标题： 关于某些CM阿贝尔簇在反循环塔中的Mordell-Weil秩 显示英文标题

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

摘要： 设$K/\mathbb{Q}$为一个虚二次扩张，令$p$为一个奇素数。 在本文中，我们研究与Hecke字符相关的CM阿贝尔簇在$K$的无限类型$(1, 0)$沿着$\mathbb{Z}_p$-反循环塔$K$的Mordell-Weil秩的增长情况。 我们的结果涵盖了$p$在$K$中的所有分解类型。 我们证明的分析方面基于我们对Hecke字符的局部和全局根数的计算，以及H. Jia对D. Rohrlich关于Hecke$L$-函数的消失阶与其根数之间关系的结果的最近推广。 然后，算术结论来自于Gross-Zagier公式和Kolyvagin方法。 显示英文摘要

摘要： 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.