Joel Bellaiche

On the sign of the Galois representations attached to unitary automorphic forms

Due to the work of many mathematicians, we (almost) know how to attach to a cuspidal cohomological automorphic representation Π for GLn of a CM field a Galois representation of dimension n, provided that Π satisfies a specific symmetry that relates it to unitary groups, namely that it is automorphic (up to a twist) of the contragredient of its conjugate. The attached Galois representation ρ satisfies an analogous symmetry condition, and thus has a sign (like a self dual absolutely irreducible can be orthogonal or symplectic). The aim of this talk is to determine this sign, in all cases (joint work with G. Chenevier), and to give some applications (to Selmer groups, and to questions on the possible reducibility of ρ).