An analogue of a conjecture of Rasmussen and Tamagawa for abelian varieties over function fields
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929404763963392 |
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| author | Melistas, Mentzelos |
| author_facet | Melistas, Mentzelos |
| contents | Let $L$ be a number field and let $\ell$ be a prime number. Rasmussen and Tamagawa conjectured, in a precise sense, that abelian varieties whose field of definition of the $\ell$-power torsion is both a pro-$\ell$ extension of $L(μ_\ell)$ and unramified away from $\ell$ are quite rare. In this paper, we formulate an analogue of the Rasmussen--Tamagawa conjecture for non-isotrivial abelian varieties defined over function fields. We provide a proof of our analogue in the case of elliptic curves. In higher dimensions, when the base field is a subfield of the complex numbers, we show that our conjecture is a consequence of the uniform geometric torsion conjecture. Finally, using a theorem of Bakker and Tsimerman we also prove our conjecture unconditionally for abelian varieties with real multiplication. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11100 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An analogue of a conjecture of Rasmussen and Tamagawa for abelian varieties over function fields Melistas, Mentzelos Number Theory Let $L$ be a number field and let $\ell$ be a prime number. Rasmussen and Tamagawa conjectured, in a precise sense, that abelian varieties whose field of definition of the $\ell$-power torsion is both a pro-$\ell$ extension of $L(μ_\ell)$ and unramified away from $\ell$ are quite rare. In this paper, we formulate an analogue of the Rasmussen--Tamagawa conjecture for non-isotrivial abelian varieties defined over function fields. We provide a proof of our analogue in the case of elliptic curves. In higher dimensions, when the base field is a subfield of the complex numbers, we show that our conjecture is a consequence of the uniform geometric torsion conjecture. Finally, using a theorem of Bakker and Tsimerman we also prove our conjecture unconditionally for abelian varieties with real multiplication. |
| title | An analogue of a conjecture of Rasmussen and Tamagawa for abelian varieties over function fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2310.11100 |