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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.17939 |
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| _version_ | 1866911222718267392 |
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| author | Zhao, Luochen |
| author_facet | Zhao, Luochen |
| contents | Let $E$ be an elliptic curve having CM by the ring of integers of an imaginary quadratic field $K$ in which $p$ splits. Following Lichtenbaum, the Bernoulli--Hurwitz numbers of $E$ (i.e., values of Eisenstein series evaluated at $E$ up to normalization) admit integral representations given by a $p$-adic measure constructed from an elliptic function. We show that the periods of this measure are in fact special values of a family of weight one Eisenstein series at the CM curve $E$ equipped with certain level data, and explicitly relate it to Katz's one-variable $p$-adic Eisenstein measure, whereby we derive period formulas of the Bernoulli--Hurwitz measure attached to any ordinary elliptic curve $\mathcal{E}$ defined over a local field. Moreover, by exploiting the modularity of these periods, and thanks to the existence of abundant weight one Hasse-type invariants, we present a novel approach to the interpolation property of the Bernoulli--Hurwitz $p$-adic zeta functions of the ordinary elliptic curve $\mathcal{E}$, and obtain a $p$-adic Kronecker's first limit formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Bernoulli--Hurwitz periods Zhao, Luochen Number Theory 11F67 (Primary) 11G15, 11S40 (Secondary) Let $E$ be an elliptic curve having CM by the ring of integers of an imaginary quadratic field $K$ in which $p$ splits. Following Lichtenbaum, the Bernoulli--Hurwitz numbers of $E$ (i.e., values of Eisenstein series evaluated at $E$ up to normalization) admit integral representations given by a $p$-adic measure constructed from an elliptic function. We show that the periods of this measure are in fact special values of a family of weight one Eisenstein series at the CM curve $E$ equipped with certain level data, and explicitly relate it to Katz's one-variable $p$-adic Eisenstein measure, whereby we derive period formulas of the Bernoulli--Hurwitz measure attached to any ordinary elliptic curve $\mathcal{E}$ defined over a local field. Moreover, by exploiting the modularity of these periods, and thanks to the existence of abundant weight one Hasse-type invariants, we present a novel approach to the interpolation property of the Bernoulli--Hurwitz $p$-adic zeta functions of the ordinary elliptic curve $\mathcal{E}$, and obtain a $p$-adic Kronecker's first limit formula. |
| title | On the Bernoulli--Hurwitz periods |
| topic | Number Theory 11F67 (Primary) 11G15, 11S40 (Secondary) |
| url | https://arxiv.org/abs/2510.17939 |