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arXiv · 2502.12648

On the Mordell-Weil rank of certain CM abelian varieties over anticyclotomic towers

Haidong Li·Ruichen Xu
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Abstract

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.

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Haidong Li, Ruichen Xu. 2025-02-18. On the Mordell-Weil rank of certain CM abelian varieties over anticyclotomic towers. https://doi.org/10.1016/j.jnt.2026.06.001

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