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

Haidong Li, Ruichen Xu

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.

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

Abstract

Let be an imaginary quadratic extension, and let be an odd prime. In this paper, we investigate the growth of Mordell-Weil ranks of CM abelian varieties associated with Hecke characters over of infinite type along the -anticyclotomic tower of . Our results cover all decomposition types of in . 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 -functions and their root numbers. The arithmetic conclusions then follow from the Gross-Zagier formula and the Kolyvagin machinery.

Paper Structure

This paper contains 23 sections, 20 theorems, 105 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Previous works
  3. Method of proof
  4. Application I: Finiteness of $A_{\varphi}(K_{\infty}^{\mathrm{ac}})$
  5. Application II: Distribution of vanishing orders among anticyclotomic twists
  6. Notations
  7. Preliminaries
  8. Hecke characters
  9. Hecke characters and their $L$-functions
  10. Galois characters
  11. Class field theory of anticyclotomic $\mathbb{Z}_p$-extensions
  12. Root number computations
  13. Background on local root numbers
  14. Local root numbers
  15. Relative root numbers
  16. ...and 8 more sections

Key Result

Theorem A

Let $A_{\varphi}$ be the abelian variety as described above, and let $d = \dim A_{\varphi}$. Let $p > 2$ be a prime number. Denote $\widetilde{W}(\varphi) := (1 - W(\varphi))/2$. Then for $n \gg 0$:

Theorems & Definitions (42)

  • Theorem A
  • Theorem 1.1: Rohrlich, Jia
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 3.1
  • ...and 32 more