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 $G = GL_2$. 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.