Arithmetic representations of fundamental groups, II: Finiteness

  title={Arithmetic representations of fundamental groups, II: Finiteness},
  author={Daniel Litt},
  journal={Duke Mathematical Journal},
  • 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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