Translator Disclaimer
1 June 2021 Arithmetic representations of fundamental groups, II: Finiteness
Daniel Litt
Author Affiliations +
Duke Math. J. 170(8): 1851-1897 (1 June 2021). DOI: 10.1215/00127094-2020-0086

Abstract

Let X be a smooth curve over a finitely generated field k, and let be a prime different from the characteristic of k. We analyze the dynamics of the Galois action on the deformation rings of mod 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 C, then the (typically infinite) set of representations of π1(Xan) into GLn(Q), which come from geometry, has no limit points. As a corollary, we deduce that if L is a finite extension of Q, then the set of representations of π1(Xan) into GLn(L), which arise from geometry, is finite.

Citation

Download Citation

Daniel Litt. "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

Information

Received: 15 April 2019; Revised: 31 October 2020; Published: 1 June 2021
First available in Project Euclid: 18 January 2021

Digital Object Identifier: 10.1215/00127094-2020-0086

Subjects:
Primary: 14F35
Secondary: 11G99

Rights: Copyright © 2021 Duke University Press

JOURNAL ARTICLE
47 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.170 • No. 8 • 1 June 2021
Back to Top