ARITHMETIC GEOMETRY LEARNING SEMINAR: SUMMER 2018: GALOIS REPRESENTATIONS ATTACHED TO MODULAR FORMS 1. Overview 1.1. The global Langlands correspondence conjecture for GLn. The global Langlands con- jecture for GLn predicts the existence of a universal group LQ, the Langlands group of Q, whose complex representations parametrize cuspidal automorphic representations of GLn(AQ). A little more precisely, there's expected to be a bijection 8 9 8 9 < (isomorphism classes of) = < (isomorphism classes of) = cuspidal automorphic ! irreducible n-dimensional ; : ; : ; representations of GLn(AQ) complex representations of LQ that is compatible with associated data (L-functions and -factors), and also compatible with the local Langlands correspondence. A precise definition of LQ is not yet clear, though there is at least one candidate [1]. In any case it's expected to be a very large group admitting surjective maps to certain other canonical groups. There's expected to be a surjection LQ ! GQ = Gal(Q=Q). Then any continuous complex representation of the Galois group GQ should give rise to a representation of LQ by composition with the map. Now complex continuous representations of GQ have finite image, since GQ is pro-finite and GLn(C) has \no small subgroups". Such representations correspond to cuspidal automorphic representations of \Galois type". Let MQ denote the (conjectural) motivic Galois group associated to the category of pure motives over Q (with coefficients extended to C). Zero-dimensional (Artin) motives span a full Tannakian subcategory, with associated Galois group GQ. Therefore there should be a surjective map MQ ! GQ. In fact, one expects surjective homomorphisms LQ !MQ ! GQ: Suppose that M is an irreducible motive over Q. Then conjecturally, M corresponds to a repre- sentation of MQ. One expects to get a representation of LQ by pullback via LQ ! MQ, and hence a cuspidal automorphism representation of GLn(AQ) for some n. In other words to each irreducible motive M should correspond a cuspidal automorphic representation πM . These are expected to be \algebraic" representations. See further below for the precise definition of \algebraic" and \Galois type". Since MQ is itself mysterious, one has little hope of associating reprsentations of MQ to auto- morphic representations of GLn. However, there are ´etalecohomology groups attached to motives over Q, and those provide l-adic representations of GQ. The Fontaine-Mazur conjecture predicts exactly the l-adic representations that are expected to arise in this way: those that are unramified at almost all primes p, and de Rham at l. Then restricting the scope of the general Langlands conjecture to algebraic cuspidal automorphic representations, one may formulate something precise: there should be a bijection 8 (isomorphism classes of) 9 8 (isomorphism classes of) 9 > > < = <> irreducible n-dimensional => algebraic cuspidal automorphic ! ; l-adic representations of G ; : representations of GL ( ) ; > Q > n AQ :> unramified at almost all p and de Rham at l ;> Date: Summer 2018. 1 ARITHMETIC GEOMETRY LEARNING SEMINAR:SUMMER 2018: GALOIS REPRESENTATIONS ATTACHED TO MODULAR FORMS2 preserving L and -factors, and compatible with local Langlands. When n = 2, the Fontaine-Mazur conjecture is a theorem, and the restricted form of global Langlands above is almost known. Let us now provide precise definitions of \algebraic" and \Galois type" as promised. In general a cuspidal automorphic representation π of GLn factors into a restricted tensor product ∼ O π = π1 ⊗ πp; p prime where π1 is a smooth representation of GLn(R) and πp a representation of GL2(Qp). Being \alge- braic" is a condition on π1. Under local Langlands for GLn(R), π1 corresponds to a represesenta- tion ρ : WR ! GLn(C); × × where the local Weil group WR is the non-trivial extension of Gal(C=R) by C . Considering C as the real points of the algebraic group S = ResC=R(Gm;C), π1 is said to be algebraic if the restriction × of ρ to C = S(R) ⊂ S(C), comes from a morphism of complex algebraic groups SC ! GLn;C. It is × of Galois type if ρ has finite image, which is equivalent to ρ(C ) = 1. The n = 1 case of the global Langlands correspondence essentially amounts to global class field theory. Over Q, this is equivalent to the Kronecker-Weber theorem that says any abelian extension of Q is a subfield of a cyclotomic field. The case n = 2 is the only other case one in which significant progress has been made. 1.2. Algebraic automorphic representations of GL2. When n = 2, the algebraic π1 fall into three types: (a) Discrete series represesentations (b) Limit of discrete series 1 (c) The principal series representation with parameter s = 2 ( () λ = 0). In terms of modular forms, such π correspond to normalized new cuspidal Hecke eigenforms f : H ! C which are: (a) Holomorphic of weight ≥ 2 (b) Holomorphic of weight 1 1 (c) Non-holomorphic Maass forms of weight 1, with Laplacian eigenvalue 4 . The algebraic global Langlands correspondence is known in cases (a) and (b), and still open in case (c). Cases (b) and (c) are of \Galois type", corresponding to Artin motives, or equivalently continuous complex represenations of GQ (with finite image). Let c 2 Gal(Q=Q) denote complex conjugation with respect to any embedding Q ,! C. A two- dimensional Galois representation ρ : Gal(Q=Q) ! GL2(F ) is called odd if det(ρ(c)) = −1, and even if det(ρ(c)) = 1. (a) Holomorphic cusp forms of weight ≥ 2. For (a) the correspondence takes the form 8 9 8 9 < holomorphic f : H ! C = < odd irreducible continuous two-dimensional = normalized new cusp Hecke eigenforms ! l-adic representations of Gal(Q=Q); : : of weight ≥ 2 ; : with distinct Hodge-Tate weights. ; • The map −! (automorphic to Galois) was established by Deligne [2], using the ´etalecoho- mology of modular curves, and building on the work of Eichler [4] and Shimura [5]. • Injectivity of the map −! is a consequence of strong multiplicity one for GL2, due to Jacquet-Langlands. • Surjection of −! is a special case of the Fontaine-Mazur conjecture. For GL2 it is a theorem of Emerton and Kisin, thus completing the correspondence. ARITHMETIC GEOMETRY LEARNING SEMINAR:SUMMER 2018: GALOIS REPRESENTATIONS ATTACHED TO MODULAR FORMS3 The case of weight 2 is special. A construction of Shimura associates to a normalized new cusp eigenform f of weight 2 an abelian variety Af which is an elliptic curve if the Fourier coefficients of f are rational. In that case the corresponding l-adic Galois representation occurs in the Tate module of the elliptic curve. The Shimura-Taniyama conjecture, now the Modularity Theorem, says f 7! Ef is a surjection onto all elliptic curves over Q, up to isogeny. In cases (b) and (c) the automorphic representations are of \Galois type", hence correspond to complex representations of GQ (with finite image). They come from 0-dimensional motives over Q, i.e. number fields. (b) Holomorphic cusp forms of weight 1. In case (b), the correspondence is also known, and takes the form 8 holomorphic 9 < = odd continuous irreducible two-dimensional normalized new cusp Hecke eigenforms ! : complex representations of Gal( = ); : of weight 1 ; Q Q • The map −! was constructed by Deligne-Serre [3], relying on the construction of Deligne for weight ≥ 2. • Injectivity again follows from strong multiplicity one for GL2. • Surjectivity follows from the Artin conjecture, in this case known due to work of Khare- Winterbenger. 1 (c) Non-holomorphic Maass cusp forms of eigenvalue 4 . Here the picture is far from clear. The conjectural correspondence is: 8 non-holomorphic (Maass) 9 < = even continuous irreducible two-dimensional normalized new cusp Hecke eigenforms ! : : 1 ; complex representations of Gal(Q=Q); of weight 1, and Laplacian eigenvalue 4 • No general construction −! is known • Galois reps with solvable image are known to be automorphic. This uses, among other things, the solvable base change theorem of Langlands-Tunnell. • When the image of the representation is non-solvable, it must be a central extension of A5. This is the so-called icosahedral case, where the correspondence is still a conjecture. 2. Seminar Plan The plan is for Zavosh to present the first three topics below. Then if there is continued interest, we may divide up the paper of Bruggeman-Zagier-Lewis (at least the first two chapters) among the willing participants, who will take turns presenting. The eventual goal is to understand which methods from the holomorphic cases (a) and (b) have any hope of being transferred to the non-holomorphic case (c). 2.1. The Eichler-Shimura isomorphism. Let Γ ⊂ PSL2(R) be an arithmetic subgroup (or more generally a Fuchsian group). Let n ≥ 0 be even, and Sn+2(Γ) denote the set of weight n + 2 holomorphic cusp forms for Γ. The Eichler-Shimura isomorphism is ∼ 1 Sn+2(Γ) −! Hpar(Γ;Vn); where Vn denotes the representation of SL2(R) with highest weight n. The right hand side is Eichler cohomology, defined as the kernel of 1 Y 1 H (Γ;Vn) ! H (Γs;Vn); s2Σ 1 1 where Σ ⊂ R[f1g is the set of cusps of Γ, Γs is the stabilizer of s in Γ, and H (Γ;Vn) ! H (Γs;Vn) is induced by restriction of cocycles. ARITHMETIC GEOMETRY LEARNING SEMINAR:SUMMER 2018: GALOIS REPRESENTATIONS ATTACHED TO MODULAR FORMS4 The representation Vn has an integral structure, i.e. a Γ-invariant lattice Mn. One may de- 1 fine integral cohomology groups Hpar(Γ;Mn) in an analogous manner, which turn out to be full- 1 rank lattices in Hpar(Γ;Vn). Under the Eichler-Shimura isomorphism they correspond to lattices Dm(Γ) ⊂ Sm(Γ). The rational vector space Dm(Γ) ⊗ Q is then closed under the action of Hecke operators, which implies the eigenvalues of normalized Hecke eigenforms are algebraic numbers.