Alternating group covers of the affine line

TL;DR

This paper proves a special case of Abhyankar's Inertia Conjecture for the alternating group A_{p+2} in characteristic p, demonstrating the existence of Galois covers with prescribed ramification properties over the projective line.

Contribution

It establishes the occurrence of all possible inertia groups for A_{p+2}-Galois covers when p ≡ 2 mod 3 and characterizes the possible upper jumps in ramification for A_{p+s}-covers under certain conditions.

Findings

Proves Abhyankar's Inertia Conjecture for A_{p+2} when p ≡ 2 mod 3.

Shows all possible inertia groups occur for these covers.

Characterizes the upper jumps in ramification for A_{p+s}-Galois covers.

Abstract

We prove Abhyankar's Inertia Conjecture for the alternating group A_{p+2} on p+2 letters when p = 2 mod 3, by showing that every possible inertia group occurs for a (wildly ramified) A_{p+2}-Galois cover of the projective k-line branched only at infinity where k is an algebraically closed field of characteristic p > 0. More generally, when 1 < s < p and gcd(p-1, s+1)=1, we prove that all but finitely many rational numbers which satisfy the obvious necessary conditions occur as the upper jump in the filtration of higher ramification groups of an A_{p+s}-Galois cover of the projective line branched only at infinity.