Wieferich primes for Drinfeld modules

TL;DR

This paper explores Wieferich primes within Drinfeld modules, revealing a link between their properties and special $L$-values, and studies their statistical distribution across different modules.

Contribution

It generalizes Thakur's theorem for the Carlitz module by connecting Wieferich primes to $L$-values and analyzes their distribution in the context of Drinfeld modules.

Findings

A monic irreducible polynomial is Wieferich if related to $L$-values valuation.

Probability of a place being Wieferich is $q^{-d}$ on average.

Generalization of Thakur's theorem for the Carlitz module.

Abstract

The aim of this paper is to discuss the notion of Wieferich primes in the context of Drinfeld modules. Our main result is a surprising connection between the proprety of a monic irreducible polynomial $\mathfrak p$ to be Wieferich and the $\mathfrak p$-adic valuation of special $L$-values of Drinfeld modules. This generalizes a theorem of Thakur for the Carlitz module.We also study statistical distributions of Wieferich primes, proving in particular that a place of degree $d$ is Wieferich with the expected probability $q^{-d}$ when we average over large enough sets of Drinfeld modules.