On The Inertia Conjecture for Alternating group Covers

TL;DR

This paper proves the wild part of Abhyankar's Inertia Conjecture for certain alternating group covers over algebraically closed fields of odd characteristic, introducing explicit constructions with minimal upper jumps.

Contribution

It establishes the wild part of the Inertia Conjecture for specific alternating groups and constructs explicit étale covers with minimal upper jumps.

Findings

Proves the wild part of Abhyankar's Inertia Conjecture for certain alternating groups.

Constructs explicit étale $A_d$-covers with minimal upper jumps.

Demonstrates the existence of such covers in odd characteristic fields.

Abstract

The wild part of Abhyankar's Inertia Conjecture for a product of certain Alternating groups is shown for any algebraically closed field of odd characteristic. For $d$ a multiple of the characteristic of the base field, a new \'etale $A_d$-cover of the affine line is obtained using an explicit equation and it is shown that it has the minimal possible upper jump.