Abhyankar's Affine Arithmetic Conjecture for the Symmetric and Alternating Groups

TL;DR

This paper proves Abhyankar's conjecture for symmetric and alternating groups over finite fields of odd characteristic, showing the existence of specific Galois extensions ramified only at infinity.

Contribution

It confirms Abhyankar's conjecture for symmetric and alternating groups in the case of finite fields with odd characteristic, expanding the understanding of Galois extensions.

Findings

Existence of Galois extensions with prescribed Galois group over function fields.

Ramification only at infinity for these extensions.

Confirmation of Abhyankar's conjecture in new cases.

Abstract

We prove that for any prime $p>2$, $q=p^\nu$ a power of $p$, $n\ge p$ and $G=S_n$ or $G=A_n$ (symmetric or alternating group) there exists a Galois extension $K/\mathbb F_q(T)$ ramified only over $\infty$ with $\mathrm{Gal}(K/\mathbb F_q(T))=G$. This confirms a conjecture of Abhyankar for the case of symmetric and alternating groups over finite fields of odd characteristic.