# Arithmetic representations of fundamental groups, II: Finiteness

@article{Litt2021ArithmeticRO,
title={Arithmetic representations of fundamental groups, II: Finiteness},
author={Daniel Litt},
journal={Duke Mathematical Journal},
year={2021}
}
• Daniel Litt
• Published 10 September 2018
• Mathematics
• Duke Mathematical Journal
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…
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