Modularity theorems for abelian surfaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916635017740288 |
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| author | Boxer, George Calegari, Frank Gee, Toby Pilloni, Vincent |
| author_facet | Boxer, George Calegari, Frank Gee, Toby Pilloni, Vincent |
| contents | We prove the modularity of a positive proportion of abelian surfaces over $\mathbf{Q}$. More precisely, we prove the modularity of abelian surfaces which are ordinary at $3$ and are $3$-distinguished, subject to some assumptions on the $3$-torsion representation (a "big image" hypothesis, and a technical hypothesis on the action of a decomposition group at $2$). We employ a 2-3 switch and a new classicality theorem (in the style of Lue Pan) for ordinary $p$-adic Siegel modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modularity theorems for abelian surfaces Boxer, George Calegari, Frank Gee, Toby Pilloni, Vincent Number Theory We prove the modularity of a positive proportion of abelian surfaces over $\mathbf{Q}$. More precisely, we prove the modularity of abelian surfaces which are ordinary at $3$ and are $3$-distinguished, subject to some assumptions on the $3$-torsion representation (a "big image" hypothesis, and a technical hypothesis on the action of a decomposition group at $2$). We employ a 2-3 switch and a new classicality theorem (in the style of Lue Pan) for ordinary $p$-adic Siegel modular forms. |
| title | Modularity theorems for abelian surfaces |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.20645 |