On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions
Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2077-2101.

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 p -extensions of E. This provides evidence towards Schneider’s conjecture on the non-vanishing of p-adic heights. For CM elliptic curves over , 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.

Soient B une variété abélienne CM simple sur un corps CM E, p un premier rationnel. On suppose que B a une réduction potentiellement ordinaire au dessus de p et est auto-duale avec signe -1. Sous quelques hypothèses supplementaires, on montre la non-annulation générique des hauteurs p-adiques (cyclotomiques) sur B le long de Z p -extensions anticyclotomiques de E. Cela confirme partiellement la conjecture de Schneider sur la non-annulation des hauteurs p-adiques. Pour les courbes elliptiques CM sur Q, le résultat était déjà connu comme conséquence de travaux de Bertrand, Gross–Zagier et Rohrlich dans les années 80. Notre preuve est basée sur des résultats de non-annulation pour les fonctions L p-adiques de Katz, et sur une formule de Gross–Zagier qui les relie à des familles de points rationnels sur B.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3381
Classification: 11G50,  11G10,  11G40
Keywords: Keywords: p-adic heights, Katz p-adic L-functions, CM abelian varieties
Burungale, Ashay A. 1; Disegni, Daniel 2

1 California Institute of Technology, 1200 E California Blvd, Pasadena CA 91125, (USA)
2 Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 84105, (Israel)
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Burungale, Ashay A.; Disegni, Daniel. On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions. Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2077-2101. doi : 10.5802/aif.3381. http://archive.numdam.org/articles/10.5802/aif.3381/

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