On a Theorem of N. Katz and Bases in Irreducible Representations

TL;DR

This paper explores the extension properties of irreducible Galois representations over local fields and relates these to bases in representations of division algebra multiplicative groups, expanding on N. Katz's theorem.

Contribution

It provides a detailed analysis of the number of extensions beyond the unique one, linking it to bases in irreducible representations of division algebra groups.

Findings

Characterization of the number of non-special extensions

Connection between extension counts and bases in division algebra representations

Insights into the structure of irreducible Galois representations

Abstract

N. Katz has shown that any irreducible representation of the Galois group of F_q((t)) has unique extension to a special representation of the Galois group of k(t) unramified outside 0 and infinity and tamely ramified at infinity. In this paper we analyze the number of not necessarily special such extensions and relate this question to a description of bases in irreducible representations of multiplicative groups of division algebras.