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

2025/02/18 by Li, Haidong, Xu, Ruichen
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.12648

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.

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