On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions
Let B be a simple CM abelian variety over a CM field E, p a rational prime. Suppose that B has potentially ordinary reduction above p and is self-dual with root number −1. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) p-adic heights on B along anticyclotomic Zp-extensions of E. This provides evidence towards Schneider's conjecture on the non-vanishing of p-adic heights. For CM elliptic curves over Q, the result was previously known as a consequence of works of Bertrand, Gross–Zagier and Rohrlich in the 1980s. Our proof is based on non-vanishing results for Katz p-adic L-functions and a Gross–Zagier formula relating the latter to families of rational points on B.
Additional Information© Association des Annales de l'institut Fourier, 2020, Certains droits réservés. Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France. http://creativecommons.org/licenses/by-nd/3.0/fr/. Reçu le: 2018-11-04; Révisé le: 2019-10-24; Accepté le: 2019-12-20; Publié le: 2021-04-15. We would like to thank D. Bertrand, K. Büyükboduk, F. Castella, M. Chida, H. Darmon, H. Hida, B. Howard, M.-L. Hsieh, M. Kakde, J. Lin, J. Nekovár, D. Prasad, D. Rohrlich, K. Rubin, C. Skinner and Y. Tian for useful conversations, correspondence or suggestions. We are grateful to the referee for thorough comments and suggestions. During part of the writing of this paper, the second author was funded by a public grant from the Fondation Mathématique Jacques Hadamard.
Published - AIF_2020__70_5_2077_0.pdf
Submitted - 1803.09268.pdf