Towards the Generalized Purely Wild Inertia Conjecture for product of Alternating and Symmetric Groups

TL;DR

This paper provides new evidence supporting the Purely Wild Inertia Conjecture, demonstrating its validity for products of simple Alternating and certain Symmetric groups in specific characteristics, and extends results towards realizing inertia groups.

Contribution

It proves the conjecture for products of simple Alternating groups in odd characteristics and certain Symmetric groups in characteristic two, advancing understanding of inertia groups in algebraic geometry.

Findings

Proves the conjecture for products of simple Alternating groups in odd characteristics.

Establishes the conjecture for certain Symmetric groups in characteristic two.

Extends results to products of perfect quasi p-groups.

Abstract

We obtain new evidence for the Purely Wild Inertia Conjecture posed by Abhyankar and for its generalization. We show that this generalized conjecture is true for any product of simple Alternating groups in odd characteristics, and for any product of certain Symmetric or Alternating groups in characteristic two. We also obtain important results towards the realization of the inertia groups which can be applied to more general set up. We further show that the Purely Wild Inertia Conjecture is true for any product of perfect quasi $p$-groups (groups generated by their Sylow $p$-subgroups) if the conjecture is established for individual groups.