Wieferich primes for Drinfeld modules
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916525628194816 |
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| author | Caruso, Xavier Gazda, Quentin Lucas, Alexis |
| author_facet | Caruso, Xavier Gazda, Quentin Lucas, Alexis |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wieferich primes for Drinfeld modules Caruso, Xavier Gazda, Quentin Lucas, Alexis Number Theory 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. |
| title | Wieferich primes for Drinfeld modules |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.11588 |