On the Ramanujan Vector Field modulo $p$
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911463163035648 |
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| author | Bianchini, Frederico |
| author_facet | Bianchini, Frederico |
| contents | For every prime $p \geq 5$, we compute the $p$-th power of the Ramanujan vector field that arises from the differential relations discovered by Ramanujan for the Eisenstein series $E_2,E_4$ and $E_6$. Our method results in explicit equations for the $p$-th power and uses classical results of Serre and Swinnerton-Dyer about modular forms modulo $p$. From this, we verify that a general conjecture by Sheperd-Barron and Ekedahl is valid for the Ramanujan vector field. Furthermore, we consider the affine realization of a certain moduli space of elliptic curves where the Ramanujan vector field is defined, and describe - in characteristic $p$ - the locus given by supersingular elliptic curves in two ways: a classical one - using equations for the supersingular polynomial - and a new one as the singular set of some vector fields. Additionally, we prove that the Ramanujan vector field is transversal to this locus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20109 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Ramanujan Vector Field modulo $p$ Bianchini, Frederico Number Theory Algebraic Geometry 11G25 (Primary), 11F33, 11G05 (Secondary) For every prime $p \geq 5$, we compute the $p$-th power of the Ramanujan vector field that arises from the differential relations discovered by Ramanujan for the Eisenstein series $E_2,E_4$ and $E_6$. Our method results in explicit equations for the $p$-th power and uses classical results of Serre and Swinnerton-Dyer about modular forms modulo $p$. From this, we verify that a general conjecture by Sheperd-Barron and Ekedahl is valid for the Ramanujan vector field. Furthermore, we consider the affine realization of a certain moduli space of elliptic curves where the Ramanujan vector field is defined, and describe - in characteristic $p$ - the locus given by supersingular elliptic curves in two ways: a classical one - using equations for the supersingular polynomial - and a new one as the singular set of some vector fields. Additionally, we prove that the Ramanujan vector field is transversal to this locus. |
| title | On the Ramanujan Vector Field modulo $p$ |
| topic | Number Theory Algebraic Geometry 11G25 (Primary), 11F33, 11G05 (Secondary) |
| url | https://arxiv.org/abs/2602.20109 |