On the Mordell-Weil rank of certain CM abelian varieties over anticyclotomic towers
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912236008636416 |
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| author | Li, Haidong Xu, Ruichen |
| author_facet | Li, Haidong Xu, Ruichen |
| contents | Let $K/\mathbb{Q}$ be an imaginary quadratic extension, and let $p$ be an odd prime. In this paper, we investigate the growth of Mordell-Weil ranks of CM abelian varieties associated with Hecke characters over $K$ of infinite type $(1, 0)$ along the $\mathbb{Z}_p$-anticyclotomic tower of $K$. Our results cover all decomposition types of $p$ in $K$.
The analytic aspect of our proof is based on our computations of the local and global root numbers of Hecke characters, together with a recent generalization by H. Jia of D. Rohrlich's result concerning the relation between the vanishing orders of Hecke $L$-functions and their root numbers. The arithmetic conclusions then follow from the Gross-Zagier formula and the Kolyvagin machinery. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12648 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Mordell-Weil rank of certain CM abelian varieties over anticyclotomic towers Li, Haidong Xu, Ruichen Number Theory Let $K/\mathbb{Q}$ be an imaginary quadratic extension, and let $p$ be an odd prime. In this paper, we investigate the growth of Mordell-Weil ranks of CM abelian varieties associated with Hecke characters over $K$ of infinite type $(1, 0)$ along the $\mathbb{Z}_p$-anticyclotomic tower of $K$. Our results cover all decomposition types of $p$ in $K$. The analytic aspect of our proof is based on our computations of the local and global root numbers of Hecke characters, together with a recent generalization by H. Jia of D. Rohrlich's result concerning the relation between the vanishing orders of Hecke $L$-functions and their root numbers. The arithmetic conclusions then follow from the Gross-Zagier formula and the Kolyvagin machinery. |
| title | On the Mordell-Weil rank of certain CM abelian varieties over anticyclotomic towers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2502.12648 |