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- W2890218790 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." @default.
- W2890218790 created "2018-09-27" @default.
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- W2890218790 date "2018-09-10" @default.
- W2890218790 modified "2023-09-27" @default.
- W2890218790 title "Arithmetic representations of fundamental groups II: finiteness" @default.
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