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

2025/02/28 by Haidong Li, Ruichen Xu
Mathematics · #math.NT

paper · pdf · doi:10.1016/j.jnt.2026.06.001

published as Journal of Number Theory 293C (2027) pp. 1-41 · 27 pages, comments are welcome! Revised according to the suggestions of the anonymous referee

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

Let K/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 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.

Related

• On the Mordell-Weil rank of certain CM abelian varieties over anticyclotomic towers
• On the Mordell-Weil Ranks of supersingular abelian varieties over\n \ℤ_p²-extensions
• On Anticyclotomic Iwasawa Theory of Hecke Characters at Ordinary Primes
• On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions
• Upper bounds for the growth of Mordell-Weil ranks in pro-p towers of Jacobians