Arithmetic representations of fundamental groups II: finiteness

TL;DR

This paper proves finiteness results for Galois representations of fundamental groups of algebraic varieties over various fields, extending classical conjectures and showing that geometric representations form finite or discrete sets.

Contribution

It establishes analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields and a weak Frey-Mazur conjecture in characteristic zero, using deformation ring analysis.

Findings

Finiteness of geometric Galois representations over algebraically closed fields.

No accumulation points for geometric representations in complex varieties.

Finite set of such representations over finite extensions of ll

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 $GL_n(\overline{\mathbb{Q}_\ell})$, 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 $\pi_1(X^{\text{an}})$ into $GL_n(L)$, which arise from geometry, is finite.