Toward Abhyankar's Inertia Conjecture for PSL_2(\ell)

TL;DR

This paper investigates the realization of PSL_2() covers of the projective line with specific ramification properties over fields of characteristic p, advancing understanding related to Abhyankar's Inertia Conjecture.

Contribution

It demonstrates the construction of PSL_2()-covers with diverse inertia groups and large ramification filtrations, providing new evidence towards Abhyankar's Inertia Conjecture.

Findings

Existence of PSL_2()-covers with various inertia groups

Realization of all sufficiently large higher ramification filtrations

Progress towards Abhyankar's Inertia Conjecture

Abstract

For \ell \neq p odd primes, we examine PSL_2(\ell)-covers of the projective line branched at one point over an algebraically closed field of characteristic p, where PSL_2(\ell) has order divisible by p. We show that such covers can be realized with a large variety of inertia groups. Furthermore, for each inertia group realized, we can realize all "sufficiently large" higher ramification filtrations.