Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjecture
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909407214829568 |
|---|---|
| author | Logan, Adam |
| author_facet | Logan, Adam |
| contents | The modularity of an elliptic curve $E/\mathbb Q$ can be expressed either as an analytic statement that the $L$-function is the Mellin transform of a modular form, or as a geometric statement that $E$ is a quotient of a modular curve $X_0(N)$. For elliptic curves over number fields these notions diverge; a conjecture of Hamahata asserts that for every elliptic curve $E$ over a totally real number field there is a correspondence between a Hilbert modular variety and the product of the conjugates of $E$. In this paper we prove the conjecture by explicit computation for many cases where $E$ is defined over a real quadratic field and the geometric genus of the Hilbert modular variety is $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08269 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjecture Logan, Adam Number Theory Algebraic Geometry 14G35 (Primary), 14Q10, 11G05, 11F41, 14Q25 (Secondary) The modularity of an elliptic curve $E/\mathbb Q$ can be expressed either as an analytic statement that the $L$-function is the Mellin transform of a modular form, or as a geometric statement that $E$ is a quotient of a modular curve $X_0(N)$. For elliptic curves over number fields these notions diverge; a conjecture of Hamahata asserts that for every elliptic curve $E$ over a totally real number field there is a correspondence between a Hilbert modular variety and the product of the conjugates of $E$. In this paper we prove the conjecture by explicit computation for many cases where $E$ is defined over a real quadratic field and the geometric genus of the Hilbert modular variety is $1$. |
| title | Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjecture |
| topic | Number Theory Algebraic Geometry 14G35 (Primary), 14Q10, 11G05, 11F41, 14Q25 (Secondary) |
| url | https://arxiv.org/abs/2411.08269 |