Vol. 16, No. 2, 2022

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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
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 5, there are families of tamely ramified covers with monodromy the symmetric group Sn or alternating group An for infinitely many n. These covers come from the moduli spaces of elliptic curves with PSL 2(𝔽)-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder about Markoff triples modulo .

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