH − rH is not torsion in Pic0 (CH ). Proof. This is easy. Note first that since L does not meet L and L , no plane H containing L can be the tangent plane to S at any point of L or L ; in other words, qH and rH are smooth points of the curve CH for all H . Now let T be as in the proof of Theorem 3.1. The assignment to each point p ∈ L of the points qp and rp ∈ Cp gives sections of the map T → L, which we claim do not differ by a translation of finite order in the fiber Cp (for almost all p). But two sections of an elliptic fibration that differ by torsion in the generic fiber can intersect only at singular points of fibers. This concludes the proof. Proof of part (a) of Theorem 1.5. Now suppose we have a quartic surface S and a line L ⊂ S. If fewer than six other lines of S meet L, we may apply part (b) of Theorem 1.5 to conclude that the points of S rational over the field of the definition of L are Zariski dense. Suppose conversely that every line of S meets at least six other lines of S. We claim in that case that S must contain a configuration of lines as in Lemma 5.4. To see this, start with any line L0 ⊂ S. Since no more than four lines on S can pass through a single point of S, we have to consider only two possibilities. Case 1: Three pairwise skew lines L1 , L2 , and L3 ⊂ S meet L0 . In this case, let M1 , . . . , M5 ⊂ S be five other lines meeting L1 (in addition to L0 ). If any one of them fails to meet both L2 and L3 , we are done. If Mi fails to meet Lj , we take for our configuration L = Lj , L = L1 , and L = Mi . Otherwise, assuming that all six lines L0 , M1 , . . . , M5 meet all three lines L1 , L2 , and L3 , we see that all of them have to lie on the unique quadric surface Q ⊂ P3 containing L1 , L2 , and L3 —but then we have nine lines in Q ∩ S, contradicting Bézout (among others). Case 2: Two triples of concurrent lines {L1 , L2 , L3 } and {M1 , M2 , M3 } meet L0 . This is even easier. Just let N be any line meeting L1 and skew to L0 . N cannot meet all three lines M1 , M2 , and M3 . (It is coplanar with them and hence meets L0 .) If it misses Mi , we take L = Mi , L = L1 , and L = N . To complete the proof of Theorem 1.5, suppose now that L, L , and L ⊂ S are a configuration as in Lemma 5.4; let K be any field over which S and all three lines are defined. For each plane H containing L and for each integer n, there is a unique point xn ∈ CH such that xn + n · qH ∼ (n + 1) · rH , also defined over K . Moreover, by Merel’s theorem and Lemma 5.4 (or also by Silverman’s theorem), for all but finitely many H these points are all distinct. We thus have for infinitely many H an infinite collection of K -rational points on CH , and once more it follows that the Zariski closure of S(K ) is all of S. 6. An example: The Fermat quartic. In light of the analysis above, it might seem unlikely that there is any quartic surface S and line L such that, for all planes
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H defined over C and containing L, the C-points of CH ∩ L are all rationally related to the hyperplane class in Pic(CH ). In fact, however, it does occur. We describe here the unique example we know of, the Fermat surface. See also the analysis of Piatetski-Shapiro and Shafarevitch [29]. To begin with, we take S ⊂ P3 the quartic given by the equation X4 − Y 4 + Z 4 − W 4 = 0 and L the line given by the equations X=Y
and
Z = W.
Any plane containing L, other than the plane Z = W , can be realized as the span of L and a third point of the form [a, −a, 1, −1] for some scalar a, that is, the plane given parametrically by [U, V , T ] −→ U + aT , U − aT , V + T , V − T . Restricting the equation of S to H gives the equation (U +aT )4 −(U −aT )4 +(V +T )4 −(V −T )4 = 8aU 3 T +8a 3 U T 3 +8V 3 T +8V T 3 , so the equation of CH is simply Fa (U, V , T ) = aU 3 + a 3 U T 2 + V 3 + V T 2 = 0. The points of CH ∩ L are given by the further equation T = 0; that is, they are the points [U, V , T ] = [1, b, 0], where b3 = a. Note that the monodromy on these as a varies is cyclic. What are the singular fibers CH of the fibration S → M? The plane Z = W corresponding to a = ∞ is certainly one, consisting of three concurrent lines meeting a point of L, and the plane X = Y corresponding to a = 0 is another singular fiber of type IV. To find the remaining ones we have simply to write out the partial derivatives of Fa (U, V , T ) and equate them all to zero. The equations 3aU 2 + a 3 T 2 = 3V 2 + T 2 = 2V T + 2a 3 U T = 0 together imply that a 4 = ±1 and that 1 U=√ T −3
and
a V = √ T. −3
There are thus eight singular fibers apart from a = 0 and a = ∞, each having two singular points (that is, consisting of a line and conic).
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We now claim that the three points pi of the intersection of CH with L differ by torsion of order 2 and that each satisfies 6 · pi ∼ ᏻCH (2). The second statement follows from the first, given that p1 + p2 + p3 ∼ ᏻCH (1), and the first is readily checked. We simply observe that two points p and q on a plane cubic curve C differ by torsion of order 2 if and only if the point of intersection Tp C ∩Tq C of the tangent lines to C at p and q lies on C. Now, the equation of the tangent line to CH at a point [µ, ν, τ ] is 3aµ2 + a 3 τ 2 · U + 3ν 2 + τ 2 · V + 2ντ + 2a 3 µτ · T = 0, and at the point [1, b, 0] for some cube root b of a this is −3b3 · U + 3b2 · V = 0. For any two distinct cube roots b of a, the resulting linear forms in U and V are independent, so that the point of intersection of any two of the tangent lines to CH at the points of CH ∩ L is just [U, V , T ] = [0, 0, 1], which is a point of CH . Again, we do not know of any other examples of a quartic surface S and a line L ⊂ S such that, for almost all planes H containing L, the points of CH ∩ L are all rationally related to the hyperplane class in Pic(CH ); nor do we know any examples at all where the points differ from each other by torsion of order greater than 2. 7. Quartic 3-folds. In this section “general” means outside a Zariski closed subset. We now use Theorem 1.5 to deduce Theorem 1.4. This is relatively simple. We just have to check that if X ⊂ Pn is any smooth quartic hypersurface and L ⊂ X any line, then for a general 3-plane P3 ⊂ Pn containing L, the surface S = X ∩ P3 and the line L satisfy the hypotheses of part (b) of Theorem 1.5. It is enough to do this in case n = 4, and it requires only a straightforward geometric argument. We start with a basic fact. Lemma 7.1. If X ⊂ P4 is a smooth quartic hypersurface, the Fano variety F1 (X) ⊂ G(1, 4) of lines on X has pure dimension 1. Proof. To begin with, the homogeneous quartic polynomial F ∈ Sym5 (C5 ) on P4 defining X gives rise to a section τF of the fifth symmetric power Sym5 (S ∗ ) of the dual of the universal subbundle S on G(1, 4), and the zero locus of this section is the Fano scheme F1 (X). This shows that it has dimension at least 1 everywhere and, since the top Chern class c5 (Sym5 S ∗ ) = 320σ3,2 = 0, that it is nonempty; it remains only to see that it cannot be 2- or higher-dimensional. To do this, let 2 = P(S|F1 (X) ) = {(L, p) : p ∈ L} ⊂ F1 (X) × X
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be the universal projective line bundle over the Fano variety F1 (X) ⊂ G(1, 4), and let ρ : 2 → X ⊂ P4 be the projection map. The tangent space to F1 (X) at a point L ∈ F1 (X) may be identified with the space of sections H 0 (L, NL/X ). In these terms, at all sufficiently general points (L, p) ∈ 2, the image of the differential dρ(L,p) : TL F1 (X) → Tp X mod Tp L is simply the image of the map H 0 (L, NL/X ) → (NL/X )p = Tp X/Tp L given by evaluation at p. Since NL/X has exactly one summand of nonnegative degree, this image is always 1-dimensional (mod Tp L), and so we may conclude that the image of the map ρ : 2 → X—that is, the union of the lines on X—is always exactly 2-dimensional. But X contains no 2-planes, and no surface in projective space other than a 2-plane may contain ∞2 lines, so we may conclude that X contains only ∞1 lines. On the basis of a naive dimension count, we expect the map ρ from the 2-dimensional variety 2 to the 3-fold X to have a 1-dimensional double point locus, that is, ∞1 pairs of distinct lines L, L ⊂ X that meet. We thus expect that each line of X meets finitely many others. (We calculate the number in just a moment.) Accordingly, we call a line L ⊂ X exceptional if it meets infinitely many other lines of X. We denote by F1e (X) ⊂ F1 (X) the locus of exceptional lines. By a straightforward dimension count, a general quartic 3-fold X has no exceptional lines, and a general X containing an exceptional line contains only finitely many; in particular, it contains nonexceptional lines as well. Our situation is that we are able to apply Theorem 1.5 directly to the surface S = H ∩X, where H is a general hyperplane section containing a nonexceptional line L, and so our concern is whether an arbitrary X may contain only exceptional lines. In fact, this is possible; the Fermat quartic is one example (we do not know any other examples). What we want to do, accordingly, is to say as much as we can about quartic 3-folds that have positive-dimensional families of exceptional lines. So, let X be a smooth quartic 3-fold, and let A ⊂ F1e (X) be an irreducible component of the Fano variety of lines on X consisting entirely of exceptional lines. For each line L ∈ A, the locus of lines meeting L contains one or more irreducible components of F1 (X), and so we must have one of the following two situations: (1) there are two irreducible components A, A ⊂ F1 (X) such that every pair of lines L ∈ A and L ∈ A meet; or (2) there is an irreducible component A ⊂ F1 (X) such that every pair of lines L, L ∈ A meet. In the first case, the surface S ⊂ X swept out by the lines of A has two rulings by lines; but the only surface in projective space with two rulings by lines is a quadric surface in P3 , and since Pic(X) = Z"ᏻX (1)#, X contains no such surfaces; thus the first case cannot occur. In the second case, let L, L ∈ A be two general lines, meeting at a point p. A third general line L ∈ A must meet both L and L ; if it does not pass through p it must lie in the plane spanned by L and L , and so this plane would have to be contained in X. Since X contains no 2-planes, we conclude that all the
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lines L ∈ A have a common point p. It follows, in particular, that all the lines L ∈ A are contained in the tangent plane Tp X and, hence, that X ∩ Tp X is simply the cone with vertex p over an irreducible plane quartic curve. Finally, since X is smooth, the Gauss map Ᏻ cannot be constant on any curve of X. Thus X ∩ Tp X can have at most isolated singularities, and hence the curve C must be smooth. We have thus established the following proposition. Proposition 7.2. Let X be a smooth quartic 3-fold, and let A ⊂ F1e (X) be an irreducible component of the Fano variety of lines on X consisting entirely of exceptional lines. Then there is a point p ∈ X such that X ∩ Tp X is the cone with vertex p over a smooth plane quartic curve, and A is simply the ruling of this cone. In particular, A ⊂ G(1, 4) ⊂ P9 has as underlying reduced scheme a smooth plane quartic curve. It is a nice exercise to check directly that in this case the component A of the Fano scheme F1 (X) has multiplicity 2. Since by the proof of Lemma 7.1 the Fano scheme is a curve of degree 320 in G(1, 4) ⊂ P9 , it follows that a quartic 3-fold X contains only exceptional lines if and only if it has exactly 40 hyperplane sections consisting of cones over quartic plane curves. Again, this is the case for the Fermat quartic; we do not know if there are others. In any event, Theorem 1.4 now follows readily from Theorem 1.5. Let X ⊂ P4 be any quartic 3-fold defined over a field K. Suppose first that X contains a nonexceptional line L; say, L is defined over a field K ⊃ K. Then for a general hyperplane H∼ = P3 ⊂ P4 containing L, the surface SH = X ∩H contains no other lines meeting L, and by part (b) of Theorem 1.5 we may deduce that SH (K ) is Zariski dense in SH ; hence X(K ) is Zariski dense in X. If on the other hand every line of X is exceptional, then by Proposition 7.2 the Fano variety of lines on X consists of a union of curves supported on plane quartic curves. If L ⊂ X is a line corresponding to a general point [L] ∈ A ⊂ F1 (X) of a component A of F1 (X)—in particular, if it is not a point of intersection of A with another component of F1 (X)—then L can meet only finitely many lines L ⊂ X corresponding to points [L ] ∈ / A. It follows that, for a general hyperplane H ∼ = P3 ⊂ P4 containing L, the surface SH = X ∩ H contains exactly three other lines meeting L. So once again the hypothesis of part (b) of Theorem 1.5 is satisfied, and we conclude that the points of X rational over the field of definition of L are Zariski dense. 8. Other elliptic surfaces. As suggested in Section 3.1, the approach via the calculation of intersection numbers in the Néron-Severi group of an associated surface works in substantially greater generality. Specifically, we prove the following theorem. Theorem 8.1. Let S be any smooth irrational surface defined over a number field K, and let π : S → P1 be an elliptic fibration (over K) all of whose fibers are irreducible; for λ ∈ P1 (K), let Eλ = π −1 (λ) be the fiber of S → P1 over λ. Let
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C ⊂ S be any smooth rational or elliptic curve of degree m ≥ 2 over P1 and of n = 0 any integer. Then for all but finitely many λ ∈ π(C(K)), nm · p ∼ ᏻEλ ⊗ ᏻS (n · C), for all p ∈ (Eλ ∩ C)(K). As before, we may immediately deduce from this the following corollary. Corollary 8.2. Let S be any smooth surface defined over a number field K, and let π : S → P1 be an elliptic fibration with irreducible fibers as in Theorem 8.1. (a) Let C ⊂ S be a smooth rational curve. Assume that S, π, and C are defined over a field K and that C is rational over K. Then the set S(K) of K-rational points of S is Zariski dense. (b) Now let C ⊂ S be a smooth curve of genus 1, and assume that S, π, and C are defined over a field K. Then there is a finite extension K of K such that the set S(K ) of K -rational points of S is Zariski dense. Part (a) of Corollary 8.2 follows from Theorem 8.1 as before. We see that for all but finitely many points p ∈ C(K), the fiber of S → P1 over π(p) has infinitely many rational points. As for part (b), we have to make an extension of our ground field K simply to ensure that the curve C has infinitely many rational points, and then the argument proceeds as before. Of course we can drop the hypothesis in Theorem 8.1 that S is irrational. Proof of Theorem 8.1. The proof is analogous to that of Theorem 3.1. We begin by making a base change. We let T be the incidence correspondence T = (p, q) : q ∈ Eπ(p) ⊂ C × S. As before, T is the fiber product T = C ×P1 S. In particular, T is an m-sheeted cover of S, branched over the union of the fibers Eλ of S → P1 such that Eλ is tangent to C. Again, T has, at worst, isolated singularities, since by the hypothesis that S is smooth and all fibers of π are irreducible it follows that all fibers are reduced as well. The surface T is also normal since it is regular in codimension 1 and since it is locally defined by one equation in the smooth irreducible variety C × X. Note that T → C has a tautologous section = (p, p) : p ∈ C ⊂ T . As a divisor, the pullback ν ∗ (C) of the curve C under the m-sheeted covering ν : T → S is thus a sum ν ∗ (C) = + R
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with R ⊂ T flat of degree m − 1 over C. As before, since all but finitely many fibers of C over P1 are reduced, R does not contain . Now let φ ∈ A1 (T ) be the class of a fiber of T → C, σ the class of the section , and ρ the class of R. The key ingredient in our proof is the following lemma. Lemma 8.3. The classes σ , ρ, and φ ∈ A1 (T ) are independent in the group A1 (T ) of Weil divisors modulo linear equivalence on T . Proof. We calculate the matrix of intersection numbers of the classes σ , ρ, and φ ∈ A1 (T ). (The intersections are defined even for singular T .) Three of these numbers are readily calculated. To begin with, φ is the class of a fiber of the map T → L, so of course φ 2 = 0; and since and R meet each fiber in 1 and m−1 points, respectively, we have (φ · σ ) = 1 and (φ · ρ) = m − 1. As before, we do not know anything about c = (σ · ρ) except that it is positive. (If T were smooth, it would have to be 2m − 2.) Finally, let −b be the self-intersection of the curve C on S. By the hypothesis that S is irrational, the canonical class KS is a nonnegative (rational) multiple of the class of a fiber of S → P1 and so has nonnegative intersection with C. It follows that b ≥ 2 if C is rational, and g ≥ 0 if C has genus 1. Table 3 Intersection Products φ
σ
ρ
φ
0
1
m−1
σ
1
−b − c
c
ρ
m−1
c
−(m − 1)b − c
Now, to calculate σ 2 and ρ 2 , we use the relation ν ∗ C = + R. It follows that
σ 2 = σ · [ν ∗ C] − ρ ,
and since, by the projection formula (which can be checked on some desingularization), ( · ν ∗ C)T = (ν∗ · C)S = (C · C)S = −b, we have σ 2 = −b − c. Similarly, from the same relation it follows that ρ 2 = ρ · [ν ∗ C] − σ .
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By the projection formula, (R · ν ∗ C)T = (ν∗ R · C)S = (m − 1)C · C S = −(m − 1)b and so ρ 2 = −(m − 1)b − c. In sum, then, we have Table 3. The determinant of this matrix is c(m − 1) + (m − 1)b + c + c(m − 1) + (b + c)(m − 1)2 , and since b is nonnegative and c positive, we are done. This concludes the proof of Theorem 8.1 and Lemma 8.3. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16] [17] [18]
W. Barth, C. Peters, and A. van de Ven, Compact Complex Surfaces, Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 1984. A. Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), 309–391. , “Variétés rationnelles et unirationnelles” in Algebraic Geometry—Open Problems (Ravello, 1982), Lecture Notes in Math. 997, Springer, Berlin, 1983, 16–33. F. Bogomolov and Yu. Tschinkel, Density of rational points on Enriques surfaces, Math. Res. Lett. 5 (1998), 623–628. , Density of rational points on elliptic K3 surfaces, preprint, http://xxx.lanl.gov/abs/ math.AG/9902092. , On the density of rational points on elliptic fibrations, J. Reine Angew. Math. 511 (1999), 87–93. F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. École Norm. Sup. (4) 25 (1992), 539–545. C. H. Clemens, Double solids, Adv. Math. 47 (1983), 107–230. A. Collino, Lines on quartic threefolds, J. London Math. Soc. (2) 19 (1979), 257–267. J.-L. Colliot-Thélène, A. N. Skorobogatov, and P. Swinnerton-Dyer, Double fibres and double covers: Paucity of rational points, Acta Arith. 79 (1997), 113–135. O. Debarre, Variétés de Fano, Astérisque 245 (1997), 4, 197–221, Séminaire Bourbaki 1996/97, exp. no. 827. V. A. Dem’janenko, “Rational points of a class of algebraic curves” in Thirteen Papers on Group Theory, Algebraic Geometry and Algebraic Topology, Amer. Math. Soc. Transl. Ser. 2 66, Amer. Math. Soc., Providence, 1968, 246–272. J. Franke, Yu. I. Manin, and Yu. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), 421–435. W. Fulton, Intersection Theory, 2d ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998. V. A. Iskovskikh, Fano threefolds, I (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 516–562, 717. , Fano threefolds, II (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 506–549. , On the rationality problem for conic bundles, Duke Math. J. 54 (1987), 271–294. V. A. Iskovskikh and Yu. I. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem (in Russian), Mat. Sb. (N.S.) 86 (1971), 140–166.
500 [19] [20] [21]
[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
HARRIS AND TSCHINKEL V. A. Iskovskikh and Yu. G. Prokhorov, Algebraic Geometry, Vol. 5: Fano Varieties, Encyclopaedia Math. Sci. 47, Springer, Berlin, 1999. K. Kodaira, On compact analytic surfaces, II, Ann. of Math. (2) 77 (1963), 563–626; III, 78 (1963), 1–40. J. Kollár, Y. Miyaoka, and S. Mori, “Rational curves on Fano varieties” in Classification of Irregular Varieties (Trento, 1990), Lecture Notes in Math. 1515, Springer, Berlin, 1992, 100–105. Yu. I. Manin, The p-torsion of elliptic curves is uniformly bounded (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 459–465. , Notes on the arithmetic of Fano threefolds, Compositio Math. 85 (1993), 37–55. L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437–449. S. Mori and S. Mukai, Classification of Fano threefolds with B2 ≥ 2, Manuscripta Math. 36 (1981/82), 147–162. , “On Fano threefolds with B2 ≥ 2” in Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, 1983, 101–129. J. P. Murre, “Classification of Fano threefolds according to Fano and Iskovskikh” in Algebraic Threefolds (Varenna, 1981), Lecture Notes in Math. 947, Springer, Berlin, 1982, 35–92. J. Noguchi, A higher-dimensional analogue of Mordell’s conjecture over function fields, Math. Ann. 258 (1981/82), 207–212. I. I. Piatetski-Shapiro and I. Shafarevitch, Torelli’s theorem for algebraic surfaces of type K3 (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572. J.-P. Serre, Lectures on the Mordell-Weil Theorem, 2d ed., Aspects Math. E15, Vieweg, Braunschweig, 1990. J. H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math. 1239, Springer, Berlin, 1987.
Harris: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138, USA; [email protected] Tschinkel: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607, USA; [email protected]
Vol. 104, No. 3
DUKE MATHEMATICAL JOURNAL
© 2000
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2 CHANG-SHOU LIN 1. Introduction. Let (S 2 , g0 ) be the unit sphere of R3 equipped with the metric g0 induced from the flat metric of R3 . For a positive smooth function f on S 2 , we consider the nonlinear equation
1 f (y)eφ − φ +ρ φ 4π S 2 f (y)e dµ
=0
on S 2 ,
(1.1)ρ
where is the Beltrami-Laplace operator of (S 2 , g0 ), dµ is the volume form with respect to g0 , and ρ > 0 is a constant. Obviously, equation (1.1)ρ is invariant under adding a constant c. Hence, we always seek solutions of (1.1)ρ , which are normalized by φ(y) dµ(y) = 0. (1.2) S2
Equation (1.1)ρ is called the mean field equation because it often arises in the context of statistical mechanics of point vortices in the mean field limits. Recently, there has been interest in (1.1)ρ because it also arises from the Chern-Simons-Higgs model vortex theory when some parameter tends to zero. (For these recent developments, we refer the readers to [5], [2], [3], [10], [11], [13], [14], [18], [19], [21], [22], and the references therein.) Clearly, equation (1.1)ρ is the Euler-Lagrange equation of the nonlinear functional Jρ (φ) =
1 2
S2
|∇φ|2 dµ − ρ log
S2
f (y)eφ dµ
(1.3)
for φ ∈ H 1 (S 2 ) satisfying (1.2). Here H 1 (S 2 ) denotes the Sobolev space of functions with L2 -integrable first derivatives. For ρ < 8π, Jρ (φ) is bounded below, and the infinimum of Jρ can be achieved by the well-known Moser-Trudinger inequality. However, for the case ρ ≥ 8π, the existence of solutions to (1.1)ρ is much more delicate. Recently, under some conditions on f , the existence of an infinimum of J8π has been proved by [10] and [18]. However, the existence of solutions to (1.1)ρ Received 3 August 1999. Revision received 28 January 2000. 2000 Mathematics Subject Classification. Primary 35J60. 501
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remains open in general for ρ > 8π. In order to study this problem, Y. Y. Li [14] initiated study of the existence of solutions by way of computing the Leray-Schauder topological degree for equation (1.1)ρ . He proved that the concentration phenomenon could occur only when ρ is equal to 8mπ, where m is a positive integer. Therefore, the topological degree is constant in each interval (8mπ, 8(m + 1)π). Furthermore, he showed that the degree is always equal to 1 as long as ρ < 8π. The main purpose of this article is to prove the following theorem. Theorem 1.1. Let f be a positive C 1 function on S 2 , and let d(ρ) denote the Leray-Schauder degree for equations (1.1)ρ and (1.2) for ρ = 8πm, where m is a positive integer. Then (i) d(ρ) = −1 for 8π < ρ < 16π, and (ii) d(ρ) = 0 for 16π < ρ < 24π. An immediate corollary of Theorem 1.1 is that equation (1.1)ρ always possesses a solution when 8π < ρ < 16π . In fact, Theorem 1.1 provides more information about equation (1.1)ρ . For example, Theorem 1.1(ii) implies that if there are solutions to equation (1.1)ρ for 16π < ρ < 24π, then (1.1)ρ always possesses two solutions, at least in the generic situation. Since d(ρ) has a gap at ρ = 8π or 16π, the concentration phenomenon actually occurs there. It is also very interesting to compare Theorem 1.1 with a previous result of Chang-Gursky-Yang (see [4]) for the case ρ = 8π. Note that when ρ = 8π, (1.1)ρ is equivalent to the Gaussian curvature equation on S 2 . Let f be a positive Morse function on S 2 satisfying f (p) = 0 for any critical point p of S 2 . It was proved in [4] that there is an a priori bound for solutions of (1.1)8π , and the Leray-Schauder degree d(8π) can be computed by the formula d(8π) = 1 −
(−1)ind(p) ,
(1.4)
p∈ −
where − = {p ∈ S 2 | p is a critical point of f such that f (p) < 0} and ind(p) is the Morse index of f at p. Note that the expression (1.4) is different from the one in [4], because we have already normalized φ so that (1.2) holds. Thus, for these functions f such that d(8π ) = ±1, equation (1.1)ρ possesses blowing-up solutions for ρi approaching 8π from above and below. The proof of Theorem 1.1 is based on the observation that the concentration phenomenon generally induces symmetry. It has been proved recently that a spherical Harnack inequality holds for blowing-up solutions of either the mean field equations on compact Riemann surfaces or the scalar curvature equation on S n . (See [6], [14] for more precise statements.) In the situation when the nonlinear equation is invariant under some group action, it is believed that concentrated solutions should possess a certain symmetry also. In this paper, we study this symmetric property of solutions and apply it to the calculation of the Leray-Schauder degree. First, we consider the case f (y) ≡ 1.
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503
Theorem 1.2. Assume f (y) ≡ 1. Then there is a universal constant C > 0 such that the a priori estimate |φ(y)| ≤ C
for y ∈ S 2
(1.5)
holds for all solutions φ of (1.1)ρ and (1.2) for 8π = ρ ≤ 16π. Furthermore, suppose that φi is a solution of (1.1)ρi and (1.2) such that maxS 2 φi = ∞ and limi→+∞ ρi = 16π . Then ρi > 16π and there exists a direction ni in R3 for each large i such that φi is axially symmetric with respect to the direction ni . Second, we consider the case f (y) ≡ exp(−γ n, y), where n is a unit vector of R3 , n, y is the inner product of n and y ∈ S 2 , and γ is a positive constant. Note that by the Kazdan-Warner identity, equation (1.1)ρ possesses no solutions when f (y) = exp(−γ n, y) and ρ = 8π. Our second symmetry result allows us to compute the topological degree of (1.1)ρ for 8π < ρ < 16π. Theorem 1.3. Let f be described as above. Then for any γ > 0, there exists ρ0 = ρ0 (γ ) > 8π such that for any solution φ of (1.1)ρ with 8π = ρ ≤ ρ0 (γ ), φ is axially symmetric with respect to the direction n. Furthermore, φ is the unique solution for each 8π = ρ ≤ ρ0 (γ ) and γ ≤ 1. Theorems 1.2 and 1.3 allow us to reduce equation (1.1)ρ to an ordinary differential equation when we come to compute the topological degree. For the proofs of both theorems, we will apply the well-known method of moving planes and its variant, the method of moving spheres. The method of moving planes was invented by Alexandrov and was later used to study the radial symmetry for positive solutions of semilinear elliptic equations by Serrin [20], Gidas-Ni-Nirenberg [12], and others. It was applied to study the concentrated behaviors of blowing-up solutions and asymptotic behavior for singular solutions. For these applications, we refer the reader to [6], [14], and the references therein. However, we should remark that for equation (1.1)ρ with those f described above, a solution φ is generally not axially symmetric. The paper is organized as follows. In Section 2, we apply the method of moving planes to prove Theorem 1.2. Here, the isoperimetric inequality due to Bandle in [1] can allow us to start the process of moving planes. For the proof of Theorem 1.3, we first apply the method of moving spheres to locate the blow-up point and then apply the method of moving planes to prove axial symmetry. This is done in Section 3. For the case f (y) = exp(−γ n, y), we prove that, if ρi > 8π, then the blow-up occurs at the minimum point of f (y). However, for ρi < 8π, the blow-up can occur at the maximum point of f . It is a very interesting question to study where the blow-up point is actually located. The complete proof of Theorem 1.1 is given in the last section. To calculate the topological degree for Theorem 1.1(ii), we need a theorem due to Wang (see [23]), which states how to count the local degree due to a nondegenerate orbit. The major part of Section 4 is devoted to studying the nondegeneracy of the orbit of solutions. It is interesting to note that in the case of ρ > 16π , the orbit is topologically homeomorphic to RP 2 , the real projective space of
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two dimensions. This is the reason why the degree d(ρ) vanishes for 16π < ρ < 24π. This nondegeneracy requires detailed analyses for the linearized equations at blowingup solutions. In particular, for the case with solutions of two blow-up points, a general form of the Pohozaev identity is employed very delicately. 2. The method of moving planes. In this section, we prove the uniform bound of solutions for equation (1.1)ρ with f ≡ 1. First suppose that there is a solution φi of (1.2) and that 1 eφi − = 0 on S 2 , (2.6) φi + ρi φi 4π S 2 e dµ such that max φi −→ +∞, S2
ρi = 8π,
lim ρi = 8π
i→+∞
(2.7) (2.8)
hold. We recall the following result of [15], which is useful in this paper (see [15, Theorems 1.1 and 1.2]). Theorem 2.1. Let f (y) = exp(−γ n, y) for y ∈ S 2 , where γ ≥ 0 and n is a unit vector of R3 . Then (i) If γ = 0 and ρ < 8π , then φ ≡ 0 is the only solution of (1.1)ρ and (1.2). (ii) If γ > 0 and ρ < 8π, then there exists a unique solution of (1.1)ρ and (1.2). Furthermore, this solution is axially symmetric with respect to the direction n. (iii) If 0 ≤ γ ≤ 1 and ρ ≤ 16π, then there exists a unique solution of (1.1)ρ and (1.2) in the class of axially symmetric functions. By Theorem 2.1, we have ρi > 8π.
(2.9)
To yield a contradiction, we apply the well-known method of moving planes. The method can work here mainly due to the concentration phenomenon of φi . To see it, we need a result to describe the concentration of φi , due to Li in [14]. Let φi be a sequence of solutions of (1.1)ρi and (1.2). Assume that {q1 , . . . , qm } is the blow-up set of φi . Set φi (y) f (y)e dµ . (2.10) ξi (y) = φi (y) − log S2
Without loss of generality, we may assume that for any blow-up point q, ξi (Pi ) = max ξi (y) −→ +∞ |y−q|≤δ
for any small δ > 0. In [14], Li proved the following result.
(2.11)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
Theorem 2.2. There exists a constant C > 0 such that eξi (Pi ) ≤C ξi (y) − log 2 ξ (P ) 2 1 + ρi f (Pi )/8 e i i |y − Pi |
505
(2.12)
for |y − Pi | ≤ δ. Furthermore, ξi − ξ¯i −→ 8π
m
G(·, ql ) − 2m
S2
l=1
G(·, y) dµ(y)
(2.13)
2 (S 2 \{q , . . . , q }), where {q , . . . , q } is the blow-up set of φ , G(·, q ) is the in Cloc 1 m 1 m i l Green function with a singularity at ql , and ξ¯i = −S 2 ξi (y) dµ(y) is the integral average of ξi .
An immediate consequence of Theorem 2.2 is the following. Let {q1 , . . . , qm } be the set of blow-up points of φi . Then Theorem 2.2 implies that lim ρi = 8mπ,
i→+∞
and for any δ > 0, there exists a constant c = c(δ) such that ξi (y) + ξi (Pi,j ) ≤ c(δ)
(2.14)
for |y − qj | ≥ δ, where Pi,j is the local maximum point of ξi near qj . In particular, ξi (Pi,j ) − ξi (Pi,l ) ≤ C
(2.15)
for 1 ≤ j , l ≤ m and for some constant C independent of i. We see later that (2.15) is very important for proving symmetry for the case with two blow-up points. Another useful tool in our approach is the isoperimetric inequality due to Bandle (see [1]). Lemma 2.3. Let q(x) be a continuous positive function defined in a simply connected domain ' of R2 . Suppose that q(x) satisfies log q(x) + q(x) ≥ 0
for x ∈ '
(2.16)
in the distribution sense and '
q(x) dx ≤ 8π.
(2.17)
Let ω ' be a subdomain such that the first eigenvalue + q(x) for the Dirichlet
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problem is nonpositive. Then ω
q(x) dx ≥ 4π.
(2.18)
Lemma 2.4. Let v0 (y) = −2 log(1 + (1/8)|y|2 ), and let ϕ(y) satisfy 2, ϕ(y) + ev0 (y) ϕ(y) = 0 in R+
ϕ(0, y2 ) = 0
for y2 ∈ R.
(2.19)
Suppose ϕ(y) → 0 as |y| → ∞. Then ϕ(y) = c∂v0 /∂y1 for some constant c ∈ R. Note that v0 is the solution of v0 + ev0 = 0 such that v0 (0) = maxR2 v0 (y) = 0, and equation (2.19) is the linearized equation at v0 . The proof of Lemma 2.3 is elementary and is omitted here. Proof of Theorem 1.2. Let φi be a sequence of solutions of (2.6) and (1.2) such that (2.7) and (2.8) hold. By Theorem 2.2, φi has only one blow-up point. Let Pi be the maximum point of φi . Without loss of generality, we may assume Pi = (1, 0, 0). By using the stereographic projection π of S 2 onto R2 , we set ρi ui (x) = ξi π −1 (x) − log 1 + |x|2 + log(4ρi ), 4π where ξi is given by (2.10). By a straightforward computation, ui (x) satisfies l ui (x) + 1 + |x|2 i eui (x) = 0 and
R2
in R2 ,
(2.20)
(2.21)
l 1 + |x|2 i eui (x) dx = 4(2 + li )π,
(2.22)
ρi − 2. 4π
(2.23)
where li =
For simplicity of notation, we denote the maximum point of ui by Pi . Obviously, Pi is contained in a small neighborhood of (1,0). By (2.9) and (2.23), we have li > 0. To apply Lemma 2.3, we set q(x) = (1 + |x|2 )li eui (x) . By elementary calculations, q(x) satisfies 4li
log q(x) = ui (x) +
1 + r2
because li > 0.
2 ≥ −q(x)
(2.24)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
507
To start the process of moving planes, we show ui (x) ≥ ui (x − ) for x1 ≥ 0,
(2.25)
where x − = (−x1 , x2 ). To see (2.25), we set w(x) = ui (x) − ui (x − ) for x1 ≥ 0. Then w(x) satisfies
where
w(x) + c(x)w(x) = 0
for x1 ≥ 0,
w(0, x ) ≡ 0 2
for x2 ∈ R,
c(x) = 1 + |x|
2 li
(2.26)
− eui (x) − eui (x ) . ui (x) − ui (x − )
2 = {(x , x ) | x > 0}. Set ' = {x ∈ R 2 | Now suppose w(x) < 0 for some x ∈ R+ 1 2 1 + − − w(x) < 0} and ' = {x | x ∈ '}. Then
l − w(x) + 1 + |x|2 i eui (x ) w(x) ≤ 0
(2.27)
for x ∈ ', which implies that the first eigenvalue of +q(x) in '− for the Dirichlet problem is nonpositive where q(x) = (1 + |x|2 )li eui (x) . By (2.19), q(x) satisfies (2.16). Thus, we have by Lemma 2.3, l 1 + |x|2 i eui (x) dx ≥ 4π. (2.28) '−
On the other hand, by Theorem 2.2 we have that for any ε > 0, there is i0 = i0 (ε) such that l 2 li ui (x) dx ≥ (2.29) 1 + |x| e 1 + |x|2 i eui (x) dx ≥ 8π − ε. B(Pi ,1/2)
2 R+
Together with (2.28) and (2.29), we have by (2.22), l 4(2 + li )π = 1 + |x|2 i eui (x) dx ≥ 4π + 8π − ε, R2
which obviously yields a contradiction because li → 0. Hence, (2.25) is established. We note that w(x) ≡ 0 because ui has a blow-up point at Pi . Thus, by the strong 2 , and we have maximum principle and the Hopf lemma, we have w(x) > 0 for x ∈ R+ ∂w/∂x1 (0, x2 ) > 0 for x2 ∈ R.
508
CHANG-SHOU LIN
For any λ ∈ R, we let 2λ = {x | x1 > λ}, Tλ = {x | x1 = λ} and let x λ = (2λ − x1 , x2 ) denote the reflection point of x with respect to Tλ . As in (2.25), we want to prove ui (x) ≥ ui (x λ ) for x ∈ 2λ and for 0 ≤ λ ≤ 2.
(2.30)
For 0 ≤ λ ≤ 2, we set wλ (x) = ui (x) − ui (x λ ) for x ∈ 2λ . Then wλ (x) satisfies l λ wλ (x) + 1 + |x|2 i eui (x) − eui (x ) l l λ = 1 + |x λ |2 i − 1 + |x|2 i eui (x ) ≤ 0
(2.31)
for x ∈ 2λ . Here we note that for x ∈ 2λ , |x|2 − |x λ |2 = 4λ(x1 − λ) ≥ 0 because λ ≥ 0. For 0 ≤ λ ≤ 2, we let λ
l eui (x) − eui (x ) cλ (x) = 1 + |x|2 i . ui (x) − ui (x λ ) By (2.20) and (2.23), we have cλ (x) ≤ c 1 + |x| −4
(2.32)
for some constant c > 0 and |x| ≥ 2. Applying the maximum principle, we can prove by (2.32) that there exists R0 > 0 such that if wλ (x) is negative somewhere in 2λ for 0 ≤ λ ≤ 2, then |xλ | ≤ R0 , where wλ (xλ ) = inf wλ (x)
(2.33)
2λ
(see [6] for the proof of (2.33)). By (2.26), wλ satisfies wλ +cλ (x)wλ (x) ≤ 0 in 2λ . Together with (2.33), (2.30) can be proved by the standard argument of the method of moving planes. For details of the proof, we refer the reader to [6]. Clearly, (2.25) yields a contradiction to the fact that ui has a local maximum point at Pi that is near (1, 0). Therefore, the uniform bound is established for ρi → 8π from above. Next, we are going to prove the uniform bound up to 16π. Now suppose φi is a solution of (2.6) and (1.2) with limi→+∞ ρi = 16π, and assume that (2.7) holds also. Then Theorem 2.2 implies that φi has exactly two blow-up points P , Q. By using the same argument as above, we conclude that P is antipodal to Q. Actually, we
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
509
want to prove more. Let Pi and Qi be local maximum points of φi near P and Q, respectively; that is, φi (Pi ) =
max φi (y),
|y−P |≤δ0
φi (Qi ) =
max φi (y).
|y−Q|≤δ0
We want to prove that Pi and Qi are antipodal for large i, and φi (y) is axially −−→ symmetric with respect to the direction Pi Qi . Now recall that by Theorem 2.2, there is a constant C > 0 such that φi (Pi ) − φi (Qi ) ≤ C.
(2.34)
−→ Suppose Pi = −Qi . Let n be the direction P Q. By (2.34), we can slightly change the direction n to another direction ni such that the following statements hold. (i) Let Pi∗ and Q∗i be the intersection of the line ni with S 2 where Pi∗ and Q∗i are close to Pi and Qi , respectively. Then eξi (Pi ) |Pi − Pi∗ |2 ∼ eξi (Qi ) |Qi − Q∗i |2 , where for two sequences of positive numbers, ai ∼ bi means that the ratio ai /bi is bounded from below and above by positive constants. (ii) Without loss of generality, we may assume Pi∗ = (0, 0, −1) and Q∗i = (0, 0, 1) and by a rotation if necessary, both Pi and Qi are contained in the half hyperplane {y ∈ S 2 | y1 > 0 and y2 = 0}. Let π be the stereographic projection of S 2 onto R2 . As before, set ρi ui (x) = ξi π −1 (x) − log 1 + |x|2 + log(4ρi ). 4π Then ui (x) satisfies (2.21) and (2.22). Clearly, the maximum point of ui is close to the image of Pi . For simplicity of notation, we still use Pi to denote the maximum point of ui ; that is, ui (Pi ) = max ui (x), R2
and we use Qi to denote the maximum point of u∗i , u∗i (Qi ) = max u∗i (x), R2
where u∗i (x) = ui
x − 2(2 + li ) log |x|. |x|2
(Note that 2+li = ρi /4π .) Obviously, both Pi and Qi → 0 as i → +∞. By Theorem 2 . For the two quan2.2, (i), and (ii), we conclude that Pi and Qi are located in R+ ∗ tities eui (Pi ) |Pi |2 and eui (Qi ) |Qi |2 , either they simultaneously have a positive lower bound or they tend to zero simultaneously. For the latter case, if ti and si are the
510
CHANG-SHOU LIN
x1 -coordinates of Pi and Qi , then both |Pi | ∼ ti and |Qi | ∼ si hold. Let vi (y) = ui Pi + e−ξi (Pi )/2 y − ui (Pi ).
(2.35)
2 (R 2 ). By passing to a Then by Theorem 2.2, vi (y) is uniformly bounded in Cloc subsequence, vi converges to v0 (y), where v0 (y) is the solution of v0 (y) = 0 in R 2 , v0 (y) + e (2.36) v0 (0) = max v0 (y), and ev0 (y) dy = 8π. R2
R2
By a result of Chen and Li in [6], v0 (y) = −2 log(1+(1/8)|y|2 ). Thus, for any small r > 0 and for eξi (Pi ) |x − Pi |2 ≤ r, we have (2.37) ui (x) = ui (Pi ) − eui (Pi ) |x − Pi |2 a + o(1) , where a is a positive constant. To yield a contradiction, we prove ui (x) ≥ ui (x − )
(2.38)
2 . Once (2.38) is established, we follow the same argument of (2.30) to prove for x ∈ R+
ui (x) ≥ ui (x λ )
(2.39)
for x ∈ 2λ and for 0 ≤ λ ≤ 1. Obviously, it yields a contradiction to the fact that ui has a maximum point at Pi . (Note that the standard argument for the method of moving planes still applies for our case because for each i, wλ (x) = ui (x) − ui (x λ ) tends to 0 as |x| → +∞.) To prove (2.38), we divide the argument into two cases. ∗
Case 1. Both eui (Pi ) |Pi |2 and eui (Qi ) |Qi |2 ≥ c > 0. By Theorem 2.2, the scaled function vi (y) = ui Pi + e−ui (Pi )/2 y − ui (Pi ) uniformly converges to v0 (y), the solution of (2.36). Two cases are discussed separately. If eui (Pi ) |Pi |2 → +∞, then by Theorem 2.2, there is a constant C such that ρi u(Pi ) 2 |Pi | ≤ C − ui (Pi ) + 4 log |Pi | (2.40) ui (x) ≤ C + ui (Pi ) − 2 log 1 + e 8 holds for x1 ≤ 0 and |x| ≤ δ0 . By the scaling in (2.35) for any R > 0, ui (x) ≥ ui (Pi ) − 2 log 1 + R 2 − C
(2.41)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
511
for |x −Pi |eu(Pi )/2 ≤ R. Since u(Pi )+2 log |Pi | → +∞, we have by (2.40) and (2.41), ui (x) ≥ ui (x − )
(2.42)
for |x − Pi | ≤ e−ui (Pi )/2 R. The similar inequality holds at Qi also. Now suppose 2 | w (x) < 0} = φ, where w (x) = u (x) − u (x − ). Then ' = {x ∈ R+ 0 0 i i − 2 li ui (x − ) 2 li ui (x) w0 (x) + 1 + |x| e w0 (x) ≤ w0 (x) + 1 + |x| − eui (x ) = 0. e By Lemma 2.3,
l 1 + |x|2 i eui (x) dx ≥ 4π.
2 R−
(2.43)
However, for any ε > 0, there is R0 = R0 (ε) and i0 = i0 (R0 ) such that if i ≥ i0 , then l 2 li ui (x) dx ≥ 1 + |x| e 1 + |x|2 i eui (x) dx ≥ 8π − ε (2.44) Bδ+
B(Pi ,e−ui (Pi )/2 R0 )
0
and
B˜ δ+
1 + |x|
0
2 li ui (x) e
dx =
B(Qi ,e
−u∗ i (Qi )/2
l ∗ 1 + |x|2 i eui (x) dx ≥ 8π − ε (2.45)
R0 )
2 | |x| ≤ δ } and B 2 | |x| ≥ δ −1 }. Clearly, ˜ + = {x ∈ R+ hold, where Bδ+0 = {x ∈ R+ 0 δ0 0 together (2.43), (2.44), and (2.45) yield a contradiction. If eui (Pi ) |Pi |2 ≤ C, then the rescaled function v˜i (y) = ui e−ui (Pi )/2 y − ui (Pi ) 2. tends to v0 (y −y0 ) in any compact set of R2 , where limi→+∞ eui (Pi )/2 Pi = y0 ∈ R+ Since v0 (y) = v0 (|y|) is decreasing in |y| for any R > 0 there is i0 = i0 (R) such that ui (x) ≥ ui (x − ) for |x| ≤ e−ui (Pi )/2 R. Therefore, '∩B(0, e−ui (Pi )/2 R) = φ. The same holds at Qi . By Lemma 2.3, l 1 + |x|2 i eui (x) dx ≥ 4π, '−
where '− = {x | x − ∈ '}. Thus, for any ε > 0, there exists i0 = i(ε) such that for i ≥ i0 , l l 1 + |x|2 i eui (x) dx ≥ 1 + |x|2 i eui (x) dx R2
'−
+ +
|x|≤e−ui (Pi )/2 R
l 1 + |x|2 i eui (x) dx
|x|≤e
−u∗ i (Qi )/2
R
l ∗ 1 + |x|2 i eui (x) dx ≥ 20π − 2ε.
Clearly, it yields a contradiction. Therefore (2.38) is proved for case 1.
512
CHANG-SHOU LIN
∗ Case 2. limi→∞ eui (Pi ) |Pi |2 + eui (Qi ) |Qi |2 = 0. Let Ni = max |w0 (x)| = |w0 (xi )| 2 R+
(2.46)
2 . The maximum can be achieved because w (x) → 0 as |x| → +∞. for some xi ∈ R+ 0 We claim that xi should be in a neighborhood of 0 or ∞. We prove it by contradiction. Suppose there is a small δ0 > 0 and c0 > 0 such that there is a subsequence of ui (still denoted by ui ) such that
|w0 (xˆi )| = sup |w0 (x)| ≥ c0 Ni , Bδc
(2.47)
0
2 | δ ≤ |x| ≤ δ −1 }. Recall that w (x) satisfies where Bδc0 = {x ∈ R+ 0 0 0
l − 0 = w0 (x) + c0 (x)w0 (x) = w0 (x) + 1 + |x 2 | i eui (x) − eui (x ) . By Theorem 2.2,
(2.48)
|c0 (x)| = o(1) 1 + |x|−4 ,
2 away from the origin. Set ¯+ where o(1) → 0 uniformly for any compact set of R
wˆ 0 (x) = Ni−1 w0 (x).
(2.49)
2 (R 2 . By the elliptic estimates, w 2 ¯+ Then |wˆ 0 (x)| ≤ 1 for x ∈ R+ ˆ 0 (x) is bounded in Cloc 2 (R 2 \{0}). Since ¯+ \{0}). By passing to a subsequence, wˆ 0 (x) converges to h(x) in Cloc 2 \{0}. c0 (x) → 0, h(x) is a bounded harmonic function that is identical to zero on ∂R+ By the reflection x → (−x1 , x2 ), h can be extended to be a bounded harmonic function in R2 \{0}. By the regularity theorem and the Liouville theorem, we conclude h(x) ≡ 0 2 because h ≡ 0 on ∂R 2 . But this yields a contradiction to (2.47). Hence, we in R+ + have proved
sup |w0 (x)| = o(1)Ni . Bδc
(2.50)
0
To yield a contradiction, we first suppose that the maximum point xi of w0 is 2 . Set located in Bδ+0 = Bδ0 ∩ R+ w˜ 0 (y) = wˆ 0 e−ui (Pi )/2 y . Then w˜ 0 (y) satisfies w˜ 0 (y) + c˜0 (y)w˜ 0 (y) = 0
for |y| ≤ δ0 eui (Pi )/2 ,
(2.51)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
where
c˜0 (y) = c0 e
−ui (Pi )/2
y e
ui (Pi )
= Ki e
−ui (Pi )/2
y
513
evi (y) − evi (y) , vi (y) − vi (y − )
vi (y) = ui (e−ui (Pi )/2 y) − ui (Pi ), and Ki (x) = (1 + |x|2 )li . Since eui (Pi ) |Pi |2 → 0, 2 (R 2 ). Thus, by Theorem 2.2, we have vi (y) converges to v0 (y) of (2.36) in Cloc −4 |c˜0 (y)| ≤ A 1 + |y| (2.52) for |y| ≤ eui (Pi )/2 δ0 and for some constant A independent of i and y. Recall that by (2.50), w˜ i (y) = o(1) for |y| = eui (Pi )/2 δ0 . Thus, by Green’s formulas, we have |w˜ 0 (x)|
− |x − y| 1 c ˜ log (y) w ˜ (y) dy + the boundary term 0 0 2π |y|≤eui (Pi )/2 δ0 |x − y| (2.53) − 1 |x − y| ≤ |c˜0 (y)||w˜ 0 (y)| dy + o(1). log u (P )/2 2π |y|≤e i i δ0 |x − y| =
Let yi = eui (Pi )/2 xi , where xi is the maximum point of wˆ 0 . Then |w˜ 0 (yi )| = 1. By elementary calculations and by (2.52), (2.53) implies −1 (2.54) 1 ≤ c 1 + |yi | for some constant c. Therefore, yi is bounded. Since |w˜ 0 (y)| ≤ 1, after passing to a subsequence if necessary, w˜ 0 (y) converges to 2 (R 2 ), where w ¯+ w˜ in Cloc ˜ satisfies 2, w˜ + ev0 (y) w(y) ˜ = 0 in R+ 2 w(y) ˜ =0 on ∂ R+ . 2 . By (2.53) again, we estimate Since yi is bounded, we have w(y) ˜ ≡ 0 in R+ −1 |w˜ 0 (y)| ≤ c 1 + |y| + o(1), (2.55)
which implies |w(y)| ˜ = O(|y|−1 ) at infinity. By Lemma 2.4, we have w(y) ˜ =c
∂v0 (y) ∂y1
(2.56)
for some constant c = 0. Thus, ∂ w˜ ∂ 2 v0 (0) (0) = c = 0, ∂y1 ∂y12
(2.57)
514
CHANG-SHOU LIN
because ∂ 2 v0 (0)/∂y12 = (1/2)v0 (0) < 0. On the other hand, ∂ w˜ 0 ui (Pi )/2 −1 ∂vi ui (Pi )/2 − Pi = N i Pi e e ∂y1 ∂y1 ∂ 2 vi = Ni−1 2 (ηi )eui (Pi )/2 (−2ti ), ∂y1
(2.58)
where ti is the x1 -coordinate of Pi and ηi → 0 as i → +∞. Therefore, by (2.56), we have lim Ni−1 eui (Pi )/2 ti > 0.
i→+∞
Let w0∗ (x) be the Kelvin transformation of w0 ; that is, − x x x ∗ w0 (x) = w0 = ui − ui = u∗i (x) − u∗i (x − ). |x|2 |x|2 |x − |2
(2.59)
(2.60)
Set wˆ 0∗ = Ni−1 w0∗ and w˜ 0∗ (y) = wˆ 0∗ (e−(uˆ i /2)(Qi ) y). By Theorem 2.2 again, (2.58) also holds at Qi ; that is, ∂ w˜ 0∗ u∗ (Qi )/2 ∂ 2 vi∗ (ηi∗ ) u∗ (Qi )/2 Qi = Ni−1 e i (−2si ), e i ∂y1 ∂y12 ∗
where si is the x1 -coordinate of Qi . Since eui (Qi )/2 si ≥ ceui (Pi )/2 ti for some constant 2 . In c > 0, (2.59) implies that w˜ 0∗ converges to w∗ (y), where w ∗ (y) ≡ 0 on R+ ∗ particular, for any R > 0, there exists i0 = i0 (R) > 0 such that w˜ 0 (y) and w˜ 0 (y) > 0 for |y| ≤ R and i ≥ i0 . Let R be large. Then the inequality l 2 li ui (x) 1 + |x| e 1 + |x|2 i eui (x) dx dx ≥ R2
'−
+ +
|x|≤e−ui (Pi )/2 R
l 1 + |x|2 i eui (x) dx
−u∗ (Q )/2 |x|≤e i i R
l ∗ 1 + |x|2 i eui (x) dx
≥ 20π − 2ε yields a contradiction. Therefore, Pi = −Qi is proved. After Pi = −Qi is established, we want to prove that φi is axially symmetric −−→ with respect to the direction ni = Pi Qi . Let π be the stereographic projection of S 2 onto R2 by taking Qi to ∞, and let ui be defined as before. The radial symmetry of ui follows from the same argument as in case 2 above. Following the notation 2 . Then w in case 2, we let w0 (x) = ui (x) − ui (x − ). Suppose w0 (x) ≡ 0 in R+ ˜ 0 (y)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
515
2 (R 2 ). Because of (2.50), (2.54), and (2.55) can be ¯+ also converges to w(y) ˜ in Cloc proved in the same way. Thus, w(y) ˜ ≡ 0 and w(y) ˜ = c(∂v0 /∂y1 )(y) for some c = 0. But ∇ w˜ 0 (0) = Ni−1 e−ui (Pi )/2 2∇ui (0) = 0 because 0 is the maximum point of ui . Obviously, it implies c = 0 and it yields a contradiction again. We conclude that ui is symmetric with respect to x1 , and the radial symmetry follows readily. By Theorem 2.1(iii), the trivial solution φ ≡ 0 is the only solution of (2.6) satisfying (1.2), provided ρ ≤ 16π . Since φi blows up at P and Q, we have ρi > 16π. Therefore, the proof of Theorem 1.2 is finished.
3. The method of moving spheres. In this section, we begin with the equation −γ n,y φi e 1 e − = 0 on S 2 , (3.1) φi + ρi e−γ n,y φi dµ 4π where n is a unit vector of R3 , γ > 0 is a constant, and limi→+∞ ρi = 8π. We want to prove that φi is axially symmetric with respect to n for large i. If ρi < 8π by Theorem 2.1, then φi is axially symmetric. Therefore, in the following, we always assume ρi > 8π . Since the function e−γ n,y is a monotone function in the variable n, y, by the Kazdan-Warner identity, equation (3.1) has no solution for ρ = 8π. Thus, ξi (Pi ) = max ξi (y) −→ +∞ S2
as i → +∞, where ξi is defined by (2.10). Without loss of generality, we may assume P = limi→+∞ Pi . By Theorem 2.2, P is the only blow-up point of φi . In [8], the authors proved that P must be a critical point of f . Note that ±n are the only critical points of e−γ n,y . In the following, we want to prove a stronger result. Lemma 3.1. P = n. Proof. Without loss of generality, we assume that n = (0, 0, 1). Suppose P = n. Let π(x) be the stereographic projection of R2 onto S 2 , which maps n to +∞. Set ρi ui (x) = ξi π −1 (x) − log 1 + |x|2 + γ + log(4ρi ). (3.2) 4π By a straightforward computation, ui (x) satisfies l 2 ui (x) + 1 + |x|2 i e2γ /(1+|x| ) eui (x) = 0 and
in R2 ,
l 2 1 + |x|2 i e2γ /(1+|x| ) eui (x) dx = 4π(2 + li ),
where li =
−2+
ρi 4π
> 0.
(3.3)
(3.4)
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CHANG-SHOU LIN
For simplicity, we still denote P to be the blow-up point of ui , and Pi is the maximum point of ui near P . Let vi (x) ≡ ui (x) + ui (Pi ). Then vi (x) satisfies, by (3.3) and (3.4),
vi (x) + λi Ki (x)evi (x) = 0,
λi
R2
Ki (x)evi (x) dx = ρi ,
where Ki (x) = (1 + |x|2 )li e2γ /(1+|x| ) and where λi = e−ui (Pi ) . By (2.12), vi (x) is 2 (R 2 \{P }) and λ → 0 as i → +∞. Thus, it is easy to see uniformly bounded in Cloc i 2 . By the Liouville that vi (x) converges to a harmonic function G in R2 \{P } in Cloc theorem, G(x) = −4 log |x − P |. To find the location of P , we apply the Pohozaev identity. For any unit vector e in R2 , we have 2
λi
Br (P )
e, ∇Ki (x) evi (x) − 1 dx =
|x−P |=r
e, ∇vi
∂vi (e, ν) − |∇vi |2 dσ. ∂ν 2 (3.5)
Recall that λi = e−u(Pi ) → 0 and, by Theorem 2.2, λi Ki (x)evi (x) converges to 8π δ(P ), the delta function at P . Thus, the left-hand side of (3.5) converges to K(P )−1 e, ∇K(P ), and (3.5) implies ∂G ∂G (e, ν) − |∇G|2 dσ ∂ν ∂ν 2 |x−P |=r ∂G 2 1 dσ = 0, = e, ν 2 |x−P |=r ∂ν
8π K(P )−1 e, ∇K(P ) =
e,
where K(x) = lim Ki (x) = e2γ /(1+|x|
2)
i→+∞
because G(x) is symmetric with respect to P . Clearly, zero is the only critical point of K. Hence, P = 0. Now suppose Pi is the maximum point of ui and, by the previous result, Pi → 0 as i → +∞. To yield a contradiction, we want to prove for any a ∈ (0, 1), ui (x) ≥ u∗i (x; a) for |x| ≤ a,
(3.6)
where u∗i (x; a) = ui
a2x |x| − 2(2 + l . ) log i a |x|2
(3.7)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
517
By a straightforward computation, we have 4 2 a a x ∗ ui (x; a) = (ui ) |x| |x|2 4 l a a 4 i 2γ |x|2 /(|x|2 +a 4 ) u∗ |x| 2(2+li ) i =− 1+ 2 e e |x| a |x| li 2 |x| ∗ 2 2 4 2 =− + a e2γ |x| /(|x| +a ) eui . a2 By (3.2), u∗i (x; a) is smooth at x = 0. Thus, u∗i satisfies 2 li ∗ 2 2 4 Du∗ + a 2 + |x| e2γ |x| /(|x| +a ) eui = 0 i a2 ∗ ui (x; a) = ui (x)
for |x| ≤ a,
(3.8)
on |x| = a.
As before, we want to compare u∗i with ui ; that is, we claim wa (x) = ui (x) − u∗i (x; a) ≥ 0
for |x| ≤ a.
(3.9)
Since both li and γ are positive, we have l l |x|2 i 2γ |x|2 /(|x|2 +a 4 ) 2 2 e ≤ 1 + |x|2 i e2γ /(1+|x| ) a + 2 a for |x| ≤ a ≤ 1. Therefore, wa (x) satisfies wa (x) + Ca (x)wa (x) l |x|2 i 2γ |x|2 /(a 4 +|x|2 ) 2 2 li 2γ /(1+|x|2 ) u∗i (x;a) = a + 2 e − 1 + |x| e ≤0 e a (3.10) for |x| ≤ a ≤ 1, where
Ca (x) = 1 + |x|
2 li 2γ /(1+|x|2 ) e
∗ eui (x) − eui (x;a) . ui (x) − u∗i (x; a)
To prove (3.9) for a = 1, we suppose w1 (x) < 0 for some x in B1 , the unit ball. Let ' = {x ∈ B1 | w1 (x) < 0}. By (3.8), we have for x ∈ ', l ∗ 2 2 w1 (x) + 1 + |x|2 i e2γ |x| /(1+|x| ) eui (x) w1 (x) l ∗ 2 2 ≤ w1 (x) + 1 + |x|2 i e2γ |x| /(1+|x| ) eui (x) − eui (x) l 2 2 2 = 1 + |x|2 i e2γ |x| /(1+|x| ) − e2γ /(1+|x| ) eui (x) ≤ 0.
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CHANG-SHOU LIN
Therefore, the first eigenvalue of +(1+|x|2 )li e2γ |x| problem on ' is nonpositive. Let
2 /(1+|x|2 )
∗
eui (x) for the Dirichlet
l ∗ 2 2 q(x) = 1 + |x|2 i e2γ |x| /(1+|x| ) eui . Obviously, q(x) satisfies 8γ 1 − r 2 4li log q(x) = uˆ i + 2 + 3 ≥ −q(x) 1 + r2 1 + r2 for x ∈ B1 . By Lemma 2.3, we have l 2 1 + |x|2 i e2γ /(1+|x| ) eui dx = |x|≥1
≥
|x|≤1
'
l ∗ 2 2 1 + |x|2 i e2γ |x| /(1+|x| ) eui (x;1) dx
q(x) dx ≥ 4π.
On the other hand, since ui blows up at Pi , we have l 2 1 + |x|2 i e2γ /(1+|x| ) eui dx ≥ 8π(1 − ε) B1
for any ε > 0, provided that i is large. Clearly, it yields a contradiction to l 2 1 + |x|2 i e2γ /(1+|x| ) eui dx = 4π(2 + li ) −→ 8π R2
as i → +∞. Hence, w1 (x) ≥ 0 in B1 . By the strong maximum principle, we have w1 (x) > 0 in B1 . So we start the process of moving spheres at a = 1. Since wa is a supersolution of wa +Ca wa ≤ 0 in Ba , the maximum principle and the Hopf lemma are continuously applied to conclude wa (x) > 0 for 0 ≤ |x| ≤ a and for 0 < a ≤ 1. Since the argument is standard now, we skip the details of the proof. Thus, (3.9) is established. By differentiating wa at the boundary |x| = a, we have ∂ui 2(2 + li ) ∂wa (x) = 2 (x) + ≤0 ∂r ∂r r for |x| = r ≤ 1. But it yields a contradiction because (∂ui /∂r)(Pi ) = 0. Therefore, Lemma 3.1 is proved. Proof of Theorem 1.3. Let φi be a solution of (3.1) with limi→+∞ ρi = 8π. Since ρi > 8π, we know that n is the blow-up point of φi by Lemma 3.1. Without loss of generality, we assume n = (0, 0, −1). Let π be the stereographic projection of S 2
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
519
onto R2 such that the north pole is mapped to the infinity. Let ui be defined as in (3.2). Then ui (x) satisfies ui (x) = 0 on R 2 , ui (x) + Ki (x)e (3.11) Ki (x)eui (x) dx = 4π(2 + li ), R2
l 2 2 Ki (x) = 1 + |x|2 i e2γ |x| /(1+|x| ) , li =
ρi − 2. 4π
(3.12) (3.13)
Let Pi be the maximum point of ui and Pi → 0 by Lemma 3.1. Without loss of generality, we assume Pi = (ti , 0) with ti ≥ 0. We first claim ti = 0.
(3.14)
w0 (x) = ui (x) − ui (x − ) > 0
(3.15)
Suppose ti > 0. We want to prove
2 . Note that K (x) satisfies for x ∈ R+ i
Ki (x) ≥ Ki (x λ )
(3.16)
for x ∈ 2λ and λ ≥ 0, where 2λ and x λ are the same notation as in Section 2. Hence, the difference wλ (x) = ui (x) − ui (x λ ) satisfies λ λ wλ (x) + Ki (x) eui (x) − eui (x ) = Ki (x λ ) − Ki (x) eui (x ) ≤ 0
(3.17)
for x ∈ 2λ and 0 ≤ λ ≤ 1. Hence, once (3.15) is established, we start the process of moving planes to show that wλ (x) ≥ 0 for x ∈ 2λ and 0 ≤ λ ≤ 1, which obviously yields a contradiction. 2 | w (x) < 0} = ∅. Recall that w (x) satisfies Now suppose that ' = {x ∈ R+ 0 0 w0 (x) + c0 (x)w0 (x) = 0, where
(3.18)
−
eui (x) − eui (x ) c0 (x) = Ki (x) . ui (x) − ui (x − )
To yield a contradiction, we want to show that c0 (x) = o(1)|x|−2 for x ∈ ' and |x| ≤ 1. We prove (3.19) by considering three cases separately.
(3.19)
520
CHANG-SHOU LIN
Case 1. We have that eui (Pi ) |Pi |2 → +∞ as i → +∞. For x ∈ ' we have −
|c0 (x)| ≤ Ki (x)eui (x ) . Since eui (Pi ) |x − − Pi |2 ≥ eui (Pi ) |Pi |2 −→ ∞, by Theorem 2.2, we have ui (x − ) ≤ C − ui (Pi ) − 4 log |Pi − x − | ≤ C − ui (Pi ) + 2 log |Pi | − 2 log |x| because
max |Pi |, |x| ≤ |P − x − |.
Hence, eui (x
−)
−1 ≤ ec eui (Pi ) |Pi |2 |x|−2 = o(1)|x|−2 .
Case 2. We have that eui (Pi ) |Pi |2 ≥ c > 0. In this case, we consider vi (y) = ui e−ui (Pi )/2 y − ui (Pi ). 2 , where v is the solution of Then vi (y) converges to v0 (y − y0 ) for some y0 ∈ R+ 0 (2.36), and
y0 = lim eui (Pi )/2 Pi . i→+∞
2 and ∂v (y − y )/∂y | Since v0 (y − y0 ) ≥ v0 (y − − y0 ) for any y ∈ R+ 0 0 1 y1 =0 > 0, we have for any R > 0, i0 = i0 (R) such that if i ≥ i0 , then
vi (y) ≥ vi (y − ) 2 and |y| ≤ R. By scaling back to u , the above implies holds for y ∈ R+ i
ui (x) ≥ ui (x − ) for |x| ≤ eui (Pi )/2 R. In particular, if x ∈ ', then lim |x − − Pi |2 eui (Pi ) −→ +∞.
i→+∞
By Theorem 2.2, (3.19) follows readily.
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
521
Case 3. We have that limi→+∞ eui (Pi ) |Pi |2 = 0. In this case, we want to prove ui (x) ≥ ui (x − )
(3.20)
for |x| ≤ eui (Pi )/2 R and i ≥ i0 , where R is any large number and i0 = i0 (R). By using the same notions of Section 2, we define wˆ 0 and w˜ 0 as (2.49) and (2.51), respectively. Clearly, (3.20) is equivalent to proving that w˜ 0 (y) converges to w(y) ˜ and w(y) ˜ = c(∂v0 /∂y1 ) for some constant c = 0. The argument just follows from the one in case 2 in the proof of Theorem 1.2. But for our case, we only have a single blow-up point. For the sake of completeness, we sketch the proof. Let Ni = max |w0 (x)|. 2 R+
We want to prove |w0 (xˆi )| = max |w0 (x)| = o(1)Ni , c Bδ
(3.21)
0
2 and |x| ≥ δ }. By using the proof of (2.50), we know that if where Bδc0 = {x | x ∈ R+ 0
|w0 (xˆi )| ≥ c Ni
(3.22)
for some c > 0, then |xˆi | → +∞ as i → +∞. Using the Kelvin transformation as in (2.60), we let y −1 ∗ . wˆ 0 (y) = Ni w0 |y|2 Then wˆ 0∗ (y) satisfies wˆ 0∗ (y) + c0∗ (y)wˆ 0∗ (y) = 0
2 in R+ ,
2 . Since |c (x)| = o(1)|x|−4 for |x| ≥ 1, and wˆ 0∗ (y) is smooth up to the boundary of R+ 0 we have y −4 |y| = o(1) |c0∗ (y)| = c0 |y|2
for |y| ≤ 1. Set g(y) = y1α ,
w¯ 0∗ (y) =
wˆ 0∗ (y) g(y)
for 0 < α < 1. By a straightforward computation, w¯ 0∗ (y) satisfies w¯ 0∗ (y) + 2∇ log g · ∇ w¯ 0∗ + c0∗ (y) + α(α − 1)y1−2 w¯ 0∗ (y) = 0.
(3.23)
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CHANG-SHOU LIN
2 , and |y | ≤ 1/|xˆ | → 0 as By (3.22), |w¯ 0∗ (y)| has a maximum point at yi ∈ R+ i i i → +∞. Without loss of generality, we assume w¯ 0∗ (yi ) > 0. By the maximum principle, (3.23) yields
− 2 ∗ 0 ≥ w¯ 0∗ (yi ) = − c0∗ (yi ) + α(α − 1)(yi,1 ) w¯ 0 (yi ) > 0, a contradiction. Thus, (3.21) is established. Therefore, the maximum point xi of |w0 (x)| is near zero. Actually, by setting w˜ 0 (y) = wˆ 0 (e−ui (Pi )/2 y), we prove eui (Pi )/2 xi is bounded. For the details, see (2.54). By (2.55), w˜ 0 (y) converges to c(∂v0 /∂y1 ) for some c = 0. By (2.57) and (2.58), we conclude that c < 0. Thus, w0 (x) > 0 for |x| ≤ e−ui (Pi )/2 R and i ≥ i0 , where R is any large number and i0 = i0 (R). Readily, (3.19) follows. 2 and 0 < α < 1. Since the Kelvin transformaLet w¯ 0 (x) = w0 (x)x1−α for x1 ∈ R+ ∗ 2 tion w0 (y) = w0 (y/|y| ) is smooth at y = 0, |w0 (x)| ≤ c0 x1 holds for 0 ≤ x1 ≤ 1 and x2 ∈ R. Hence, w¯ 0 (x) tends to zero uniformly for x2 ∈ R as x1 → 0. Thus, the infinitum of w¯ 0 is achieved in '; that is, w¯ 0 (xˆi ) = inf w¯ 0 (x) < 0 '
for some xˆi . By a straightforward compution, w¯ 0 satisfies w¯ 0 + 2αx1−1
∂ w¯ 0 + c0 (x) + α(α − 1)x1−2 w¯ 0 = 0. ∂x1
(3.24)
By the maximum principle at xˆi , (3.24) yields 0 ≤ w¯ 0 (xˆi ) = − c0 (xˆi ) + α(α − 1)(xˆi,1 )−2 w¯ 0 (xˆi ) < 0 by (3.24), a contradiction. Therefore the claim (3.14) is proved. After (3.14) is established, the radial symmetry of ui can be proved by the same argument at the last stage of the proof of Theorem 1.2. Therefore, the radial symmetry of ui is proved and the uniqueness follows from Theorem 2.1(iii). Thus, Theorem 1.3 is completely proved. 4. Proof of Theorem 1.1. In this section, we apply previous symmetric results to compute the Leray-Schauder degree d(ρ) for 8π < ρ < 16π and 16π < ρ < 24π . Throughout this section, we always consider equation (1.1)ρ with f (y) ≡ exp(−γ n, y) for a fixed unit vector n, and we are only concerned with solutions that are axially symmetric with respect to n. Let π be the stereographic projection of S 2 onto R2 , which maps −n to infinity. For any axially symmetric solution φ of
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
523
(1.1)ρ and (1.2) for ρ > 8π , we set u by (3.2). Following conventional notation, we let u(r; s) be the unique solution of u + 1 u + 1 + r 2 l e2γ r 2 /(1+r 2 ) eu = 0, r (4.1) u(0; s) = s and u (0; s) = 0, where l > 0 and γ ≥ 0 are constants. To compute d(ρ), it is important to prove the nonsingularity of the linearized equation and to determine the number of negative eigenvalues. The linearized equation of (4.1) at u(r) is called nonsingular if the equation l 2 2 ϕ + 1 + r 2 e2γ r /(1+r ) eu ϕ = 0
in R2
(4.2)
2 2 possesses no bounded nontrivial solutions. For any C function ψ on S satisfying S 2 ψ(y) dµ = 0, we set φ(y) ψ(y) dµ −1 2 f (y)e ϕ(x) = ψ π (x) − S φ(y) dµ S 2 f (y)e
for x ∈ R2 , where f (y) = exp(γ n, y). Then ϕ(x) is a bounded nontrivial solution of (4.2) if and only if ψ(x) is a nontrivial solution of the linearized equation (1.1)ρ : f (y)eφ S 2 f (y)eφ ψ(y) dµ f (y)eφ ψ ψ + ρ − =0 (4.3) φ ( S 2 f (y)eφ dµ)2 S 2 f (y)e dµ on S 2 . Thus, the linearized equation of (4.1) at u is nonsingular if and only if the linearized equation of (1.1)ρ at φ is nonsingular. It is easy to see that u(r; s) exists for all r > 0 and always satisfies the asymptotic behavior u(r; s) = −β(s) log r + O(1)
at ∞
(4.4)
for some constant β(s), which continuously depends on the parameter s. Integrating (4.1), β(s) can be computed through the following formulas: l 2 2 (4.5) 1 + |x|2 e2γ |x| /(1+|x| ) eu(|x|;s) dx. 2πβ(s) = R2
Note that by (2.23), we seek solutions u(r; s) such that β(s) = 2(2 + l)
(4.6)
for some s. We recall results about radial solutions from [9] and [16]. First, [9, Theorem 1.1] states the following.
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CHANG-SHOU LIN
Lemma 4.1. Let u(r; s) be the solution of (4.1) and let β(s) be given by (4.5). Then (i) If 0 < l ≤ 1, then lims→+∞ β(s) = 4 and lims→−∞ β(s) = 2(2 + 2l). (ii) If l > 1, then lims→+∞ β(s) = 4l and lims→−∞ β(s) = 2(2 + 2l). Note that by Lemma 4.1, equation (4.1) always possesses a radial solution satisfying (4.6) for 0 < l < 2. By Theorem 2.1, this solution is unique for 0 ≤ γ ≤ 1 and 0 ≤ l ≤ 2. For γ = 0, by (2.20) and (2.23), the solution u0 (r; l) = −(2 + l) log(1 + r 2 ) + log 4(2 + l) of (4.1) corresponds to the trivial solution φ ≡ 0 on S 2 . Since the Laplacian of S 2 has eigenvalues k(k + 1) for a nonnegative integer k, by the remark above, the linearized equation (4.1) at u0 (r; l) is nonsingular for 0 < l ≤ 2. Thus, u0 (0; l) = log 4(2 + l) is not the minimum of β(s) for 0 < l ≤ 2 because if β(s0 ) achieves the minimum of β(s), then the linearized equation of (4.1) at u(r; s0 ) must be singular (see (4.10) below). Set β0 = min β(s). s∈R
Then we have β0 < 2(l + 2) for l = 2. By the continuity of β(s) on l, we have β0 < 2(l + 2)
(4.7)
for l close to 2. Since by Lemma 4.1, 2(2 + l) < lim β(s) < lim β(s) for l > 2, s→+∞
s→−∞
together with (4.7), we conclude that there are at least two solutions of β(s) = 2(2+l). Hence, (4.1) possesses at least two radial solutions satisfying (4.6) for γ = 0, l > 2, and l close to 2. Hence, we have proved the first part of the following result. Corollary 4.2. Fix a unit vector n in R3 . There exists a small δ0 > 0 such that for f (y) = 1 and 16π < ρ ≤ 16π +δ0 , (1.1)ρ possesses at least an axially symmetric nontrivial solution φ satisfying (1.2). Furthermore, any sequence of axially symmetric nontrivial solutions of (1.1)ρi and (1.2) with f (y) = 1 must blow up at ±n. Proof. The existence part has been proved already. Now suppose that φi is an axially symmetric nontrivial solution of (1.1)ρi and (1.2) with ρi > 16π and limi→+∞ ρi = 16π . By Theorem 2.2, either φi blows up at ±n simultaneously or φi is uniformly bounded for all i. Assume that the latter case occurs. By passing to a subsequence, φi converges to φ in C 2 (S 2 ), where φ satisfies eφ 1 =0 − φ + 16π φ 4π S 2 e dµ φ dµ = 0. S2
on S 2 , (4.8)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
525
By Theorem 2.1, we have φ ≡ 0. Since φi accumulates at φ ≡ 0, the linearized equation of (4.8) at φ = 0 must be singular; that is, there is a nontrivial solution w of 2 w + 4w = 0 on S , w dµ = 0. S2
This yields a contradiction to the fact that 4 is not an eigenvalue of the Laplacian on S 2 . Therefore, φi must blow up at ±n. Lemma 4.3. There exists a small 0 < δ1 ≤ δ0 such that axially symmetric nontrivial solutions to (1.1)ρ and (1.2) with f = 1 are unique for 16π < ρ ≤ 16π + δ1 . The proof of Lemma 4.3 is long and requires a delicate application of the Pohozaev identity. Let u(r; s) be the unique solution of (4.1) and let ϕ(r; s) = (∂u/∂s)(r; s). Then ϕ satisfies the linearized equation at u, ϕ + 1 ϕ + 1 + r 2 l e2γ r 2 /(1+r 2 ) eu ϕ = 0, r (4.9) ϕ(0) = 1, ϕ (0) = 0. By an elementary argument, we prove that for any fixed s, either limr→+∞ (ϕ(r)/log r) = 0 or ϕ(r) is uniformly bounded for r ∈ [0, ∞). Obviously, the latter case is equivalent to limr→+∞ ϕ (r)r = 0. Integrating (4.9), we have by (4.5), ∞ l 2 2 ˙ =− − β(s) 1 + r 2 e2γ r /(1+r ) eu(r;s) ϕ(r; s)r dr = lim rϕ (r; s), (4.10) 0
r→+∞
˙ ˙ where β(s) = dβ/ds. Thus, β(s) = 0 for some s if and only if ϕ(r; s) is uniformly bounded in [0, ∞). The behavior of ϕ(r; s) is described in [16, Lemma 3.1 and Theorem 1.5], which is stated as (i) in the following lemma. Note that (ii) of the lemma is equivalent to (iii) of Theorem 2.1. (See [16, Section 4] for a proof.) Lemma 4.4. (i) For γ = 0 and l > 0, ϕ(r; s) has at least two zeros for all s. Moreover, ϕ(r; s) has exactly two zeros and limr→+∞ ϕ(r; s) = +∞ whenever 4l ≤ β(s) < 4(1 + l). (ii) For 0 < γ ≤ 1 and 0 < l ≤ 2, ϕ(r; s) has exactly two zeros and limr→+∞ ϕ(r; s) = +∞ whenever β(s) = 2(2 + l). From now until the end of the proof of Lemma 4.6, we always assume γ = 0. In this case, we claim There exists a δ1 > 0 such that if 2 < l ≤ 2 + δ1 , s ≥ δ1−1 and ˙ > 0. β(s) ≥ 4 + 2l, then β(s)
(4.11)
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CHANG-SHOU LIN
Proof of Lemma 4.3. We first prove that There exists a small δ1 > 0 such that if 2 < l ≤ 2 + δ1 , s ≥ δ1−1 and ˙ = 0. β(s) ≥ 4 + 2l, then β(s)
(4.12)
Suppose that (4.12) does not hold. Then there is a sequence of solutions ui (r) = u(r; si ) of (4.1) with γ = 0, li → 2 and si → +∞ such that ϕi (r) ≡ ϕ(r; si ) is bounded in [0, ∞) for each i, and β(si ) ≥ 2(2+li ). For simplicity, we let βi = β(si ). By Lemma 4.4(i), ϕi must change sign at least twice. Let ri denote the first zero of ϕi (r). By scaling, we let ϕˆi (r) = ϕi (e−ui (0)/2 r). Then ϕˆi satisfies ϕˆ + 1 ϕˆ + 1 + e−ui (0) r 2 li evi (r) ϕi = 0, i i r ϕˆ (0) = 1 and ϕˆ (0) = 0, i
i
where vi (r) = ui (e−ui (0)/2 r) − ui (0). Since vi (r) converges to v0 (r) ≡ −2 log(1 + 2 , ϕˆ (r) converges to ϕˆ (r) ≡ (8 − r 2 )/(8 + r 2 ) in C 2 , where ϕˆ (r) is r 2 /8) in Cloc i 0 0 loc the solution of ϕˆ (r) + 1 ϕˆ (r) + ev0 (r) ϕˆ0 (r) = 0, 0 r 0 (4.13) ϕˆ (0) = 1 and ϕˆ (0) = 0. 0
0
Since ϕˆ0 (r) < 0 for 0 < r < +∞, we have eui (0) ri2 −→ +∞
(4.14)
as i → +∞. To yield a contradiction, we want to prove ri −→ 0
(4.15)
as i → ∞. We prove (4.15) by using the Pohozaev identity. Suppose ri ≥ δ0 > 0 for some δ0 > 0. Set ru i rui + 4 + r 2 Ki (r)eui (r) , 2 P˜i (r) = rϕi ru i + 2 + r 2 Ki (r)eui (r) ϕi ,
Pi (r) =
where Ki (r) = (1 + r 2 )li . By a straightforward computation, Pi (r) and P˜i (r) satisfy r tKi (t)eui (t) t dt > 0 Pi (r) = 0
(4.16) (4.17)
(4.18)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
527
for r > 0, and P˜i (r) =
r 0
Pi (t)ϕi (t) dt
= Pi (r)ϕi (r) −
r 0
Pi (t)ϕi (t) dt.
(4.19)
Let r = ri . Then by (4.19), ri u (ri ) ri ui + 4 ϕi (ri ) = − − i 2
ri 0
Pi (t)ϕi (t) dt > 0.
(4.20)
Thus, ri u i +4 < 0. By scaling, ϕi (e−ui (0) r) converges to (8 − r 2 )/(8 + r 2 ) uniformly in any compact set of [0, ∞) as i → +∞. From here, it is easy to see ϕi (ri ) → −c < 0 for some c > 0 (actually, it can be proved that c = 1, but this is not important in the following argument) and ri u i (ri ) → −4 as i → ∞. We want to prove ri u (ri ) + 4 ≤ c0 e−ui (0) i
(4.21)
for some constant c0 > 0. To see it, by (4.18), we have r Pi e−ui (0)/2 s = tKi (t)eui (t) t dt 0
= 2li
l −1 t 2 1 + t 2 i eui (t) t dt
0
= 2li e where
r
−ui (0)
s
(4.22)
l −1 τ 2 1 + e−ui (0) τ 2 i evi (τ ) τ dτ,
0
r = e−ui (0)/2 s, vi (τ ) = ui e−ui (0)/2 τ − ui (0).
Note that vi (τ ) → log 1/(1 + (τ 2 /8)2 ). Thus, eui (0) Pi (e−ui (0)/2 s) converges to 2li
s 0
τ3
2 ds.
1 + τ 2 /8
Also, if we set ϕˆi (τ ) = ϕi (e−ui (0) τ ), then ϕi (e−ui (0) r)e−ui (0) = ϕˆi (τ ) → −32τ /(8 + τ 2 )2 . Hence, lim e
i→∞
ui (0)
e−ui (0) R 0
R
= −128 0
Pi (t)ϕi (t) dt
t (8 + t 2 )2
t 0
s3 ds dt. (1 + s 2 /8)2
(4.23)
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CHANG-SHOU LIN
Let Ri → +∞ such that both vi (s) and ϕˆi (s) uniformly converge for s ≤ Ri . Set ti = e−ui (0)/2 Ri . Thus, for all t ≥ ti , we have eui (0)/2 t → +∞. By Theorem 2.2, eui (t) = O(1)t −4 e−ui (0) for t ≥ ti . Thus, for ri ≥ t ≥ ti , we have ri ri ui (s) tϕ (t) ≤ Ki (s)e s|ϕi (s)| ds ≤ Ki (s)eui (s) s ds i t
≤ c e−ui (0)
t
ri t
(4.24)
s −3 ds = O(1)e−ui (0) t −2 ,
where |ϕi (s)| ≤ 1 are used. Since ri ≥ δ0 > 0 for some δ0 > 0, we have Pi (t) ≤ Pi (ri ) ≤ cri u (ri ) + 4 + O e−ui (0) for 0 ≤ t ≤ ri , where the last inequality is due to (4.16) and ri ≥ δ0 . Hence, by (4.23), (4.24), and (4.20) gives ti ri ri u (ri ) + 4 ≤ c Pi (s) ϕi (s) ds + |Pi (s)| ϕi (s) ds
ti
0
≤ c1 e ≤ c1 e
−ui (0)
−ui (0)
+ ri u (ri ) + 4
ri ti
e
−ui (0) −3
s
ds
−u (0) −2 i + ri u (ri ) + 4 e ti .
Because e−ui (0) ti−2 → 0, the inequality (4.21) follows from the above. By the expression of Pi and ri ≥ δ0 > 0, (4.21) gives Pi (ri ) = O e−ui (0) . On the other hand, since Pi (t) is increasing in t and by (4.22), Pi (ti )eui (0) −→ +∞
as i −→ +∞,
(4.26) obviously yields a contradiction. Thus, we have proved ri → 0. Let u∗i (r) = u(1/r) − βi log r and ϕi∗ (r) = ϕi (1/r). Then u∗i and ϕi∗ satisfy u∗ + and
u∗ ∗ + Ki∗ (r)eui = 0, r
ϕ∗ ∗ ϕi∗ + i + Ki∗ (r)eui (r) ϕi∗ (r) = 0, r ϕ ∗ (0) = lim ϕi and ϕ ∗ (0) = 0, i
(4.25)
r→+∞
i
(4.26)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
529
where Ki∗ (r) = (1 + r 2 )li r βi −2(2+li ) . Note limi→+∞ βi − 2(2 + li ) = 0. Since
1 0
∗ Ki∗ (r)eui (r) r dr
∞
= 1
Ki (r)eui (r) r dr −→ 4
as i → +∞, u∗i (0) must tend to +∞. Let ri∗ be the first zero of ϕi∗ (r). By an argument similar to that for (4.15), we can prove ri∗ → 0. Hence, ϕi (r) has a zero at (ri∗ )−1 that tends to +∞ as i → +∞. By (4.15), ϕi has a second zero at rˆi . Let ξi denote the zero of ϕi in the interval (ri , rˆi ). Without loss of generality, we may assume ξi is bounded. Otherwise, we may apply the Kelvin transformation and follow the same argument to yield a contradiction. By Theorem 2.2, we have eui (r) ∼ r −4 e−ui (0) for r ∈ [ti , ξi ]. Thus, ri2 Ki (ri )eui (ri ) ≥ c0
e−ui (0) . ri2
(4.27)
We claim eui (0) ri4 ≥ c1
(4.28)
for some positive constant c1 independent of i. To see it, we use the general form of the Pohozaev identity. For any λ > 0, we set Pλ (r; ui ) =
ru i (ru i + λ) + r 2 Ki (r)eui (r) . 2
(4.29)
By a straightforward computation, Pλ satisfies d Pλ (r; ui ) = rKi (r) + (4 − λ)Ki (r) reui (r) . dr
(4.30)
For any δ > 0, we let λ = 4 + δ(e−ui (0) /ri2 ). Thus, d Pλ (r; ui ) ≤ 0 dr
for r ≤ c2 (δ)
e−ui (0)/2 . ri
(4.31)
If eui (0) ri4 → 0 as i → ∞, we have ri ≤ o(1)
e−ui (0)/2 e−ui (0)/2 ≤ c2 (δ) . ri ri
Hence, Pλ (r; ui ) is decreasing in r for r ∈ [0, ri ]. In particular, Pλ (ri ; ui ) < 0.
(4.32)
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CHANG-SHOU LIN
Recall ri u i (ri ) + 4 < 0 by (4.20). By (4.32), we have ri u i (ri ) ri u i (ri ) + 4 ≤ 0. = Pλ (ri ; ui ) − 2 2
ri u (ri ) e−ui (0) δ ri2 K(ri )eui (ri ) + i 2 ri2 Thus,
ri2 K(ri )eui (ri ) = o(1)
e−ui (0) , ri2
a contradiction to (4.27). Hence, the claim (4.28) is proved. Now, by equation (4.9), r r rϕi ≤ Ki (t)eui (t) |ϕi (t)|t dt ≤ Ki (t)eui (t) t dt ri
≤ c e−ui (0) Hence,
ri
r ri
t −3 dt ≤ c e−ui (0) ri−2 .
ϕi (r) ≤ c e−ui (0) ri−3
for ri ≤ r ≤ ξi .
By (4.28), we have 1 ∼ −ϕi (ri ) = ϕi (ξi ) − ϕi (ri ) =
ξi ri
ϕi (s) ds ≤ c e−ui (0) ri−3 ≤ c1 ri −→ 0,
which is obviously impossible. Therefore, (4.12) is proved. Suppose that (4.11) is false; that is, β (s) < 0 for large s by (4.12). Then β(s) > 4l for 2 < l ≤ 2 + δ1 and for large s. On the other hand, Lemma 4.4 states that ˙ 2, which yields a contradiction. Therefore, (4.11) is established. Clearly, uniqueness follows immediately from Corollary 4.2 and (4.11). Lemma 4.5. let ϕ(r; s) be the solution of (4.9) with 2 < l ≤ 2 + δ1 and β(s) = 2(2 + l). Then ϕ(r; s) has exactly three zeros and limr→+∞ ϕ(r; s) = −∞. Proof. By (4.10) and (4.11), we have limr→+∞ rϕ (r) < 0. Hence, limr→+∞ ϕ (r; s) = −∞. Since ϕ(r; s) is known to have two zeros at least, it implies ϕ has three zeros at least. Let τ1 < τ2 < τ3 be the first three zeros of ϕ. If δ1 tends to zero, the second zero τ2 must tend to +∞. This can be proved by the same argument as in the proof of Lemma 4.3, where a contradiction results when ξi∗ is assumed to be bounded. Hence, we may assume τ2 ≥ 1. Let u(r) ˆ = u(1/r; s)−2(2 +l) log r, the Kelvin transformation of u. By a straightforward computation, uˆ satisfies the same equation as u does. Since u(r) ˆ = −2(2 + l) log r +O(1) at ∞, u(r) ˆ ≡ u(r) follows readily from the uniqueness of Lemma 4.3. It implies that ϕ(r) ˆ = ϕ(1/r; s) satisfies (4.9), but not the initial conditions. Hence, ϕ are ϕˆ are linearly independent. By the Liouville theorem, between any two zeros of ϕ(r) ˆ for 0 ≤ r ≤ 1, there must be a zero of ϕ(r; s). Since ϕ(r; s) has only one zero in [0,1], ϕ(r) ˆ has at most two zeros in [0,1]. Therefore, the proof of Lemma 4.5 is complete. Let φ(y; n) denote the unique solution of (1.1)ρ with f ≡ 1 in Lemma 4.3. Since φ(y; −n) is also a solution that is axially symmetric with respect to n, φ(y; −n) ≡ φ(y; n) on S 2 by Lemma 4.3. Clearly, equation (1.1)ρ with f ≡ 1 is invariant under the action of O(3), the group of orthorgonal transformations in R3 . Set O(φ) ≡ {φ(y; n) | |n| = 1} as the orbit of φ. By the remark above, O(φ) is homeomorphic to RP 2 , the real projective space of two dimensions. The orbit O(φ) is called nondegenerate if the null space of the linearized equation (4.3) at φ is equal to the tangent space of O(φ). To analyze (4.3), we may assume n = (0, 0, 1) for simplicity. Let π be the standard stereographic projection of S 2 onto R2 , and let u(r) be the corresponding solution in R2 . To prove nondegeneracy is equivalent to proving that equation (4.2) possesses exactly two independent bounded solutions because O(φ) is a two-dimensional manifold. By separation of variables, it
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is equivalent to finding bounded solutions of ϕk (r)+
l ϕk k 2 ϕk − 2 + 1+r 2 eu(r) ϕk (r) = 0 r r
for r ∈ [0, ∞). By a straightforward computation, we see that 1 + r2 2+l + r w(r) ≡ u (r) 4 2
(4.36)k
(4.37)
always satisfies (4.36)k for k = 1, and w(r) is uniformly bounded in [0, ∞) because ru (r) → −2(2 + l) as r → +∞. Hence, w(r) w(r) x1 and x2 are null vectors of r r the linearized equation (4.3) at φ.
(4.38)
To count the number of negative eigenvalues of (4.3), we also count the number of negative eigenvalues of (4.36)k for each integer k. The summation of the number of negative eigenvalues of (4.36)k over k = 1, 2, . . . , is equal to the total number of negative eigenvalues of (4.3). By Lemma 4.5, ϕ(r; s) has exactly three zeros and limr→+∞ ϕ(r; s) = −∞, which gives us that in the class of axially symmetric functions on S 2 , there are exactly three negative eigenvalues. In the following, we should prove that w(r) of (4.37) changes sign only once. Thus, in the class of {ψ(z)x1 , ψ(z)x2 | ψ(z) ∈ C 2 (S 2 )}, there are only two negative eigenvalues, where S 2 = {(x1 , x2 , z) | x12 + x22 + z2 = 1}. Therefore, the following result is expected to hold. Lemma 4.6. let φ(y; n) be the solution of (1.1)ρ and (1.2) in Lemma 4.3. Then the orbit O(φ) is nondegenerate and the linearized equation (4.3) at φ has a twodimensional null space and five negative eigenvalues, that is, λ5 < 0 = λ6 = λ7 < λ8 . Proof. By Lemma 4.5 and the remark above, we have to prove (i) w(r) of (4.37) changes sign only once; and (ii) for each k ≥ 2 (4.36)k possesses no bounded solutions in [0, ∞), and any solution ϕk satisfying ϕk (0) = ϕk (0) = 0 has no zero in [0, ∞). We note that the nondegeneracy of the orbit follows from Lemma 4.5, (4.38), and the first part of (ii). The number of negative eigenvalues comes from the remark above and from the second part of (ii). To prove (i), we note that w(0) = 0 and w (0) = u (0) + (2 + l)/2 = −eu(0) /2 + (2 + l)/2 < 0 if l is close to 2. Thus, w(r) < 0 for r small. By using the identity u(r) ≡ u(1/r)−2(2+l) log r in [0, ∞), we have w(1/r) = −w(r). Hence, w(r) > 0 for large r. Now suppose that w(r) has more than one zero. Then w(r) must have three zeros at least. Let r1 , r3 be the first and the last zero of w(r), and let r1 < r2 < r3 be another zero. Since w(r) is bounded near ∞, w(r) and ϕ(r; s) of Lemma 4.5 are linearly independent. Thus, by the Sturm-Liouville comparison theorem, ϕ(r; s)
TOPOLOGICAL DEGREE FOR MEAN FIELD EQUATIONS ON S 2
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must have a zero in intervals [0, r1 ), (r1 , r2 ), (r2 , r3 ) and (r3 , ∞). But this yields a contradiction to Lemma 4.5, which states that ϕ has exactly three zeros. Hence, (i) is proved. To prove (ii), we note that for k ≥ 2, the inequality l 2 1 + r 2 eu(r) ≤ 2 1 + o(1) r
(4.39)
holds for 0 ≤ r < +∞, where o(1) → 0 as l → 2. The inequality (4.39) can be checked for 0 ≤ r ≤ e−u(0)/2 R for large R because eu(r) = (1 + o(1))eu(0) (1 + eu(0) r 2 /8)−2 for 0 ≤ r ≤ e−u(0)/2 R, where o(1) → 0 as R → +∞. For e−u(0)/2 R ≤ r ≤ 1, eu(r) = O(1)e−u(0) r −4 = o(1)r −2 by Theorem 2.2. Hence, (4.39) obviously holds. For r ≥ 1, (4.39) follows by the identity u(r) ≡ u(1/r) − 2(2 + l) log r. Note that by (4.39), the coefficient of zero order of (4.36)k is negative for k ≥ 2. Then, the maximum principle, it implies any solution ϕk of (4.36)k never changes sign and |ϕk (x)| → +∞ as |x| → +∞. Thus, Lemma 4.6 is proved completely. Next we consider the case 0 < γ ≤ 1 and 8π < ρ ≤ 8π + ρ0 (γ ). By Theorem 1.3, solutions of (4.1) with β = 2(2 + l) are unique. By Lemma 4.4(ii), the solution ϕ of (4.9) changes sign exactly twice. As in Lemma 4.6, we compute the number of negative eigenvalues of the linearized equation of (4.1) at φ with f (y) = exp(−γ n, y) through counting the number of zeros of ϕk + where K(r) = (1 + r 2 )l e2γ r
ϕk k 2 ϕk − 2 + K(r)eu(r) ϕk = 0, r r
2 /(1+r 2 )
((4.40)k )
.
Lemma 4.7. For any γ > 0, there exists a small ρ0 (γ ) > 0 such that the linearized equation (1.1)ρ with f (y) = exp(γ n.y) and 8π ≤ ρ ≤ 8π + ρ0 (γ ) is nondegenerate. Moreover, λ4 < 0 < λ5 , where λi is the ith eigenvalue of the linearized equation (4.3). Proof. We want to prove that (i) the unique solution ϕ1 of (4.40)1 satisfying ϕ1 (0) = 0 and ϕ1 (0) = 1 has exactly one zero and limr→+∞ ϕ1 (r) = −∞, and (ii) for any integer k ≥ 2, any solution ϕk (r) of (4.40)k with ϕk (0) = ϕk (0) = 0 has no zeros in (0, ∞) and |ϕk (r)| → +∞ as r → +∞. Clearly, the nonsingularity of the linearized equation follows from Lemma 4.4(ii) and the second part of (i) and (ii) above. The number of negative eigenvalues follows from Lemma 4.4(ii) and the first part of (i) and (ii) above. Since ϕ1 satisfies ϕi (0) = 0 and ϕ1 (0) = 1, ϕ1 (r) > 0 for small r. Now suppose ϕ1 (r) > 0 for all r > 0. Recall that the derivative u (r) satisfies u −
u (r) + K(r)eu u (r) + K (r)eu = 0. r2
(4.41)
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By (4.40)1 and (4.41), we have ϕ1 (r)u
(r)r − ϕ1 (r)u (r)r
r
ϕ1 u − u ϕ1 t dt
= 0
r
=−
(4.42)
ϕ1 K (t)e
u(t)
t dt < 0,
0
because both K (t) and ϕ1 (t) are positive. If ϕ1 (r) tends to a constant as r → ∞, then rϕ1 (r) → 0 as r → +∞. In this case, the left-hand side of (4.42) tends to zero as r → +∞, which yields a contradiction. If ϕ1 (r) is unbounded as r → +∞, then ϕ1 (r) > 0 for large r. Since u (r) > 0 for large r, the left-hand side of (4.42) is positive for large r, which again yields a contradiction. Hence, ϕ1 (r) must change sign at least once. Now suppose that either ϕ1 (r) has two zeros for r ∈ (0, ∞) or ϕ1 (r) has one zero and ϕ1 (r) remains bounded as r → +∞. In the latter case, we can show that limr→+∞ ϕ1 (r) = 0. Thus, we can assume that ϕ1 (r) has two zeros ri < si for a sequence of li → 0. (Note that si could be +∞.) By Lemma 3.1, ui (x) blows up at the origin. Since eui (0) ri2 → +∞ by a scaling argument, we have by Theorem 2.2,
l o(1) 2 2 1 + r 2 i e2γ r /(1+r ) eui (r) = 2 r
(4.43)
for r ≥ ri , where o(1) → 0 as i → +∞. Since ϕ1 has two zeros at ri and si , ϕ1 must have a local minimum ti ∈ (ri , si ); that is, ϕ1 (ti ) = min ϕ1 (r) < 0. ri ≤r≤si
Applying the maximum principle to (4.40)1 , we have
1 2 li 2γ ti2 /(1+ti2 ) ui (ti ) 0 ≤ ϕ1 (ti ) = 2 − 1 + ti e ϕ(ti ) < 0 e ti by (4.43), which yields a contradiction. Thus, ϕ1 (r) changes sign only once and ϕ1 (r) → −∞ as r → +∞.This proves (i). For the case k ≥ 2, any nontrivial solution ϕk (r) of (4.40)k for k ≥ 2, satisfying ϕk (0) = ϕk (0) = 0, must not change sign, and |ϕk (r)| → +∞ as r → +∞. If ϕk (ri ) = 0 and ri is the first zero of ϕi in (0, ∞], then (4.43) holds also, and by appling the maximum principle as above, (4.40)k yields a contradiction. Thus, ϕk (r) does not change sign. This finishes the proof of Lemma 4.7. To finish the proof of Theorem 1.1, we need a result of Wang in [23], which tells us how to compute the local degree due to a nondegenerate orbit. Let φ be a nontrivial solution of (1.1)ρ with f ≡ 1. Recall O(φ) is the orbit of φ.
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Lemma 4.8. let O(φ) be a nondegenerate orbit. Then the local degree contributed by O(φ) is equal to (−1)m χ (O(φ)), where χ(O(φ)) is the Euler characteristic, and m is the number of negative eigenvalues of the linearized equation. For a proof, we refer the readers to [23]. Now we are in the position to complete the proof of Theorem 1.1. Proof of Theorem 1.1. First we compute d(ρ) for 8π < ρ < 16π for equation (1.1)ρ . Since d(ρ) is independent of the function f , we choose f (y) = e−γ n,y for some direction n and 1 ≥ γ > 0. By Theorem 1.2 and Lemma 4.7, we know that if ρ is close to 8π , then (1.1)ρ possesses a unique solution φρ . Since the linearized equation of φρ is nonsingular, d(ρ) = (−1)m , where m is the number of negative eigenvalues of the linearized operator. Note that φρ is always normalized to satisfy (1.2). Then m = 4 − 1 = 3 by Lemma 4.7. Thus, d(ρ) = −1 for 8π < ρ < 16π. For 16π < ρ < 24π , we consider f (y) ≡ 1 on S 2 . Let ρ ≥ 16π be close to 16π. By Theorem 1.2 and Lemma 4.3, we know that for any two nontrivial solutions φ1 and φ2 of (1.1)ρ and (1.2), there exists a g ∈ O(3) such that φ1 (y) ≡ φ2 (gy) on S 2 . By Lemma 4.3, we have that the orbit O(φ) = {φg | φ is a nontrivial solution of (2.6)} is nondegenerate. Note that O(φ) is homeomorphic to RP 2 . Thus, χ(O(φ)) = 1. By Wang’s theorem and Lemma 4.6, we know the contribution of the degree due to O(φ) is equal to (−1)4 χ (O(φ)) = 1. Thus, d(ρ) = −1 + 1 = 0, where −1 is the local degree due to the trivial solution. Hence, Theorem 1.1 is proved completely. Acknowledgments. I wish to thank Professor Louis Nirenberg for showing me the result of Wang in [23], and the referees for their helpful comments. References [1] [2]
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C. Bandle, Isoperimetric Inequalities and Applications, Monogr. Stud. Math. 7, Pitman, Boston, 1980. E. Caglioti, P.-L. Lions, C. Marchioro, and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys. 143 (1992), 501–525. , A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, II, Comm. Math. Phys. 174 (1995), 229–260. S.-Y. A. Chang, M. J. Gursky, and P. C. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. Partial Differential Equations 1 (1993), 205–229. S. Chanillo and M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry, Comm. Math. Phys. 160 (1994), 217–238. W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615–622.
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CHANG-SHOU LIN C.-C. Chen and C.-S. Lin, Estimate of the conformal scalar curvature equation via the method of moving planes, II, J. Differential Geom. 49 (1998), 115–178. , Singular limits of a nonlinear eigenvalue problem in two dimension, preprint. K.-S. Cheng and C.-S. Lin, On the conformal Gaussian curvature equation in R2 , J. Differential Equations 146 (1998), 226–250. W. Ding, J. Jost, J. Li, and G. Wang, The differential equation u = 8π − 8πheu on a compact Riemann surface, Asian J. Math. 1 (1997), 230–248. , Existence results for mean field equations, preprint. B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry of positive solutions of nonlinear elliptic equations in Rn ” in Mathematical Analysis and Applications, Part A, Adv. Math. Supp. Stud. 7a, Academic Press, New York, 1981, 369–402. M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math. 46 (1993), 27–56. Y. Y. Li, Harnack type inequality: The method of moving planes, Comm. Math. Phys. 200 (1999), 421–444. Y. Y. Li and I. Shafrir, Blow-up analysis for solutions of −u = V eu in dimension two, Indiana Univ. Math. J. 43 (1994), 1255–1270. C. S. Lin, Uniqueness of conformal metrics with prescribed total curvature in R2 , to appear in Calc. Var. Partial Differential Equations. , Uniqueness of solutions of the mean field equation on S 2 , to appear in Arch. Rational Mech. Anal. M. Nolasco and G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds, Arch. Rational Mech. Anal. 145 (1998), 161–195. L. M. Polvani and D. G. Dritschel, Wave and vortex dynamics on the surface of a sphere, J. Fluid Mech. 255 (1993), 35–64. J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304 –318. M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), 109–121. G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys. 37 (1996), 3769–3796. Z. Q. Wang, Symmetries and the calculations of degree, Chinese Ann. Math. Ser B. 10 (1989), 520–536.
Department of Mathematics, Chung-Cheng University, Minghsiung, Chia-Yi, Taiwan
