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 a CM number field F, we deform nearly ordinary Galois representations $$\overline{\rho }:{{\,\mathrm{Gal}\,}}(\overline{F}/F) \rightarrow G(k)$$ to analytic families $$X_{\overline{\rho }}$$ of Galois representations $$\Gamma _F \rightarrow G(\overline{\mathbb {Q}}_p)$$ lifting $$\overline{\rho }$$ such that the set of points of $$X_{\overline{\rho }}$$ which are geometric (in the sense of the Fontaine–Mazur conjecture) with parallel Hodge–Tate weights is contained in an analytic subvariety of $$X_{\overline{\rho }}$$ with positive codimension. 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 $$G = {{\,\mathrm{GL}\,}}_2$$ . 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.