Abhyankar's Inertia Conjecture for Some Sporadic Groups

TL;DR

This paper verifies Abhyankar's Inertia Conjecture for specific sporadic groups by demonstrating the realizability of all possible inertia groups and most ramification invariants in Galois covers of the affine line.

Contribution

It confirms the conjecture for certain sporadic groups and shows that nearly all ramification invariants are realizable for these groups, especially for M11.

Findings

All possible inertia groups occur for some G-Galois covers.

All but eight ramification invariants are realizable for M11-Galois covers.

Most sporadic groups satisfy the conjecture in particular characteristics.

Abstract

This paper verifies Abhyankar's Inertia Conjecture for certain sporadic groups $G$ in particular characteristics by showing that all possible inertia groups occur for $G$-Galois covers of the affine line. For a larger set of sporadic groups, all but finitely many possible ramification invariants are shown to occur. In particular, we prove that all but eight of the possible ramification invariants are realizable for $\text{M}_{11}$-Galois covers of the affine line.