ON THE MODULARITY OF ENDOMORPHISM ALGEBRAS

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1 ON THE MODULARITY OF ENDOMORPHISM ALGEBRAS FRANÇOIS BRUNAULT Abstract. We show that any homomorphism between Jacobians of modular curves arises from a linear combination of Hecke modular correspondences. The proof uses the adelic language and is based on a study of the actions of GL 2 and Galois on the étale cohomology of the tower of modular curves. We also make this result explicit for Ribet s twisting operators on modular abelian varieties. Résumé. Nous montrons que tout homomorphisme entre jacobiennes de courbes modulaires provient d une combinaison linéaire de correspondences modulaires de Hecke. La preuve utilise le langage adélique et se base sur une étude des actions de GL 2 et de Galois sur la cohomologie étale de la tour des courbes modulaires. Nous rendons également explicite ce résultat pour les endomorphismes des variétés abéliennes modulaires définis par Ribet. It is natural to ask whether the endomorphism algebra of the Jacobian of a modular curve is generated by the Hecke operators. Ribet showed in [6] that if N is prime, the algebra (End J 0 (N)) Q is generated by the Hecke operators T n with n prime to N, answering positively a question of Shimura. Mazur [5] subsequently showed an integral refinement of Ribet s result, namely that the algebra End J 0 (N) is generated by the Hecke operators T p with p prime, p N, and by the Atkin-Lehner involution w N. For general N, the obvious generalization of Ribet s result does not hold, since the Hecke operators generate a commutative subalgebra, while End J 0 (N) is not commutative in general. The reason behind non-commutativity is the existence of certain degeneracy operators, giving rise to old modular forms (which do not exist in the case of weight 2 and prime level). For arbitrary N, Kani [3] showed that if Γ is a congruence subgroup intermediate between Γ 1 (N) and Γ 0 (N), and J Γ is the Jacobian of the modular curve associated to Γ, then the algebra End(J Γ ) Q is generated by the Hecke operators together with explicit degeneracy operators. The purpose of this note is to develop an alternative approach to these questions using the adelic language. We show that after tensoring with Q, any homomorphism between Jacobians of modular curves arises from a linear combination of Hecke modular correspondences. The cost of our abstract approach is that our results are less explicit in nature: the Hecke double coset algebra is a complicated object whose structure is not known in general. On the other hand, our results are stronger in that we consider homomorphisms instead of endomorphisms (Theorem 1), and in that we allow for homomorphisms defined over abelian number fields (Theorem 2). I hope to convince the reader that the adelic language provides a convenient point of view for studying these questions. Ribet showed in [7] that the endomorphism algebra of a modular abelian variety A f is generated over the Hecke field of f by a finite set of endomorphisms coming from the inner twists of f. In the last section of this paper, we explain how to write these endomorphisms in terms of Hecke correspondences, giving thus some substance to Theorem 2. Date: December 19, Mathematics Subject Classification. Primary 11F41; Secondary 11F25, 11F70, 11F80, 14G32. Key words and phrases. Modular curves, Hecke correspondences, endomorphism algebras, automorphic representations, Galois representations. 1

2 2 F. BRUNAULT I would like to thank Gabriel Dospinescu for pointing out to me the relevance of Burnside s theorem, Filippo A. E. Nuccio and Vincent Pilloni for interesting discussions related to this paper, and Eknath Ghate for useful advice. 1. Statement of the main result Let A f denote the ring of finite adèles of Q, and let G = GL 2 (A f ). For any compact open subgroup K of G, let M K denote the open modular curve over Q associated to K, and let M K denote the compactification of M K. If M K is geometrically connected, we denote by J K the Jacobian variety of M K. Let K, K be compact open subgroups of G. We denote by T K,K = Z[K/G/K ] the free abelian group on K/G/K. In the case M K and M K are geometrically connected, we have a canonical map ρ J T K,K Hom Q (J K, J K ) (see Section 2). Theorem 1. Let K, K be compact open subgroups of G such that the modular curves M K and M K are geometrically connected. Then ρ J ( T K,K ) Q = Hom Q (J K, J K ) Q. Remark 1. According to the Langlands philosophy, the Galois representations associated to algebraic varieties are expected to be automorphic. In fact, this conjectural correspondence should be functorial: not only the Galois representations, but also the morphisms between them should have an automorphic explanation. Theorem 1 can be seen as a very simple case of this principle. Remark 2. Let J be the Jacobian of a smooth projective curve X. It is known that every endomorphism of J arises from an effective linear combination of correspondences on X. In the case X is a modular curve, a general correspondence on X could arise from a cover associated to a noncongruence subgroup. Our result says that congruences subgroups are enough to generate the endomorphism algebra. 2. Modular curves in the adelic setting Let K be a compact open subgroup of G = GL 2 (A f ). The complex points of the modular curve M K are given by M K (C) = GL + 2(Q)/(H G)/K where GL + 2(Q) acts on H G by γ (τ, g) = (γ(τ), γg), and K acts on G by right multiplication. The set of connected components of M K (C) is in bijection with Ẑ / det(k). More precisely, let χ Gal(Q ab /Q) Ẑ denote the cyclotomic character, and let F be the finite abelian extension of Q associated to χ 1 (det(k)). Then the structural morphism M K Spec Q factors through Spec F, and the curve M K over Spec F is geometrically connected. We refer to F as the base field of M K. Let K, K be compact open subgroups of G, and let g G. We define a correspondence T (g) between M K and M K by the following diagram M K gk g 1 α α T (g) M K M K given on the complex points by α([τ, h]) = [τ, h] and α ([τ, h]) = [τ, hg]. The correspondence T (g) extends to the compactifications and induces a map T (g) = α α Ω 1 (M K ) Ω 1 (M K ).

3 MODULARITY OF ENDOMORPHISM ALGEBRAS 3 The map T (g) depends only on the double coset KgK, and there is a canonical map ρ Ω T K,K Hom Q (Ω 1 (M K ), Ω 1 (M K )) sending the characteristic function of KgK to T (g). We let T K,K = ρ Ω ( T K,K ). Assume that M K and M K are geometrically connected. For any g G, we define similarly T (g) = α α J K J K. Note that the homomorphism T (g) is a priori defined over the base field of M K gk g 1, but its differential at the origin maps the tangent space Ω 1 (M K ) into Ω 1 (M K ), hence it is defined over Q. We therefore get a map ρ J T K,K Hom(J K, J K ). Since Hom(J K, J K ) acts faithfully on the tangent spaces, the map ρ J factors through T K,K. Summing up, we have a commutative diagram ρ Ω ρ J λ T K,K T K,K Hom(J K, J K ) Hom Q (Ω 1 (M K ), Ω 1 (M K )) where λ denotes the differential at the origin. In the case K = K, we define T K = T K,K and T K = T K,K. The convolution product endows T K with the structure of a unitary ring. We refer to T K as the Hecke double coset algebra. Note that T K,K has the structure of a ( T K, T K )-bimodule. Define 3. Proof of Theorem 1 Ω = lim K Ω 1 (M K ) Q, where the direct limit is taken with respect to the pull-back maps. The space Ω is endowed with an action of G and the subspace Ω K of K-invariants coincides with Ω 1 (M K ) Q. According to the multiplicity one theorem, the space Ω decomposes as a direct sum of distinct irreducible admissible representations of G: Ω = Ω(π). π Π Let Π(K) be the set of those representations π Π satisfying Ω(π) K 0. We have a direct sum decomposition Ω 1 (M K ) Q = π Π(K) Ω(π) K, and the spaces Ω(π) K are pairwise non-isomorphic simple T K Q-modules [4, p. 393]. Lemma 1. The canonical map is an isomorphism. ρ K T K Q End Q (Ω(π) K ) π Π(K) Proof. The map ρ K is injective by definition of T K. The surjectivity follows from Burnside s Theorem [1, 5, N 3, Cor. 1 of Prop. 4, p. 79]. Lemma 2. Let K, K be compact open subgroups of G. For any π Π, the bimodule T K,K maps Ω(π) K into Ω(π) K. Let R = Π(K) Π(K ). The map ρ K,K T K,K Q Hom Q (Ω(π) K, Ω(π) K ) π R is an isomorphism of (T K, T K )-bimodules.

4 4 F. BRUNAULT Proof. The map ρ K,K is injective by definition of T K,K. For the surjectivity, let K be a compact open subgroup of G such that R Π(K ). We have a commutative diagram T K,K T K,K Q π R Hom(Ω(π) K, Ω(π) K ) Hom(Ω(π) K, Ω(π) K ) T K,K Q π R Hom(Ω(π) K, Ω(π) K ). Since the right-hand map is surjective, it suffices to show that the maps ρ K,K and ρ K,K are surjective. Choosing K = K K, we are reduced to show the lemma in the cases K K and K K. Moreover, since T K,K is a (T K, T K )-bimodule, and thanks to Lemma 1, it suffices to show that for any π R, the map ρ π T K,K Q Hom Q (Ω(π) K, Ω(π) K ) is non-zero. In the case K K, the image of the double coset K 1 K = K under ρ π is the inclusion map of Ω(π) K into Ω(π) K, which is non-zero. In the case K K, the image of the double coset K 1 K = K under ρ π is the trace map from Ω(π) K to Ω(π) K. Since the restriction of the trace map to Ω(π) K is the multiplication by the index (K K), the trace map is non-zero as required. Now let us consider the direct limit of the étale cohomology groups of M K : H = lim K H 1 ét (M K Q Q, Z l ) Q l. The space H is endowed with two commuting actions of G and Γ Q = Gal(Q/Q), and we have H K = H 1 ét (M K Q Q, Z l ) Q l. We will now see how to separate these two actions. Let us fix an embedding of Q into Q l. Definition 3. For any π Π, let V π = Hom G (Ω(π), H). Note that V π is a Q l -vector space endowed with an action of Γ Q. Lemma 4. The Galois representation V π is 2-dimensional, and we have a G Γ Q -equivariant isomorphism H Ω(π) Q V π. π Π In particular, for any compact open subgroup K of G, we have a T K [Γ Q ]-equivariant isomorphism H K Ω(π) K Q V π. π Π(K) Proof. Let us fix an isomorphism Q l C. By the comparison theorem between Betti and étale cohomology, we have H lim K H 1 B(M K (C), Q l ). On the other hand, we have a C-linear isomorphism Ω 1 (M K (C)) Ω 1 (M K (C)) H 1 B(M K (C), C) (ω, ω ) [ω + c ω ]

5 MODULARITY OF ENDOMORPHISM ALGEBRAS 5 where c denotes the complex conjugation on M K (C). It follows that H (Ω Ω) Q l. Since Hom G (Ω(π), Ω(π )) = { Q if π = π, 0 if π π, we deduce that V π has dimension 2. Finally, there is a canonical map Ω(π) V π H, and the space H decomposes as the direct sum of the images of these maps. Remark 3. The isomorphisms in Lemma 4 have motivic origin, see [9, 2.2.4, 2.2.5]. The following lemma is well-known (see the proof of [7, Thm 4.4]). Lemma 5. The representation V π is irreducible, and we have Hom ΓQ (V π, V π ) = { Q l if π = π, 0 if π π. Proof of the main theorem. Let K, K be compact open subgroups of G such that M K and M K are geometrically connected. Consider the composite map ρét ρ J 1 T K,K Q l Hom(J K, J K ) Q l Hom ΓQ (H K, H K ). Since these maps are injective, it suffices to show that ρét is surjective, and for this it is enough to compare the dimensions. Let R = Π(K) Π(K ). By Lemma 2, we have dim T K,K = (dim Ω(π) K )(dim Ω(π) K ). π R On the other hand, using Lemmas 4 and 5, we get Hom ΓQ (H K, H K ) = Hom(Ω(π) K, Ω(π) K ) Q l, π R and thus dim Hom ΓQ (H K, H K ) = dim T K,K as desired. 4. Generalization to abelian extensions Let K be a compact open subgroup of G, and let F be the base field of M K. Let F be a finite abelian extension of Q containing F, and let U F be the subgroup of U = det(k) defined by U F = χ(gal(q ab /F )). Then we have a canonical isomorphism M K F F M KF where K F is the subgroup of K defined by K F = {g K det(g) U F }. Let K, K be compact open subgroups of G such that the base fields of M K and M K are equal to a fixed finite abelian extension F of Q. Definition 6. Let T = (X, α, α ) be a finite correspondence between M K and M K, seen as curves over Q. We say that T is defined over F if the following diagram commutes: X α α T M K M K µ δ Spec F δ

6 6 F. BRUNAULT Lemma 7. Let U F = χ(gal(q ab /F )) and let g G. The correspondence T (g) = KgK is defined over F if and only if det(g) Q >0 U F. We denote by T K,K ;F the subgroup of T K,K generated by those correspondences T (g) which are defined over F. Note that we have a canonical map ρ J T K,K ;F Hom F (J K, J K ) where J K (resp. J K ) denotes the Jacobian variety of M K (resp. M K ) over F. Theorem 2. Let K, K be compact open subgroups of GL 2 (A f ), and let F be a finite abelian extension of Q containing the base fields of M K and M K. Then the canonical map is an isomorphism. ρ J T KF,K F ;F Q Hom F (J K, J K ) Q Proof. By the above discussion, it is sufficient to prove the theorem in the case the base fields of M K and M K are equal to F. Let Γ = Gal(F /Q). For any γ Γ, let T K,K ;γ denote the subgroup of T K,K generated by those correspondences T (g) satisfying det(g) Q >0 (ˆγU F ), where ˆγ Ẑ is any element satisfying χ 1 (ˆγ) F = γ. Since the elements of T K,K ;γ are γ-linear, we have a direct sum decomposition T K,K = T K,K ;γ. γ Γ By the proof of Theorem 1, we have an isomorphism (1) ρét T K,K Q l HomΓQ (H K, H K ). We now wish to identify those elements of Hom ΓQ (H K, H K ) which come from T K,K ;γ. Let Σ denote the set of embedding of F into Q. We have M K Q Q = M K F,σ Q σ Σ inducing a direct sum decomposition H K = σ Σ Hσ K with H K σ = H 1 ét (M K F,σ Q, Z l ) Q l. Note that the action of Γ Q on H K permutes the components Hσ K according to the rule γ Hσ K = Hγσ K for any γ Γ Q. Fixing an element σ 0 Σ, we have an isomorphism Ind Γ Q Γ F H K σ 0 H K where Γ F = Gal(Q/F ). By Frobenius reciprocity, we have (2) Hom ΓQ (H K, H K ) Hom ΓF (H K σ 0, H K ). Moreover T K,K ;γ maps H K σ into H K σγ. Combining the isomorphisms (1) and (2), we get T K,K ;γ Q l Hom ΓF (Hσ K 0, Hσ K 0 γ) Taking γ = 1, we get a commutative diagram ρét (γ G). ρ J 1 T K,K ;F Q l Hom F (J K, J K ) Q l Hom ΓF (H K σ 0, H K where ρét is an isomorphism. We conclude as in the proof of Theorem 1. We now give some consequences for modular abelian varieties. The following Corollary shows that every endomorphism of a modular abelian variety defined over an abelian number field arises from the Hecke double coset algebra. Corollary 1. Let K be a compact open subgroup of G. Let F be a finite abelian extension of Q containing the base field of M K. µ σ 0 )

7 MODULARITY OF ENDOMORPHISM ALGEBRAS 7 (1) Let A/F be an abelian subvariety of J K /F. Define T A = {T T KF ;F ρ J (T ) leaves stable A}. Then the canonical map T A Q End F (A) Q is surjective. (2) Let A/F be an abelian variety which is a quotient of J K /F. Define T A = {T T KF ;F ρ J (T ) factors through A}. Then the canonical map T A Q End F (A) Q is surjective. Proof. Let us prove (1). Let ι A J K denote the inclusion map. Let p J K A be a homomorphism such that p ι = [n] A for some integer n 0. Let φ End F (A). Define ψ = ι φ p End F (J K ). By Theorem 2, there exists T T KF ;F Q such that ρ J (T ) = ψ. Note that ψ leaves stable A, so that T T A Q, and we have ψ A = [n]φ. The proof of (2) is similar. We emphasize that Corollary 1 is true even for elliptic curves with complex multiplication, as long as their endomorphisms are defined over an abelian extension of Q. For example, the elliptic curve E = X 0 (32) has complex multiplication by Z[i] defined over Q(i). Let K = K 0 (32) Q(i) = {( a b ) GL c d 2(Ẑ) c 0 (32), ad 1 (4)}. The matrix ( ) normalizes K, and the canonical map T K;Q(i) End Q(i) E Z[i] maps the coset K ( ) K = K ( ) to the element i [2, p. 3], hence T K;Q(i) End Q(i) E. To conclude this section, let me mention some open questions. Questions. (1) Do Theorems 1 and 2 hold integrally? (2) Do Theorems 1 and 2 hold for modular curves in positive characteristic? (3) The analogue of J K in weight k > 2 is the (Chow) motive associated to the space of cusp forms of weight k and level K [8]. Do the results presented here extend to these motives? Do they extend to automorphic forms on more general groups? 5. Comparison with Ribet s result Let f = n 1 a n q n be a newform of weight 2 on Γ 1 (N), and let A f /Q be the modular abelian variety associated to f. The abelian variety A f is simple over Q and the algebra End Q (A f ) Q is isomorphic to the Hecke field K f of f. Ribet determined in [7] the structure of the endomorphism algebra End Q (A f ) Q. In particular, he proved that this algebra is generated over K f by finitely many endomorphisms coming from the inner twists of f. Our goal in this section is to write these endomorphisms of A f in terms of Hecke correspondences, making thus Corollary 1 explicit for these endomorphisms. Let us first recall Ribet s construction [7, 5]. We assume that f doesn t have complex multiplication. Let Γ denote the set of automorphisms γ of K f such that f γ = f χ γ for some Dirichlet character χ γ. Let m denote the least common multiple of N and the conductors of the characters χ γ. Then h = (n,n)=1 a n q n is an eigenform on the group Γ 0 (m 2 ) Γ 1 (m). Let J denote the Jacobian variety of the modular curve associated to this group. By Shimura s construction [7, 2], there exists an optimal quotient ν J A h associated to h. The abelian varieties A f and A h are isogenous. In particular, their endomorphism algebras are isomorphic. For every γ Γ, Ribet constructs an endomorphism η γ of A h as follows. Write f γ = f χ. Let r denote the conductor of χ. For every u Z, there is an endomorphism α u/r of J acting on the space of cusp forms as g g(z + u/r). Define η γ = χ 1 (u) ν α u/r Hom(J, A h ) Q u Z/rZ

8 8 F. BRUNAULT where χ 1 (u) K f is seen as an element of End Q (A h ) Q. Then η γ factors through ν and induces an endomorphism η γ of A h. Since α u/r is defined over Q(ζ r ), we have η γ End Q(ζr)(A h ) Q. Let us now turn to the adelic language. Consider the group K = K 0 (m 2 ) K 1 (m) = {( a c b d ) GL2(Ẑ) c 0(m2 ), d 1(m)} and its subgroup K = K Q(ζr). The modular curve M K and its Jacobian J = J K are defined over the field Q(ζ r ), and we have a canonical isomorphism J J Q(ζr). Since the elements χ 1 (u) belong to K f, they certainly come from the Hecke algebra of K. Therefore, there exist elements λ u in T K ;Q(ζ r) Q such that ρ J (λ u ) factors through A h and induces χ 1 (u) on A h. Lemma 8. For every u Z, we have α u/r = ρ J (T ( 1 u/r 0 1 )). Proof. By Lemma 7, the correspondence T ( 1 u/r ) is defined over Q(ζ 0 1 r). Moreover the matrix ( 1 u/r ) normalizes 0 1 K, so that T ( 1 u/r ) acts on 0 1 Ω1 (M K ) by sending a cusp form g to g(z + u/r). It follows that α = ρ u/r Ω ( T ( 1 u/r )), hence the Lemma. 0 1 We now define X γ = r 1 u=0 λ u T ( 1 u/r 0 1 ) T K ;Q(ζ r) Q. Proposition 9. The endomorphism ρ J (X γ ) factors through A h and induces the endomorphism η γ on A h. Proof. This follows from the definition of λ u and Lemma 8. References [1] N. Bourbaki. Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples. Springer, Berlin, Second revised edition of the 1958 edition. [2] M. Dickson and M. Neururer. Products of Eisenstein series and Fourier expansions of modular forms at cusps. Preprint, [3] E. Kani. Endomorphisms of Jacobians of modular curves. Arch. Math. (Basel), 91(3): , [4] R. P. Langlands. Modular forms and l-adic representations. In Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages Lecture Notes in Math., Vol Springer, Berlin, [5] B. Mazur. Modular curves and the Eisenstein ideal. Publ. Math. Inst. Hautes Études Sci., 47:33 186, [6] K. A. Ribet. Endomorphisms of semi-stable abelian varieties over number fields. Ann. Math. (2), 101: , [7] K. A. Ribet. Twists of modular forms and endomorphisms of abelian varieties. Math. Ann., 253(1):43 62, [8] A. J. Scholl. Motives for modular forms. Invent. Math., 100(2): , [9] A. J. Scholl. Integral elements in K-theory and products of modular curves. In The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), volume 548 of NATO Sci. Ser. C Math. Phys. Sci., pages Kluwer Acad. Publ., Dordrecht, ÉNS Lyon, Unité de mathématiques pures et appliquées, 46 allée d Italie, Lyon, France address: francois.brunault@ens-lyon.fr URL: