Tamely ramified covers of the projective line with alternating and symmetric monodromy

Bell, Renee; Booher, Jeremy; Chen, William Y.; Liu, Yuan

doi:10.2140/ant.2022.16.393

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Tamely ramified covers of the projective line with alternating and symmetric monodromy

Renee Bell, Jeremy Booher, William Y. Chen and Yuan Liu

Vol. 16 (2022), No. 2, 393–446

DOI: 10.2140/ant.2022.16.393

Abstract

Let $k$ be an algebraically closed field of characteristic $p$ and $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite étale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p \geq 5$ , there are families of tamely ramified covers with monodromy the symmetric group $S_{n}$ or alternating group $A_{n}$ for infinitely many $n$ . These covers come from the moduli spaces of elliptic curves with ${PSL}_{2} (𝔽_{ℓ})$ -structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder about Markoff triples modulo $ℓ$ .

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Keywords

finite fields, tame fundamental group, Markoff triples, tamely ramified covers, characteristic $p$, covers of curves

Mathematical Subject Classification

Primary: 11G20, 14H30

Milestones

Received: 11 August 2020

Revised: 28 April 2021

Accepted: 13 June 2021

Published: 27 April 2022

Authors

Renee Bell
Université Paris-Sud, Orsay France	Department of Mathematics University of Pennsylvania Philadelphia, PA United States

Jeremy Booher
School of Mathematics and Statistics University of Canterbury Christchurch New Zealand

William Y. Chen
Department of Mathematics Columbia University New York, NY United States

Yuan Liu
Department of Mathematics University of Michigan Ann Arbor, MI United States

Vol. 16, No. 2, 2022