Galois deformation spaces with a sparsity of automorphic points

Abstract

Let $k/{\mathbb{F}}_p$ denote a finite field. For any split connected reductive group $G/W(k)$ and certain CM number fields $F$, we deform certain Galois representations $\overline{\rho }:Gal(\overline{F}/F)\to G(k)$ to continuous families $X_{\overline{\rho }}$ of Galois representations $Gal(\overline{F}/F)\to G(\overline{{\mathbb{Q}}_p})$ lifting $\overline{\rho }$ such that the space of points of $X_{\overline{\rho }}$ which are geometric (in the sense of the Fontaine-Mazur conjecture) with parallel Hodge-Tate weights has positive codimension in $X_{\overline{\rho }}$. Thus the set of points in $X_{\overline{\rho }}$ which could (conjecturally) be associated to automorphic forms is sparse. This generalizes a result of Calegari and Mazur for $F/\mathbb{Q}$ quadratic imaginary and $G=G{L}_2$. The sparsity of automorphic points for $F$ a CM field contrasts with the situation when $F$ is a totally real field, where automorphic points are often provably dense.

39 pages