The Galois representation associated to modular forms pt. 2 Erik Visse

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1 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 The website for this seminar could be found at nadimpallisvrn/new.html but seems to have disappeared. The author is in no way an expert on the presented material and any mistake is his own. The talk is based on material from the following sources: (roughly in order of reliance 1 ) [Edi11], [Edi07], [DI95] and [Con07]. The first part of this talk was given by Chloe Martindale on Tuesday May 20. Notes from her lecture are referred to as [Mar15], even though those contain precisely as much original content as this lecture does. However, I have tried keeping her notation for ease of the participants to the seminar. The goal of this lecture is to sketch the existence of Galois representations associated to modular forms as formulated in Theorem Preliminaries Definition 1.1. Recall the Hecke operators and diamond 2 operators where defined last week on modular forms. They can equivalently be defined T n : f ( ) ( ) d m Z d gcd(m,n dk 1 a mn/d 2 q m 1 0 and d : f f [σd ] where σ d 0 d mod N. Definition 1.2. A cuspidal form that is an eigenform for all Hecke operators and diamond operators is called an eigenform. Those with a 1 (f) = 1 are called normalized. Proposition 1.3. The Hecke operators satisfy the relations and T nm = T n T m T p n = T p n 1T p p k 1 p T p n 2. For a modular form f = n 0 a nq n one has T p (f) = a p f. Definition 1.4. K f is the number field generated over Q by all coefficients of f in its Fourier expansion. Fact 1.5. For every prime l, there is equality K λ = K f Q Q l where the product ranges over all primes λ of K f lying over l. λ l 1 Which is mostly coincidental and no particular value of any one treatment over any other is claimed. 2 Chloe used the notation [d] k for d 1

2 Definition 1.6. Let ρ : Gal(Q/Q) GL(V ) be an l-adic Galois representation and let p l be a prime number. Let p be a prime of Q lying over p. Let I p/p be the image of I p in Gal(Q/Q) under the embedding Gal(Q p /Q p ) Gal(Q/Q). We call the representation ρ unramified at p if for all primes p, I p/p lies in the kernel of ρ. Consequently all lifts of a topological generator of Gal(F p /F p ) have equal image in GL(V ). We call a lift of such a topological generator the Frobenius-element in Gal(Q p /Q p ) denoted Frob p. 2 The theorem The following theorem was first proven by Eichler and Shimura for k = 2. Deligne added the cases k > 2 and Deligne together with Serre found the theorem for k = 1. For this lecture we completely ignore the case k = 1 and often only give ideas for k = 2 which one supposedly could generalize to k 2. Theorem 2.1. Let f be a normalized cuspidal eigenform of type (k, N, ε). For every prime number l and any embedding λ of K into Q l, there is a continuous twodimensional representation ρ f,λ : Gal(Q/Q) GL(V λ ) over Q l that is unramified outside Nl and such that for each prime number p Nl the characteristic polynomial of Frob p acting on V λ is det(x Frob p V λ ) = X 2 a p (f)x + ε(p)p k 1. Proposition 2.2. The representations whose existence are guaranteed by the theorem above are unique up to isomorphism. Furthermore, they are irreducible. Proof. Omitted. It is known that noncuspidal eigenforms also lead to Galois representations, but those are always reducible and are therefore determined by their 1-dimensional components and belong to the realm of class field theory. This is why we restrict ourselves to cuspidal eigenforms. There are two essentially equivalent ways of constructing these representations. We will treat both separately in only minor detail. Last week we ve seen two definitions of cusp forms of weight k with respect to a congruence subgroup Γ: modular forms of weight k with respect to Γ that vanish at all cusps, and global sections of the some sheaf Ω 1 X Γ (C) ω (k 2) on the modular curve X Γ (C) provided that Γ acts sufficiently nice. To make sure we are in the sufficiently nice case, from now on we require N 5. For N < 5, one can do similar constructions as below by replacing it by 5N, and then one has to verify the properties of the constructed representation for the prime(s lying over) 5 separately. 2

3 3 Construction using the Jacobian In this section we only consider k = 2. Let J = J 1 (N) be the Jacobian variety of the modular curve X 1 (N)(C). We set W l = Q lim m J(Q)[l m ] where J(Q)[l m ] denotes the l m -torsion of the Q-points of J and the tensor product is taken as Z-modules. Remark that the right-hand factor of W l is also called the Tate module of J(Q) and W l is a Q l -module. We have seen how the Hecke operators act on J in last week s lecture, namely induced from the action on X 1 (N). Similarly, d for d (Z/NZ) acts on X 1 (N) by (E, P ) (E, dp ) and this also induces an action on J. Let T be the subring of End(J) generated by all Hecke operators and diamond operators. Then T Ql = T Q l acts on W l. Lemma 3.1. W l is a free module over T Ql of rank 2 and Gal(Q/Q) acts continuously on it. Proof. We omit the proof, but for clarity do remark that the action of Gal(Q/Q) is induced from the action on J(Q). Choosing a basis for W l over T Ql, we get a representation Gal(Q/Q) GL 2 (T Ql ) which is not quite what we wanted. However, using the map T K f given by T T (f) and remembering Fact 1.5, we can compose maps to get a representation ρ f,λ : Gal(Q/Q) GL 2 (K f,λ ). Proposition 3.2. These representations satisfy the properties mentioned in Theorem 2.1. Proof. We only sketch how one finds the characteristic polynomial. The Jacobian J = J 1 (N) has a model over Z[1/N] that we call J. For p N, we have the Eichler Shimura relation in End ( ) J Fp that reads T p = Frob + p Ver, where Ver is the adjoint of Frob called Verschiebung and satisfies Frob Ver = p = Ver Frob. The endomorphism Frob induces Frob p on F p -points, from which one easily finds the equality Frob 2 p T p Frob p +p p = 0 in End(J Fp ). Hence ρ f,λ (Frob p ) is a zero of the polynomial X 2 T p X+p p. To show that this in fact is the characteristic polynomial (acting on W l ), we must calculate the trace of ρ f,λ (Frob p ). Details can be found in [Con07]. To find the characteristic polynomial over K f,λ, one then applies the composition of maps GL 2 (W l ) GL 2 (K f,λ ) that was also used above and one remarks p f = ε(p)p k 2. 3

4 4 Construction using cohomology The goal of this section is to construct a 2-dimensional representation Gal(Q/Q) GL(W l ) for some W l over T Ql from which the steps above can be taken to get a representation of the desired kind. The construction in this section is of a more abstract nature than that from the previous section and will probably appeal to a different audience. It should be possible to generalize the construction using the Jacobian variety to higher k 2, but another advantage of the construction in the current section does make possible the application that is mentioned in the last section. The W l to be constructed shall come from (étale) cohomology of some (étale0 sheaf on X 1 (N). Note that by X 1 (N) we denote the Z[1/N] model (that we haven t shown existed, but it does) of the projective curve whose C-points give the Riemann surface X 1 (N)(C). We have already seen that Y 1 (N) parametrizes elliptic curves with a given point of order N. In fact, this can be stated as the existence of an elliptic curve p : E Y 1 (N) together with a chosen section of order N. In general an elliptic curve over a scheme S is a morphism of schemes p : E S that is proper, smooth, of relative dimension 1, has geometrically connected fibres and is of genus 1. Let Z E denote the constant sheaf on E(C). Via p, this induces a sheaf R 1 p Z E that is the first derived direct image of Z E under p. Let j : Y 1 (N) X 1 (N) be the open immersion. Then we define a sheaf on X 1 (N) by F k = j Sym k 2 (R 1 p Z E ). Remark 4.1. The appearance of the sheaf F k might look alien, but it is in fact a natural object to consider as the first degree cohomology carries a Hodge decomposition C H 1 (X 1 (N)(C), F k ) = S k (Γ 1 (N)) S k (Γ 1 (N)) where the terms are of type (k 1, 0) and (0, k 1) respectively. In case k = 2, the sheaf F 2 is the constant sheaf Z on X 1 (N) and the Hodge decomposition above basically boils down to the Kodaira Spencer relation from last week s lecture. [Mar15] At this point the prime l comes in. We define F k,l = F k Q l. Alternatively F k,l can defined similarly to F k above, but with the constant sheaf (Q l ) E on the étale site Eét instead of the constant sheaf Z E. Comparing cohomology assures us of H 1 (X 1 (N)(C), F k,l ) = H 1 ét (X 1(N) Q,ét, F k,l ). Now we set W l = H 1 ét (X 1(N) Q,ét, F k,l ) and one can check all the desired properties. Remark 4.2. Instead of taking a complicated (étale) sheaf, one could also opt to take a more complicated scheme. Just as S k (Γ 1 (N)) could be embedded into 4

5 BIBLIOGRAPHY C H 1 (X 1 (N)(C), F k ), one can also let E k 2 be the (k 2)-fold fiber power of E Y 1 (N) and E k 2 a certain projective model of it over X 1 (N) that is smooth over C and then embed S k (Γ 1 (N)) into H 1 (E k 2, C). And indeed, setting also makes our construction work. W l = H k 1 ét (E k 2 Q,ét, Q l) 5 Application The following theorem is a direct consequence of the construction using étale cohomology, due to (Deligne s proof of) the Weil conjectures and therefore attributed to Deligne in the case k 2 and Deligne Serre for k = 1. Theorem 5.1. Let f = n>0 a n(f)q n be a normalized cuspidal eigenform of level N and weight k. For any prime p N, we have a p (f) 2 p (k 1)/2. These bounds where conjectured by Ramanujan and are therefore called Ramanujan bounds. Bibliography [Con07] Brian Conrad. Modular forms, Cohomology, and the Ramanujan Conjecture. still in draft version, [DI95] Diamond and Im. Modular forms and modular curves. In Seminar on Fermat s Last Theorem, volume 17, pages Canadian Mathematical Society Conference Proceedings, [Edi07] Bas Edixhoven. Modular Curves. available on leidenuniv.nl/~edixhovensj/talks/2007/ictp-trieste.pdf, Lecture notes from the Summer School and Conference on Automorphic Forms and Shimura Varieties in Trieste. [Edi11] Bas Edixhoven. Modular curves, modular forms, lattices, Galois representations. In Jean-Marc Couveignes and Bas Edixhoven, editors, Computational Aspects of Modular Forms and Galois Representations, volume 176 of Ann. Math., pages Princeton University Press, [Mar15] Chloe Martindale. The Galois Representation Associated to Modular Forms (Part I). available on TalkLanglands.pdf, Lecture notes from this seminar. 5