ON THE MODULARITY OF ENDOMORPHISM ALGEBRAS
|
|
- Amos Pierce
- 5 years ago
- Views:
Transcription
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:
FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationRaising the Levels of Modular Representations Kenneth A. Ribet
1 Raising the Levels of Modular Representations Kenneth A. Ribet 1 Introduction Let l be a prime number, and let F be an algebraic closure of the prime field F l. Suppose that ρ : Gal(Q/Q) GL(2, F) is
More informationTATE CONJECTURES FOR HILBERT MODULAR SURFACES. V. Kumar Murty University of Toronto
TATE CONJECTURES FOR HILBERT MODULAR SURFACES V. Kumar Murty University of Toronto Toronto-Montreal Number Theory Seminar April 9-10, 2011 1 Let k be a field that is finitely generated over its prime field
More informationKleine AG: Travaux de Shimura
Kleine AG: Travaux de Shimura Sommer 2018 Programmvorschlag: Felix Gora, Andreas Mihatsch Synopsis This Kleine AG grew from the wish to understand some aspects of Deligne s axiomatic definition of Shimura
More informationOn the equality case of the Ramanujan Conjecture for Hilbert modular forms
On the equality case of the Ramanujan Conjecture for Hilbert modular forms Liubomir Chiriac Abstract The generalized Ramanujan Conjecture for unitary cuspidal automorphic representations π on GL 2 posits
More information15 Elliptic curves and Fermat s last theorem
15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine
More informationModularity of Abelian Varieties
1 Modularity of Abelian Varieties This is page 1 Printer: Opaque this 1.1 Modularity Over Q Definition 1.1.1 (Modular Abelian Variety). Let A be an abelian variety over Q. Then A is modular if there exists
More informationThe special L-value of the winding quotient of level a product of two distinct primes
The special L-value of the winding quotient of level a product of two distinct primes Amod Agashe Abstract Let p and q be two distinct primes and let J e denote the winding quotient at level pq. We give
More informationl-adic MODULAR DEFORMATIONS AND WILES S MAIN CONJECTURE
l-adic MODULAR DEFORMATIONS AND WILES S MAIN CONJECTURE FRED DIAMOND AND KENNETH A. RIBET 1. Introduction Let E be an elliptic curve over Q. The Shimura-Taniyama conjecture asserts that E is modular, i.e.,
More informationComputer methods for Hilbert modular forms
Computer methods for Hilbert modular forms John Voight University of Vermont Workshop on Computer Methods for L-functions and Automorphic Forms Centre de Récherche Mathématiques (CRM) 22 March 2010 Computer
More informationSERRE S CONJECTURE AND BASE CHANGE FOR GL(2)
SERRE S CONJECTURE AND BASE CHANGE OR GL(2) HARUZO HIDA 1. Quaternion class sets A quaternion algebra B over a field is a simple algebra of dimension 4 central over a field. A prototypical example is the
More informationAbelian Varieties over Q with Large Endomorphism Algebras and Their Simple Components over Q
Abelian Varieties over Q with Large Endomorphism Algebras and Their Simple Components over Q Elisabeth Eve Pyle B.S. (Stanford University) 1988 C.Phil. (University of California at Berkeley) 1992 A dissertation
More informationVARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES
VARIATION OF IWASAWA INVARIANTS IN HIDA FAMILIES MATTHEW EMERTON, ROBERT POLLACK AND TOM WESTON 1. Introduction Let ρ : G Q GL 2 (k) be an absolutely irreducible modular Galois representation over a finite
More informationFields of definition of abelian varieties with real multiplication
Contemporary Mathematics Volume 174, 1994 Fields of definition of abelian varieties with real multiplication KENNETH A. RIBET 1. Introduction Let K be a field, and let K be a separable closure of K. Let
More informationThe Galois Representation Associated to Modular Forms (Part I)
The Galois Representation Associated to Modular Forms (Part I) Modular Curves, Modular Forms and Hecke Operators Chloe Martindale May 20, 2015 Contents 1 Motivation and Background 1 2 Modular Curves 2
More informationDieudonné Modules and p-divisible Groups
Dieudonné Modules and p-divisible Groups Brian Lawrence September 26, 2014 The notion of l-adic Tate modules, for primes l away from the characteristic of the ground field, is incredibly useful. The analogous
More informationFrom K3 Surfaces to Noncongruence Modular Forms. Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015
From K3 Surfaces to Noncongruence Modular Forms Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015 Winnie Li Pennsylvania State University 1 A K3 surface
More informationComputing coefficients of modular forms
Computing coefficients of modular forms (Work in progress; extension of results of Couveignes, Edixhoven et al.) Peter Bruin Mathematisch Instituut, Universiteit Leiden Théorie des nombres et applications
More informationOn the generation of the coefficient field of a newform by a single Hecke eigenvalue
On the generation of the coefficient field of a newform by a single Hecke eigenvalue Koopa Tak-Lun Koo and William Stein and Gabor Wiese November 2, 27 Abstract Let f be a non-cm newform of weight k 2
More informationProof. We omit the proof of the case when f is the reduction of a characteristic zero eigenform (cf. Theorem 4.1 in [DS74]), so assume P (X) has disti
Some local (at p) properties of residual Galois representations Johnson Jia, Krzysztof Klosin March 5, 26 1 Preliminary results In this talk we are going to discuss some local properties of (mod p) Galois
More informationGalois Representations
9 Galois Representations This book has explained the idea that all elliptic curves over Q arise from modular forms. Chapters 1 and introduced elliptic curves and modular curves as Riemann surfaces, and
More informationof S 2 (Γ(p)). (Hecke, 1928)
Equivariant Atkin-Lehner Theory Introduction Atkin-Lehner Theory: Atkin-Lehner (1970), Miyake (1971), Li (1975): theory of newforms (+ T-algebra action) a canonical basis for S k (Γ 1 (N)) and hence also
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationLocal root numbers of elliptic curves over dyadic fields
Local root numbers of elliptic curves over dyadic fields Naoki Imai Abstract We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension
More informationVisibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one
Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one Amod Agashe February 20, 2009 Abstract Let E be an optimal elliptic curve over Q of conductor N having analytic rank one, i.e.,
More informationOn the notion of visibility of torsors
On the notion of visibility of torsors Amod Agashe Abstract Let J be an abelian variety and A be an abelian subvariety of J, both defined over Q. Let x be an element of H 1 (Q, A). Then there are at least
More informationThree-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms
Three-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms U-M Automorphic forms workshop, March 2015 1 Definition 2 3 Let Γ = PSL 2 (Z) Write ( 0 1 S =
More informationMULTIPLICITIES OF GALOIS REPRESENTATIONS IN JACOBIANS OF SHIMURA CURVES
MULTIPLICITIES OF GALOIS REPRESENTATIONS IN JACOBIANS OF SHIMURA CURVES BY KENNETH A. RIBET Mathematics Department, University of California, Berkeley CA 94720 USA Let p and q be distinct primes. The new
More informationThe Galois Representation Attached to a Hilbert Modular Form
The Galois Representation Attached to a Hilbert Modular Form Gabor Wiese Essen, 17 July 2008 Abstract This talk is the last one in the Essen seminar on quaternion algebras. It is based on the paper by
More informationA note on trilinear forms for reducible representations and Beilinson s conjectures
A note on trilinear forms for reducible representations and Beilinson s conjectures M Harris and A J Scholl Introduction Let F be a non-archimedean local field, and π i (i = 1, 2, 3) irreducible admissible
More informationMA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26
MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26 1. Abelian Varieties of GL 2 -Type 1.1. Modularity Criteria. Here s what we ve shown so far: Fix a continuous residual representation : G Q GLV, where V is
More informationIntroductory comments on the eigencurve
Introductory comments on the eigencurve Handout # 5: March 8, 2006 (These are brief indications, hardly more than an annotated list, of topics mentioned in my lectures. ) 1 The basic Hecke diagram As before
More informationA Version of the Grothendieck Conjecture for p-adic Local Fields
A Version of the Grothendieck Conjecture for p-adic Local Fields by Shinichi MOCHIZUKI* Section 0: Introduction The purpose of this paper is to prove an absolute version of the Grothendieck Conjecture
More informationResidual modular Galois representations: images and applications
Residual modular Galois representations: images and applications Samuele Anni University of Warwick London Number Theory Seminar King s College London, 20 th May 2015 Mod l modular forms 1 Mod l modular
More informationNon CM p-adic analytic families of modular forms
Non CM p-adic analytic families of modular forms Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. The author is partially supported by the NSF grant: DMS 1464106. Abstract:
More informationOn the modular curve X 0 (23)
On the modular curve X 0 (23) René Schoof Abstract. The Jacobian J 0(23) of the modular curve X 0(23) is a semi-stable abelian variety over Q with good reduction outside 23. It is simple. We prove that
More information14 From modular forms to automorphic representations
14 From modular forms to automorphic representations We fix an even integer k and N > 0 as before. Let f M k (N) be a modular form. We would like to product a function on GL 2 (A Q ) out of it. Recall
More informationInduction formula for the Artin conductors of mod l Galois representations. Yuichiro Taguchi
Induction formula for the Artin conductors of mod l Galois representations Yuichiro Taguchi Abstract. A formula is given to describe how the Artin conductor of a mod l Galois representation behaves with
More informationLifting Galois Representations, and a Conjecture of Fontaine and Mazur
Documenta Math. 419 Lifting Galois Representations, and a Conjecture of Fontaine and Mazur Rutger Noot 1 Received: May 11, 2001 Revised: November 16, 2001 Communicated by Don Blasius Abstract. Mumford
More informationFiniteness of the Moderate Rational Points of Once-punctured Elliptic Curves. Yuichiro Hoshi
Hokkaido Mathematical Journal ol. 45 (2016) p. 271 291 Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves uichiro Hoshi (Received February 28, 2014; Revised June 12, 2014) Abstract.
More informationThe Galois representation associated to modular forms pt. 2 Erik Visse
The Galois representation associated to modular forms pt. 2 Erik Visse May 26, 2015 These are the notes from the seminar on local Galois representations held in Leiden in the spring of 2015. The website
More informationMATH G9906 RESEARCH SEMINAR IN NUMBER THEORY (SPRING 2014) LECTURE 1 (FEBRUARY 7, 2014) ERIC URBAN
MATH G9906 RESEARCH SEMINAR IN NUMBER THEORY (SPRING 014) LECTURE 1 (FEBRUARY 7, 014) ERIC URBAN NOTES TAKEN BY PAK-HIN LEE 1. Introduction The goal of this research seminar is to learn the theory of p-adic
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationEndomorphism Rings of Abelian Varieties and their Representations
Endomorphism Rings of Abelian Varieties and their Representations Chloe Martindale 30 October 2013 These notes are based on the notes written by Peter Bruin for his talks in the Complex Multiplication
More informationREPRESENTATION THEORY, LECTURE 0. BASICS
REPRESENTATION THEORY, LECTURE 0. BASICS IVAN LOSEV Introduction The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite
More informationA MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET. March 7, 2017
A MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET DIPENDRA PRASAD March 7, 2017 Abstract. Following the natural instinct that when a group operates on a number field then every term in the
More informationMod p Galois representations attached to modular forms
Mod p Galois representations attached to modular forms Ken Ribet UC Berkeley April 7, 2006 After Serre s article on elliptic curves was written in the early 1970s, his techniques were generalized and extended
More informationGalois representations and automorphic forms
Columbia University, Institut de Mathématiques de Jussieu Yale, November 2013 Galois theory Courses in Galois theory typically calculate a very short list of Galois groups of polynomials in Q[X]. Cyclotomic
More informationWhat is the Langlands program all about?
What is the Langlands program all about? Laurent Lafforgue November 13, 2013 Hua Loo-Keng Distinguished Lecture Academy of Mathematics and Systems Science, Chinese Academy of Sciences This talk is mainly
More informationModularity of p-adic Galois representations via p-adic approximations
Journal de Théorie des Nombres de Bordeaux 16 (2004), 179 185 Modularity of p-adic Galois representations via p-adic approximations Dedicated to the memory of my mother Nalini B. Khare 30th August 1936
More informationLecture 4: Examples of automorphic forms on the unitary group U(3)
Lecture 4: Examples of automorphic forms on the unitary group U(3) Lassina Dembélé Department of Mathematics University of Calgary August 9, 2006 Motivation The main goal of this talk is to show how one
More information1.2 The result which we would like to announce here is that there exists a cuspidal automorphic representation u of GL 3;Q (not selfdual) such that th
A non-selfdual automorphic representation of GL 3 and a Galois representation Bert van Geemen and Jaap Top Abstract The Langlands philosophy contemplates the relation between automorphic representations
More informationEXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.
EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is
More informationCLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES
CLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES FRANK GOUNELAS 1. Class Field Theory We ll begin by motivating some of the constructions of the CM (complex multiplication) theory for
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik On projective linear groups over finite fields as Galois groups over the rational numbers Gabor Wiese Preprint Nr. 14/2006 On projective linear groups over finite fields
More informationHecke eigenforms in the Cuspidal Cohomology of Congruence Subgroups of SL(3, Z)
Hecke eigenforms in the Cuspidal Cohomology of Congruence Subgroups of SL(3, Z) Bert van Geemen, Wilberd van der Kallen, Jaap Top and Alain Verberkmoes October 25, 1996 Abstract In this paper, Hecke eigenvalues
More informationSéminaire BOURBAKI Novembre ème année, , n o p-adic FAMILIES OF MODULAR FORMS [after Hida, Coleman, and Mazur]
Séminaire BOURBAKI Novembre 2009 62ème année, 2009-2010, n o 1013 p-adic FAMILIES OF MODULAR FORMS [after Hida, Coleman, and Mazur] by Matthew EMERTON INTRODUCTION The theory of p-adic families of modular
More informationKenneth A. Ribet University of California, Berkeley
IRREDUCIBLE GALOIS REPRESENTATIONS ARISING FROM COMPONENT GROUPS OF JACOBIANS Kenneth A. Ribet University of California, Berkeley 1. Introduction Much has been written about component groups of Néron models
More informationl-adic Representations
l-adic Representations S. M.-C. 26 October 2016 Our goal today is to understand l-adic Galois representations a bit better, mostly by relating them to representations appearing in geometry. First we ll
More information9 Artin representations
9 Artin representations Let K be a global field. We have enough for G ab K. Now we fix a separable closure Ksep and G K := Gal(K sep /K), which can have many nonabelian simple quotients. An Artin representation
More informationDRINFELD LEVEL STRUCTURES AND LUBIN-TATE SPACES. Przemysªaw Chojecki
DRINFELD LEVEL STRUCTURES AND LUBIN-TATE SPACES Przemysªaw Chojecki Le but principal de cet exposé est de demontrer la regularité des anneaux representants les foncteurs des deformations des groupes formels
More information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More informationON THE MODIFIED MOD p LOCAL LANGLANDS CORRESPONDENCE FOR GL 2 (Q l )
ON THE MODIFIED MOD p LOCAL LANGLANDS CORRESPONDENCE FOR GL 2 (Q l ) DAVID HELM We give an explicit description of the modified mod p local Langlands correspondence for GL 2 (Q l ) of [EH], Theorem 5.1.5,
More informationψ l : T l (A) T l (B) denotes the corresponding morphism of Tate modules 1
1. An isogeny class of supersingular elliptic curves Let p be a prime number, and k a finite field with p 2 elements. The Honda Tate theory of abelian varieties over finite fields guarantees the existence
More informationA p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1
A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection
More informationOn Hrushovski s proof of the Manin-Mumford conjecture
On Hrushovski s proof of the Manin-Mumford conjecture Richard PINK and Damian ROESSLER May 16, 2006 Abstract The Manin-Mumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski
More informationCHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0. Mitya Boyarchenko Vladimir Drinfeld. University of Chicago
CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0 Mitya Boyarchenko Vladimir Drinfeld University of Chicago Some historical comments A geometric approach to representation theory for unipotent
More informationEKNATH GHATE AND VINAYAK VATSAL. 1. Introduction
ON THE LOCAL BEHAVIOUR OF ORDINARY Λ-ADIC REPRESENTATIONS EKNATH GHATE AND VINAYAK VATSAL 1. Introduction In this paper we study the local behaviour of the Galois representations attached to ordinary Λ-adic
More informationRaynaud on F -vector schemes and prolongation
Raynaud on F -vector schemes and prolongation Melanie Matchett Wood November 7, 2010 1 Introduction and Motivation Given a finite, flat commutative group scheme G killed by p over R of mixed characteristic
More informationCLASS FIELD THEORY WEEK Motivation
CLASS FIELD THEORY WEEK 1 JAVIER FRESÁN 1. Motivation In a 1640 letter to Mersenne, Fermat proved the following: Theorem 1.1 (Fermat). A prime number p distinct from 2 is a sum of two squares if and only
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics MOD p REPRESENTATIONS ON ELLIPTIC CURVES FRANK CALEGARI Volume 225 No. 1 May 2006 PACIFIC JOURNAL OF MATHEMATICS Vol. 225, No. 1, 2006 MOD p REPRESENTATIONS ON ELLIPTIC
More informationQuestion 1: Are there any non-anomalous eigenforms φ of weight different from 2 such that L χ (φ) = 0?
May 12, 2003 Anomalous eigenforms and the two-variable p-adic L-function (B Mazur) A p-ordinary p-adic modular (cuspidal) eigenform (for almost all Hecke operators T l with l p and for the Atkin-Lehner
More informationON GALOIS GROUPS OF ABELIAN EXTENSIONS OVER MAXIMAL CYCLOTOMIC FIELDS. Mamoru Asada. Introduction
ON GALOIS GROUPS OF ABELIAN ETENSIONS OVER MAIMAL CYCLOTOMIC FIELDS Mamoru Asada Introduction Let k 0 be a finite algebraic number field in a fixed algebraic closure Ω and ζ n denote a primitive n-th root
More informationTwists and residual modular Galois representations
Twists and residual modular Galois representations Samuele Anni University of Warwick Building Bridges, Bristol 10 th July 2014 Modular curves and Modular Forms 1 Modular curves and Modular Forms 2 Residual
More information1.5.4 Every abelian variety is a quotient of a Jacobian
16 1. Abelian Varieties: 10/10/03 notes by W. Stein 1.5.4 Every abelian variety is a quotient of a Jacobian Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety.
More informationIsogeny invariance of the BSD conjecture
Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p
More informationRIMS. Ibukiyama Zhuravlev. B.Heim
RIMS ( ) 13:30-14:30 ( ) Title: Generalized Maass relations and lifts. Abstract: (1) Duke-Imamoglu-Ikeda Eichler-Zagier- Ibukiyama Zhuravlev L- L- (2) L- L- L B.Heim 14:45-15:45 ( ) Title: Kaneko-Zagier
More informationOn the Irreducibility of the Commuting Variety of the Symmetric Pair so p+2, so p so 2
Journal of Lie Theory Volume 16 (2006) 57 65 c 2006 Heldermann Verlag On the Irreducibility of the Commuting Variety of the Symmetric Pair so p+2, so p so 2 Hervé Sabourin and Rupert W.T. Yu Communicated
More informationReciprocity maps with restricted ramification
Reciprocity maps with restricted ramification Romyar Sharifi UCLA January 6, 2016 1 / 19 Iwasawa theory for modular forms Let be an p odd prime and f a newform of level N. Suppose that f is ordinary at
More informationLemma 1.1. The field K embeds as a subfield of Q(ζ D ).
Math 248A. Quadratic characters associated to quadratic fields The aim of this handout is to describe the quadratic Dirichlet character naturally associated to a quadratic field, and to express it in terms
More informationMod p Galois representations of solvable image. Hyunsuk Moon and Yuichiro Taguchi
Mod p Galois representations of solvable image Hyunsuk Moon and Yuichiro Taguchi Abstract. It is proved that, for a number field K and a prime number p, there exist only finitely many isomorphism classes
More informationProblems on Growth of Hecke fields
Problems on Growth of Hecke fields Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A. A list of conjectures/problems related to my talk in Simons Conference in January 2014
More informationAutomorphic Galois representations and Langlands correspondences
Automorphic Galois representations and Langlands correspondences II. Attaching Galois representations to automorphic forms, and vice versa: recent progress Bowen Lectures, Berkeley, February 2017 Outline
More informationALGEBRA 11: Galois theory
Galois extensions Exercise 11.1 (!). Consider a polynomial P (t) K[t] of degree n with coefficients in a field K that has n distinct roots in K. Prove that the ring K[t]/P of residues modulo P is isomorphic
More informationCOUNTING MOD l SOLUTIONS VIA MODULAR FORMS
COUNTING MOD l SOLUTIONS VIA MODULAR FORMS EDRAY GOINS AND L. J. P. KILFORD Abstract. [Something here] Contents 1. Introduction 1. Galois Representations as Generating Functions 1.1. Permutation Representation
More informationCOMPLEX MULTIPLICATION: LECTURE 15
COMPLEX MULTIPLICATION: LECTURE 15 Proposition 01 Let φ : E 1 E 2 be a non-constant isogeny, then #φ 1 (0) = deg s φ where deg s is the separable degree of φ Proof Silverman III 410 Exercise: i) Consider
More informationTheta divisors and the Frobenius morphism
Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following
More informationTAMAGAWA NUMBERS OF ELLIPTIC CURVES WITH C 13 TORSION OVER QUADRATIC FIELDS
TAMAGAWA NUMBERS OF ELLIPTIC CURVES WITH C 13 TORSION OVER QUADRATIC FIELDS FILIP NAJMAN Abstract. Let E be an elliptic curve over a number field K c v the Tamagawa number of E at v and let c E = v cv.
More informationarxiv: v5 [math.nt] 2 Aug 2017
NON-OPTIMAL LEVELS OF A REDUCIBLE MOD l MODULAR REPRESENTATION HWAJONG YOO arxiv:1409.8342v5 [math.nt] 2 Aug 2017 Abstract. Let l 5 be a prime and let N be a square-free integer prime to l. For each prime
More informationTHE TATE MODULE. Seminar: Elliptic curves and the Weil conjecture. Yassin Mousa. Z p
THE TATE MODULE Seminar: Elliptic curves and the Weil conjecture Yassin Mousa Abstract This paper refers to the 10th talk in the seminar Elliptic curves and the Weil conjecture supervised by Prof. Dr.
More informationUp to twist, there are only finitely many potentially p-ordinary abelian varieties over. conductor
Up to twist, there are only finitely many potentially p-ordinary abelian varieties over Q of GL(2)-type with fixed prime-to-p conductor Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA 90095-1555,
More informationON ISOTROPY OF QUADRATIC PAIR
ON ISOTROPY OF QUADRATIC PAIR NIKITA A. KARPENKO Abstract. Let F be an arbitrary field (of arbitrary characteristic). Let A be a central simple F -algebra endowed with a quadratic pair σ (if char F 2 then
More information1.6.1 What are Néron Models?
18 1. Abelian Varieties: 10/20/03 notes by W. Stein 1.6.1 What are Néron Models? Suppose E is an elliptic curve over Q. If is the minimal discriminant of E, then E has good reduction at p for all p, in
More informationRiemann surfaces with extra automorphisms and endomorphism rings of their Jacobians
Riemann surfaces with extra automorphisms and endomorphism rings of their Jacobians T. Shaska Oakland University Rochester, MI, 48309 April 14, 2018 Problem Let X be an algebraic curve defined over a field
More informationVISIBILITY FOR ANALYTIC RANK ONE or A VISIBLE FACTOR OF THE HEEGNER INDEX
VISIBILITY FOR ANALYTIC RANK ONE or A VISIBLE FACTOR OF THE HEEGNER INDEX Amod Agashe April 17, 2009 Abstract Let E be an optimal elliptic curve over Q of conductor N, such that the L-function of E vanishes
More informationArakelov theory and height bounds
Abstract Arakelov theory and height bounds Peter Bruin Berlin, 7 November 2009 In the work of Edixhoven, Couveignes et al. (see [5] and [4]) on computing two-dimensional Galois representations associated
More informationThe Local Langlands Conjectures for n = 1, 2
The Local Langlands Conjectures for n = 1, 2 Chris Nicholls December 12, 2014 1 Introduction These notes are based heavily on Kevin Buzzard s excellent notes on the Langlands Correspondence. The aim is
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More informationarxiv: v1 [math.rt] 11 Sep 2009
FACTORING TILTING MODULES FOR ALGEBRAIC GROUPS arxiv:0909.2239v1 [math.rt] 11 Sep 2009 S.R. DOTY Abstract. Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of
More informationHIGHLY REDUCIBLE GALOIS REPRESENTATIONS ATTACHED TO THE HOMOLOGY OF GL(n, Z)
HIGHLY REDUCIBLE GALOIS REPRESENTATIONS ATTACHED TO THE HOMOLOGY OF GL(n, Z) AVNER ASH AND DARRIN DOUD Abstract Let n 1 and F an algebraic closure of a finite field of characteristic p > n + 1 Let ρ :
More information