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Construction of Modular Galois Representations

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Construction of Modular Galois Representations Johannes Anschütz Michael Füerer Kleine AG: Construction of modular Galois representations Februar , Overview e aim of this Kleine AG is to present the following theorem due to Deligne. P n eorem: Let f 2 Sk(Γ1(N ); χ) be a normalized eigenform of weight k ≥ 2. Let f = anq its Fourier expansion and define Kf := Q(a1; a2; ::) ⊆ C. en Kf is a number field and if λ is a place of Kf dividing the prime ` 2 Z, there exists a continuous two-dimensional representation ρf ;λ : Gal(Q/Q) . GL2(Kf ;λ) which is unramified at all primes p - `N and satisfies − 2 − k−1 det(X ρf ;λ(Frobp)) = X ap X + p χ(p) for each arithmetic Frobenius Frobp at p. In particular, : Tr(ρf ;λ(Frobp)) = ap So basically, for suitable modular forms there is associated a Galois representation with pre- scribed traces. is emphasizes once more the arithmetic significance of modular forms. e proof of the above theorem will heavily use geometry and cohomology of modular curves. e constructions and proofs in the theorem are involved and unfortunately not clearly outlined in our references,¹ so we present a short summary here (see also http: //vbrt.org/writings/l-adic-talk.pdf for another one). As a first and easy reduction it suffices (by taking the contragredient representation) to con- struct a two-dimensional Kf ;λ-representation Vf ;λ of Gal(Q/Q) unramified for p - `N and satisfying k−1 2 det(1 − Fp X j Vf ;λ) = 1 − ap X + p χ(p)X ¹ [Del] deals with the case N = 1 to capture the discriminant and [Con] presents some arguments only in the weight two case. −1 for each geometric Frobenius Fp = Frobp at p. e construction of Vf ;λ proceeds in two steps. e first step uses étale cohomology of a modular curve to introduce a representation V` having our final representation Vf ;λ as a quotient. Namely, let a : Y1(N ) . Spec(Q) be the modular curve over Q of level Γ1(N ) with universal elliptic curve f : E . Y1(N ). Now define V 1 ⊗ Q; k−2 1 Q . 1 ⊗ Q; k−2 1 Q ` := Im(Hc (Y1(N ) Q Sym (R f∗ `)) H (Y1(N ) Q Sym (R f∗ `))) k−2 1 as the image of the cohomology with compact support of the local system Sym (R f∗Q`) in the usual cohomology (talk ). By construction, this Q`-vector space carries a Gal(Q/Q)- action. e second step uses the² Hecke algebra T away from `N . As we will see V` is actually a module³ over the algebra T ⊗Q Q` with T -action defined over Q (talk ). Hence the T -action commutes with the Gal(Q/Q)-action. Moreover, the algebra T surjects onto the field Kf by sending a Hecke operator Tp to ap (talk ). Using the decomposition Y Kf ⊗Q Q` Kf ;λ0 λ0j` we can conclude that T ⊗Q Q` surjects onto Kf ;λ as well. Finally, we set V ⊗ V f ;λ := Kf ;λ T ⊗QQ` ` which turns out to be the representation we were looking for. To prove Deligne’s theorem we have to do a lot of work, namely we first have to investigate the T ⊗Q Q`-module V` and then to relate the action of a geometric Frobenius Fp at p to the action of T . At first, known comparison theorems between étale and singular cohomology imply that V` arises by tensoring the Q-vectorspace V 1 an; k−2 1 anQ . 1 an; k−2 1 anQ := Im(Hc (Y1(N ) Sym (R f∗ )) H (Y1(N ) Sym (R f∗ ))) with Q` (talk ). Here (−)an denotes the analytification functor sending a variety over Q to its C-valued points endowed with the classical topology. Like V`, the space V carries a T -action (talk ), and this is identified with the T -action on V` aer tensoring with Q`. e structure of V as a T -module can now be read off from the T -equivariant Eichler-Shimura isomorphism sh: Sk(Γ1(N )) ⊕ Sk(Γ1(N )) V ⊗Q C using classical theory (talk and ). At this point we understand the T -module V rather explicitly and we can turn our interest towards the Galois action. e action of a geometric Frobenius Fp on V` can fortunately be analyzed by means of the congruence relation − h i k−1 2 − − h i k−1 −1 1 Tp X + p p X = (1 Fp X )(1 p p Fp ) ² Note that there is not “the” Hecke algebra. Hecke operators are acting on various spaces, and different actions may give different Hecke algebras. ³ free of rank 2 over T ⊗Q Q`/ Ann(V`) which should be understood as an equality in EndT (V`)[X ] (talk ). Here hpi 2 EndT (V`) denotes the diamond operator for p acting on V` (talk ). Finally, our last ingredient comes into play. Namely, there exists a (modified) Poincaré duality pairing [·; ·]: V` ⊗ V` . Q` T h i k−1 −1 which is -equivariant and for which Fp and p p Fp are adjoint (talk ). Comparing characteristic polynomials (over T ) of adjoints (over Q`) yields then the equality k−1 2 2 2 det(1 − Tp X + hpip X j Sk(Γ1(N ))) = det(1 − Fp X j V`) from which our theorem follows easily by tensoring with Kf ;λ. For simplicity, we will assume N ≥ 5 throughout. e reason for this is that for N ≤ 4 the moduli functors for elliptic curves with Γ1(N ) structures are not representable by schemes, i.e. the modular curves do not exist. is can be circumvented by using some ad-hoc argu- ments [Con], but we do not want to care about this. Caution! e k we use will always be the weight of our modular form f . In some of the references, modular forms of weight k + 2 are considered, so some k’s in the references may be off by 2 compared to ours. The talks Each talk should take minutes. Talk 1 (Jonathan Zachhuber): Modular curves and modular forms. e first talk will introduce level structures and modular curves over Q together with their analytic counter- parts over C. e spaces V and V` will be defined in this talk. • Define generalized elliptic curves [DR, Déf. .]. Introduce Γ1(N )-level structures and state representability of the moduli functors, e.g. following [DR]. More explicitely: a point of exact order N , also called a Γ1(N )-level structure, on a generalized elliptic curve E . S is a monomorphism Z/N Z . E[N ] /S such that the image meets each irreducible component in each geometric fiber. e functor that associates to a Z[1/N ]-scheme S the set of isomorphism classes of pairs (E; φ), where E is a generalized elliptic curve over S and φ is a point of exact order N on E, is representable by a proper smooth curve X1(N ) over Z[1/N ].⁴ ⁴ A representing object is defined as a Deligne-Mumford stack in [DR, Déf. .., Constr. ..] (puing H = Γ00(N ) in their notation). at it is in fact a scheme follows from the discussion in [DR,§..] together with [KM, Cor. ..] resp. [Con, Lem. ...] and the fact that the only automorphism of a Néron polygon fixing a chosen point of exact order N is the identity. is last statement is easy to see using the description of automorphisms of Néron polygons in [DR,§..]. • Define Y1(N ) as the open subset of X1(N ) parametrizing smooth curves with level structure. an an • Also state that the analytifications Y1(N ) and X1(N ) represent the analogous mod- uli functors for analytic spaces over C (see [Con, m. ...]) and that they can be realized as quotients of the upper half plane H resp. the extended upper half plane ∗ 1 H = H [ P (Q) by the group Γ1(N ) ([Con, m. ...]; note that there X1(N ) is ∗ defined as H /Γ1(N )). 1 an • Define ω = f∗Ω for f : E . X1(N ) being the universal generalized elliptic E/X1(N )an an curve over X1(N ) and define the space of cusp forms Sk(Γ1(N )) as 0 an ⊗(k−2) 1 H (X1(N ) ; ω ⊗ Ω ): X1(N )an State that it recovers the classical definition as functions on the upper half plane if we an view X1(N ) as a quotient of the upper half plane (this is proved in [Con, Lem. ..., m. ...], but need not be presented in detail). • Define e1 an k−2 1 V = H (Y1(N ) ; Sym R f∗Q); V e1 ⊗ Q; k−2 1 Q ` = Hét(Y1(N ) Z[1/N ] Sym R f∗ `) 1 k−2 1 e and the sheaf F = R ae(Sym R f∗Q`). Here, we write H for the image of compactly supported cohomology in usual cohomology. Similarly, by R1aewe mean the image of 1 1 R a! in R a∗. Explain why F is a local system on Spec Z[1/N `] unramified outside N ` with generic fiber V` and why V ⊗Q Q` V` [Con, m. ..., §..]. Since this talk contains mainly definitions and statements, the speaker is encouraged to present some more details or even full proofs of whatever he likes. Talk 2 (Konrad Fischer): Hecke operators and eigenforms. e important action of the Hecke and diamond operators which has not been further described in the outline above will be presented in this talk. Both also have classical definitions as operators acting on modular forms. • State the representability of the Y1(N; p) moduli functor [Con, m.
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