Even Galois Representations and the Cohomology of Gl(2,z) Avner Ash and Darrin Doud

In this paper a Galois representation will be a continuous representation ρ : GQ → GL(n,F) where F is either a topological field of characteristic 0 or a finite field. When the characteristic of F is not two, we say that ρ is odd if the image of complex conjugation is conjugate to a diagonal matrix with alternating 1’s and −1’s on the diagonal. If F has characteristic two, every Galois representation is considered to be odd. When n = 2, ρ is odd, and F is a finite field, Serre’s conjecture [13] (now a theorem of Khare and Wintenberger [10, 11]) states that ρ is attached to a modular form that is an eigenform of the Hecke operators. This means that the characteristic polynomial of the image of a Frobenius element at an unramified prime ` under ρ equals a certain polynomial created from the eigenvalues of the Hecke operators at `. Other papers [2, 3, 9] conjecture a similar attachment for n ≥ 2, with modular forms replaced by elements of arithmetic cohomology groups. Work of Scholze [12] proves that any eigenclass of the Hecke operators in the cohomology of a congruence subgroup of SL(n,Z) with coefficients in a finite-dimensional admissible module M over a finite field F has an attached Galois representation. For a field F of characteristic 0, the analogous theorem was already proven in [8] by Harris, Lan, Taylor and Thorne. Caraiani and Le Hung [5] showed that the representation guaranteed by Scholze’s theorem must be odd. (“Admissible” means that if F has characteristic 0 then M is an algebraic representation, and if F has positive characteristic, then the matrices used to define the Hecke operators act on M via reduction modulo some fixed integer. ) In this paper, we attach certain even Galois representations to eigenclasses in arithmetic cohomology groups. The details of our main result may be seen in Theorem 11.1, at the end of the paper. Following [5] we know that we will need to use a non-admissible, infinite dimensional coefficient module for the cohomology. We also have to be careful with the exact definition of “attachment”, which we now explain. Let f be a modular form of weight k ≥ 0 on the upper half plane, with level Γ1(N) and nebentype θ, and suppose that f is an eigenform for the Hecke operators T` and T`,` for all ` N . Denote the eigenvalue of T` by a`, and the eigenvalue of T`,` by A`. When k ≥ 2, and f is holomorphic, there is a Galois representation ρ such that for all ` N , det(I − ρ(Frob`)X) = 1− a`X + `A`X, where (in this case), A` is easily seen to be equal to ` k−2θ(`).