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

TL;DR

This paper studies how the Mordell-Weil ranks of certain CM abelian varieties grow over anticyclotomic towers of imaginary quadratic fields, using root number calculations, the Gross-Zagier formula, and Kolyvagin's methods.

Contribution

It provides new results on the rank growth of CM abelian varieties over anticyclotomic extensions, extending previous work to all decomposition types of the prime.

Findings

Rank growth is established for all prime decomposition types.

Root number computations are used to predict rank behavior.

Results connect L-function vanishing with Mordell-Weil rank increases.

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.