Tamely Ramified Covers of the Projective Line with Alternating and Symmetric Monodromy

TL;DR

This paper classifies which finite groups can occur as monodromy groups of tamely ramified covers of the projective line minus three points over an algebraically closed field, revealing infinite families with symmetric and alternating groups for primes p ≥ 5.

Contribution

It demonstrates the existence of infinite families of tamely ramified covers with symmetric and alternating monodromy groups, linked to moduli spaces of elliptic curves and recent work on Markoff triples.

Findings

Infinite families with monodromy groups S_n and A_n for infinitely many n.

Construction of covers from moduli spaces of elliptic curves with PSL_2(ℱ_ℓ)-structure.

Utilization of recent advances in Markoff triples modulo ℓ.

Abstract

Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite \'{e}tale 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 $p\ge 5$, 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 $PSL_2(\mathbb{F}_\ell)$-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder, about Markoff triples modulo $\ell$.