• DOI:10.1215/00127094-2020-0086
• Corpus ID: 119133934

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 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.