Density of automorphicpoints in deformation rings of polarized global Galois representations

Hellmann, Eugen

Abstract

Conjecturally, the Galois representations that are attached to essen- tially selfdual regular algebraic cuspidal automorphic representations are Zariski- dense in a polarized Galois deformation ring. We prove new results in this direc- tion in the context of automorphic forms on definite unitary groups over totally real fields. This generalizes the infinite fern argument of Gouvea-Mazur and Chenevier, and relies on the construction of non-classical p-adic automorphic forms, and the computation of the tangent space of the space of trianguline Ga- lois representations. This boils down to a surprising statement about the linear envelope of intersections of Borel subalgebras