Galois deformation spaces with a sparsity of automorphic points

Abstract

Let k/F p denote a finite field. For any split connected reductive group G/W(k) and a CM number field F, we deform nearly ordinary Galois representations ρ : Gal(F/F) → G(k) to analytic families X ρ of Galois representations F → G(Q p ) lifting ρ such that the set of points of X ρ which are geometric (in the sense of the Fontaine-Mazur conjecture) with parallel Hodge-Tate weights is contained in an analytic subvariety of X ρ with positive codimension. Thus, the set of points in X ρ which could (conjecturally) be associated to automorphic forms is sparse. This generalizes a result of Calegari and Mazur for F/Q quadratic imaginary and G = GL 2 . The sparsity of automorphic points for a CM field F contrasts with the situation when F is a totally real field, where automorphic points are often provably dense.