Euler-Kronecker constants of modular forms: beyond Dirichlet $L$-series
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915044765204480 |
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| author | Charlton, Steven Medvedovsky, Anna Moree, Pieter |
| author_facet | Charlton, Steven Medvedovsky, Anna Moree, Pieter |
| contents | 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 $ΔE_4$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Euler-Kronecker constants of modular forms: beyond Dirichlet $L$-series Charlton, Steven Medvedovsky, Anna Moree, Pieter Number Theory 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 $ΔE_4$ 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. |
| title | Euler-Kronecker constants of modular forms: beyond Dirichlet $L$-series |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.01803 |