Eisenstein series modulo $p^2$
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916627464847360 |
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| author | Ahlgren, Scott Hanson, Michael Raum, Martin Richter, Olav K. |
| author_facet | Ahlgren, Scott Hanson, Michael Raum, Martin Richter, Olav K. |
| contents | We study congruences for Eisenstein series on $\mathrm{SL}_2(\mathbb{Z})$ modulo $p^2$, where $p \geq 5$ is prime. It is classically known that all Eisenstein series of weight at least $4$ are determined modulo $p^2$ by those of weight at most $p^2-p+2$. We prove that up to powers of $E_{p-1}$, each such Eisenstein series is in fact determined modulo $p^2$ by a modular form of weight at most $2p-4$. We also determine $E_2$ modulo $p^2$ in terms of a modular form of weight $p+1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_16917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eisenstein series modulo $p^2$ Ahlgren, Scott Hanson, Michael Raum, Martin Richter, Olav K. Number Theory We study congruences for Eisenstein series on $\mathrm{SL}_2(\mathbb{Z})$ modulo $p^2$, where $p \geq 5$ is prime. It is classically known that all Eisenstein series of weight at least $4$ are determined modulo $p^2$ by those of weight at most $p^2-p+2$. We prove that up to powers of $E_{p-1}$, each such Eisenstein series is in fact determined modulo $p^2$ by a modular form of weight at most $2p-4$. We also determine $E_2$ modulo $p^2$ in terms of a modular form of weight $p+1$. |
| title | Eisenstein series modulo $p^2$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.16917 |