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arXiv · 2412.11588

Wieferich primes for Drinfeld modules

Xavier Caruso·Quentin Gazda·Alexis Lucas
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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.

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Xavier Caruso, Quentin Gazda, Alexis Lucas. 2024-12-16. Wieferich primes for Drinfeld modules. https://arxiv.org/abs/2412.11588

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