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 ${ℤ}_{p}$-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 $B$.

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 ${\mathbf{Z}}_{p}$-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 $\mathbf{Q}$, 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 $B$.

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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