Artin Conjecture for p-adic Galois Representations of Function Fields

TL;DR

This paper investigates the Artin conjecture for p-adic Galois representations over function fields, demonstrating its validity in a specific p-adic region but not universally, and establishing non-rationality of certain L-functions.

Contribution

It provides the first evidence of the conjecture's partial validity in positive characteristic and proves the non-rationality of geometric unit root L-functions.

Findings

Artin conjecture holds in a non-trivial p-adic disk

Conjecture fails in the full p-adic plane

Geometric unit root L-functions are non-rational

Abstract

For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.