Arithmetic representations of fundamental groups II: finiteness

Abstract

Let $X$ be a smooth curve over a finitely generated field $k$, and let $\ell$ be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation rings of mod $\ell$ representations of the geometric fundamental group of $X$. Using this analysis, we prove analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero. For example, we show that if $X$ is a normal, connected variety over $\mathbb{C}$, the (typically infinite) set of representations of ${\pi }_1(X^{\text{an}})$ into $G{L}_n(\overline{{\mathbb{Q}}_{\ell }})$, which come from geometry, has no limit points. As a corollary, we deduce that if $L$ is a finite extension of ${\mathbb{Q}}_{\ell }$, then the set of representations of ${\pi }_1(X^{\text{an}})$ into $G{L}_n(L)$, which arise from geometry, is finite.

30 pages, comments welcome!