Let X be a smooth curve over a finitely generated field k, and let ℓ be a prime different from the characteristic of k. We analyze the dynamics of the Galois action on the deformation rings of mod ℓ representations of the geometric fundamental group of X. Using this analysis, we prove several finiteness results for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey–Mazur conjecture for function fields in characteristic 0. For example, we show that if X is a normal, connected variety over , then the (typically infinite) set of representations of into , which come from geometry, has no limit points. As a corollary, we deduce that if L is a finite extension of , then the set of representations of into , which arise from geometry, is finite.
"Arithmetic representations of fundamental groups, II: Finiteness." Duke Math. J. 170 (8) 1851 - 1897, 1 June 2021. https://doi.org/10.1215/00127094-2020-0086