The Galois Representation Attached to a Hilbert Modular Form

Transcription

1 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 Takeshi Saito on Hilbert modular forms and p-adic Hodge theory, but I also used talk notes by Gerard van der Geer and Theo van den Bogaart. However, I made some changes in the presentation, which may have led to the introduction of errors. The reader be warned. 1 Hilbert modular forms and automorphic representations Notation 1.1 (First part We fix the following data. F/Q, a totally real number field with [F : Q] = n > 1. I = {τ 1,...,τ n } = Hom(F, R, the embeddings of F into R. O F, the ring of integers of F. D 1 = {b F Tr F/Q (O F b Z}, the codifferent ideal. v O F, a fixed place which we only need and define if n is even. (k = (k 1,...k n, w an n + 1-tuple of integers such that w k i 2 and k i w mod 2. X = P 1 (C P 1 (R, the union of the upper and the lower half planes, X + the upper half plane. Adelic Hilbert modular forms We quickly recall the definition of adelic modular forms, following Saito. Let X I be the n-fold product of X with the left GL 2 (R I -action for ( ( ( a b c d = a i b i c i d i i and z = (z i i. Note that γ.z = ( a b az + b c d z = cz + d = ( a i z i + b i c i z i + d i XI i GL 2 (A F = GL 2 (R I GL 2 (A F,f. 1

2 Via the embeddings (τ i and the diagonal F A F,f, we obtain a natural left action of GL 2 (F on X I GL 2 (A F,f by γ.(z, g = (γz, γg. There is also a right action of GL 2 (A F,f on X I GL 2 (A F,f by right multiplication on the second factor. A function X I GL 2 (A F,f C is called holomorphic if it induces a locally constant map GL 2 (A F,f g (z f(z,g Hol(X I, C. There is a right GL 2 (F-action and a left GL 2 (A F,f -action on Hol(X I GL 2 (A F,f, C, which are defined as follows. Let f Hol(X I GL 2 (A F,f, C, γ = ( a b c d GL2 (F, z X I and g GL 2 (A F,f : (γ f(z, g = Let g GL 2 (A F,f, w+k 2 det(γ 2 (cz + d k f(γz, γg = ( (g f(z, g = f(z, gg. i det(γ i w+k i 2 2 (c i z i + d i k i f(γz, γg. For K GL 2 (A F,f open compact subgroup, the space of adelic Hilbert modular form of multiweight (k on K is M (k C,K = {f Hol(XI GL 2 (A F,f, C γ f = f, g f = f γ GL 2 (F g GL 2 (A F,f }. The union (direct limit over all open compact K is denoted by M (k C. As explained in Hai s talk, such adelic Hilbert modular forms have a Fourier expansion, which, however, I do not intend to recall. Let us just say that for a Hecke eigenform (to be defined in a moment the Fourier coefficients are (up to some normalisation factor equal to Hecke eigenvalues. We let S (k C,K and S(k C be the cuspidal subspaces, i.e. the subspaces where all 0-th Fourier coefficients vanish. For the applications to Galois representations we introduce one special open compact subgroup for each integral ideal n O F. Let ˆT = ÔF D 1 Ô F be a lattice of A 2 F,f. Let K 1 (n = {g GL 2 (A F,f g ˆT = ˆT, g ( 1 0 ( 1 0 mod n ˆT }. Hecke operators Here we present two points of view on Hecke operators. Let g GL 2 (A F,f. To g we attach the operator T g defined as follows: S (k C,K res S (k C,g 1 Kg K g S (k C,K gkg 1 Tr S (k C,K, where the trace map is given by f h K/K gkg 1 h f, supposing, of course, that K is such that this sum is finite. This description of Hecke operators is nice because it will be very similar to the description on Shimura curves to be given later on. But, there is also the equivalent double coset point 2

3 of view: Let T = T g = KgK or let T be any other subset K-invariant from the left and the right. Then we have/put T : f h T/K We define two important types of Hecke operators: h f. Let p be a prime of F and π p a uniformiser of (O F p. Let T p := T g with g = ( π p Let p be a prime of F and π p a uniformiser of (O F p. Let ( πp 0 R p := T g with g = 0 π p. (It may be that one has to impose some conditions on K. But for sure, the definition is correct with K 1 (n and (p, n = 1. Let L C be a field containing the Galois closure of F over Q. Fact 1.2 There are L-structures S (k L,K and S(k L Moreover, each S (k L,K in S(k C,K and S(k is a finite dimensional L-vector space. Definition 1.3 The Hecke algebra of S (k L,K is defined as C, respectively. T (k L,K = T ( (k ( (k p, R p End L S L,K p OF L-algebra = T p, R p End C S C,K p OF L-algebra. Fact 1.4 The Hecke algebra T (k L,K is a finite dimensional commutative L-algebra. Hence, there exist Hecke eigenforms, i.e. elements of S (k C,K that are eigenvectors for all elements of the Hecke algebra. Let f be a Hecke eigenform. The system of eigenvalues attached to f is described by the L-algebra homomorphism Θ f : T (k L,K C, T λ T, where λ T is the eigenvalue of T, i.e. Tf = Tf = λ T f. As already said above, if f is suitably normalised, the eigenvalue of T p is equal to the Fourier coefficient at p (times the norm of p, according to Saito. But, we will not need Fourier coefficients here (not explicitly, at least. We let L(f = im(θ f, the coefficient field of f (with respect to L. It is a finite extension of L due to the finite dimensionality of the Hecke algebra. In particular, if L is a number field (e.g. the Galois closure of F, then so is L(f. 3

4 The automorphic representation attached to a Hilbert newform We let π f be the GL 2 (A F,f -orbit of f in S (k C attached to f. and call it the automorphic GL 2(A F,f -representation Fact 1.5 As GL 2 (A F,f -representations (over C we have an isomorphism S (k C = π f. f newform The term newform here only refers to the fact that we do not distinguish between systems of eigenvalues such that the Θ f differ only at finitely many p. Each π f can be defined over L(f. To be precise, we sometimes write π f,l(f. Then we have by definition π f,l(f L(f C = π f. We do not need the following for the sequel but list it nevertheless: We call two Hecke eigenforms f 1 and f 2 Galois conjugate if there exist embeddings ι i : L(f i C such that ι 1 (f 1 = ι 2 (f 2, i.e. ι 1 Θ f1 = ι 2 Θ f2. The GL 2 (A F,f -orbit of ι(f ι : L(f C, ι L = id defines an automorphic representation π L(f over L such that π L(f L C = ι π ι(f with the product running through the ι as above. This yields an isomorphism of GL 2 (A F,f -representations (over L 2 Main Theorem S (k L = We now come to Galois representations. f newform up to Galois conjugacy π L(f. We fix algebraic closures Q and Q p for all p and we consider F as a subfield of Q (i.e. we fix an embedding and F p as a subfield of Q p (also by fixing an embedding for every prime p of F. We choose embeddings whose restriction to F is equal to ι p : Q Q p F F p Q p via the natural (resp. fixed embeddings. Note that the choice of ι p corresponds to the choice of a prime ideal for each finite extension F M Q which are compatible with intersection. We also obtain an embedding of absolute Galois groups Gal(Q p /F p Gal(Q/F, σ ι 1 p σ ι p. Note that this definition makes sense, since Q/F is a normal extension. If we have two such embeddings ι p1 and ι p2, then the two embeddings of Galois groups are conjugate by ι p1 ι 1 p 2. 4

5 Call F p the residue field of p. We have the natural exact sequence 0 I p Gal(Q p /F p Gal(F p /F p 0, where I p is the inertia group at p. Of course, we suppose that p divides the rational prime p. The right hand side map is the natural one. By Frob p we denote the arithmetic Frobenius element in Gal(F p /F p, i.e. the one given by x x q with q = #F p. (There is always some confusion about geometric and arithmetic Frobenius elements. I prefer the latter. We also denote by Frob p any preimage in Gal Qp /F p, which is, of course, not well defined. So we have to handle it with care, but we will... Theorem 2.1 Let f S (k C,K 1 (n be a newform corresponding to the L-algebra homomorphism Θ f : T (k L,K 1 (n C. Let Λ be a prime of L(f. Then there is a Galois representation, i.e. a continuous group homomorphism which satisfies: ρ f,λ = ρ : Gal(Q/F GL 2 (L(f Λ It is unramified outside nl with (l = Λ Z, i.e. ρ(i p = 0 for all p with (p, nl = 1; hence, ρ(frob p is well-defined for these p. Tr(ρ(Frob p = Θ f (T p for all (p, nl. det(ρ(frob p = Θ f (R p Nm(p for all (p, nl. This theorem is due to many people, in particular Carayol, Blasius, Rogawski and Taylor. I think it is proved in the above generality, but I have not checked it. In this lecture we will need the additional assumption (if [F : Q] is even that π f is discrete series at v. I won t explain what this means. The theorem is, in fact, more precise. The restriction of ρ to Gal(Q p /F p can be described at all places p, not only the unramified ones. This can be formulated in terms of Weil-Deligne representations (see the seminar a year ago. For the places above l, this is the result proved in Saito s article. 3 Quaternionic automorphic forms and epresentations Notation 3.1 (Second part B, the quaternion algebra (unique up to isomorphism over F which is split at τ 1 and ramified at τ 2,...,τ n (and v, if n is even. I.e. we have with H the Hamiltonian quaternion algebra. G = Res F/Q B, the Weil restriction. B Q R = Mat 2 (R H H }{{} n 1 copies 5

6 L/Q, a Galois number field containing F and splitting B, i.e. Note that Recall the Shimura curve G(A f = p B F L = Mat 2 (L. p p,p v of level K. It has a model M F over F. We let GL 2 (F p (B F F v. M K (C = G(Q\ ( X G(A f /K M(C = G(Q\ ( X G(A f. Kay defined several objects. We shall only list them here (maybe, we even slightly change them, but will not recall the precise definitions. P (k K,L, a constructible sheaf of L-vector spaces on M K(C. P (k L, a sheaf of L-vector spaces on M(C. P (k λ, an étale sheaf of L λ-vector spaces on M F for some maximal ideal λ O F such that V (k K, a locally free O M K (C-module of rank 1. V (k, a locally free O M(C -module of rank 1. Let W (k K H i (M(C, P (k L L L λ = H i et (M F Q, P (k λ. := V(k K O MK (C Ω1 M K (C. Let W (k := V (k OM(C Ω 1 M(C. The principal result from Kay s talk is the following theorem. Theorem 3.2 (Analog of Eichler-Shimura There is an isomorphism: H 1 (M(C, P (k L L C = H 0 (M(C, W (k H 0 (M(C, W (k. A similar result holds at finite level K. Definition 3.3 We call S (k C,K = H0 (M K (C, W (k K the space of quaternionic automorphic forms of level K and multi-weight (k. Analogously, we let (taking lim K S (k C = H0 (M(C, W (k. This definition is unsatisfactory, it should be made explict. But that is impossible during this talk, for time reasons (also time reasons during preparation... I think/hope that one will find a description similar to that of adelic Hilbert modular forms. 6

7 Hecke operators The definition is analogous to the one given for adelic Hilbert modular forms. Let g G(A f. We start on the Shimura curve (on C-points, but also on the model over F ; the map g is right multiplication by g on the second factor: M K M K g 1 Kg where the outer maps are the natural projections. g M gkg 1 K M K, On the quaternionic automorphic forms and, more generally, on cohomology these maps induce an operator T g, as follows: H i (M K, π H i (M K g 1 Kg, g H i (M gkg 1 K, π H i (M K,. Of course, the T g also give maps on lim K, i.e. a G(A f -action, in particular on S (k K and on the H1 (. The naturality of all maps in the above theorem makes the following theorem believable. Theorem 3.4 The map from Theorem 3.2 is compatible with the G(A f -action. Let p v. Since B F F p = GL 2 (F p, it makes sense to define T p and R p as for Hilbert modular forms, i.e. as T g for g = ( π p ( 0 πp or g = 0 π p, respectively. However, we won t need them for the sequel. Automorphic representation of G(A f An automorphic representation of G(A f is an irreducible constituent of S (k C = H0 (M(C, W (k. 4 The Jacquet-Langlands correspondence and hint on the proof Theorem 4.1 (Jacquet-Langlands Let f be an adelic Hilbert newform and π f the associated automorphic GL 2 (A F,f -representation such that π f,v is discrete series (if [K : Q] is even. Then there exists a unique automorphic representation π f of G(A f such that π f,p = π f,p as GL 2 (F p - representations for all p v. Moreover, π f has a model over L(f, denoted by π f,l(f (as has π f, see above. Theorem 4.2 (Multiplicity one There is an isomorphism of G(A f -representations (over C S (k C = H0 (M(C, W (k = f newform, not discrete series at v Corollary 4.3 There is an isomorphism of G(A f -representations (over C H 1 (M(C, P (k L L C = f newform, not discrete series at v π f. (π f π f. 7

8 Hence, H 1 (M(C, P (k L L L(f contains π f,l(f precisely twice, since π f,l(f = π f,l(f and all other constituents are non-isomorphic to π f,l(f. ( Corollary 4.4 (a Hom G(Af π f,l(f, H 1 (M(C, P (k L L L(f = L(f L(f (since π f,l(f is absolutely irreducible. (b Let Λ be a prime ideal of L(f dividing λ. Then ( Hom G(Af π f,l(f L(f L(f Λ, H 1 (M(C, P (k L L L(f Λ = L(fΛ L(f Λ. Now we use the comparison from above (after tensoring with L(f Λ over L λ : H i (M(C, P (k L L L(f Λ = H i et (M F Q, P (k λ L λ L(f Λ. Corollary 4.5 The 2-dimensional L(f Λ -vector space ( Hom G(Af π f,l(f L(f L(f Λ, H 1 et(m F Q, P (k λ L λ L(f Λ = L(fΛ L(f Λ carries a continuous linear Gal(Q/F-action. This is the Gal(Q/F-representation that we are looking for. Unfortunately, we cannot check the claimed properties in this talk. 8