Arithmetic representations of fundamental groups, II: Finiteness

Abstract

Let X be a smooth curve over a finitely generated field k, and let ℓ be a prime different from the characteristic of k. We analyze the dynamics of the Galois action on the deformation rings of mod ℓ representations of the geometric fundamental group of X. Using this analysis, we prove several finiteness results for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey–Mazur conjecture for function fields in characteristic 0. For example, we show that if X is a normal, connected variety over $\mathbb{C}$, then the (typically infinite) set of representations of ${\mathrm{\pi }}_1(X^{\mathrm{an}})$ into ${\mathrm{GL}}_n(\overline{{\mathbb{Q}}_{\ell }})$, 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 ${\mathrm{\pi }}_1(X^{\mathrm{an}})$ into ${\mathrm{GL}}_n(L)$, which arise from geometry, is finite.