$P$-adic $L$-functions: $t$-modules and Dirichlet-Goss $L$-series

Daniel Krell Calvo

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arXiv · 2606.08085

$P$-adic $L$-functions: $t$-modules and Dirichlet-Goss $L$-series

Daniel Krell Calvo

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Abstract

We prove that the order of vanishing of a natural twist of the $P$-adic Carlitz zeta values at the positive "even" integers is always $1$. We also obtain the same result for $P$-adic Pellarin $L$-series, and for almost all $P$-adic Dirichlet-Goss $L$-series. To do so, we demonstrate a $P$-adic version of a reduced variant of the class formula for Anderson $t$-modules proven by Angl\`es, Ngo Dac and Tavares-Ribeiro in 2020.

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Daniel Krell Calvo. 2026-06-06. $P$-adic $L$-functions: $t$-modules and Dirichlet-Goss $L$-series. https://arxiv.org/abs/2606.08085

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