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arXiv · 2412.01803

Euler-Kronecker constants of modular forms: beyond Dirichlet $L$-series

Steven Charlton·Anna Medvedovsky·Pieter Moree
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Abstract

The Euler-Kronecker constants related to congruences of Fourier coefficients of modular forms that have been computed so far, involve logarithmic derivatives of Dirichlet $L$-series as most complicated functions (to the best of our knowledge). However, generically the more complicated Artin $L$-series will make their appearance. Here we work out some simple examples involving an Artin $L$-series related to an ${\mathfrak S}_3$, respectively~${\mathfrak S}_4$ extension. These examples are related to a mod-2 congruence for $X_0(11)$, respectively a mod-59 congruence for $\Delta E_4$ conjectured by Serre and Swinnerton-Dyer and proved by Haberland. The latter example solves a problem posed by Ciolan, Languasco and the third author in 2023.

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Steven Charlton, Anna Medvedovsky, Pieter Moree. 2024-12-02. Euler-Kronecker constants of modular forms: beyond Dirichlet $L$-series. https://arxiv.org/abs/2412.01803

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