On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions
[Sur la non-annulation des hauteurs p-adiques sur les variétés abéliennes CM, et l’arithmétique des fonctions L p-adiques de Katz]
Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2077-2101.

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.

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.

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DOI : https://doi.org/10.5802/aif.3381
Classification : 11G50,  11G10,  11G40
Mots clés : hauteurs p-adiques, fonctions L p-adiques de Katz, variétés abéliennes CM
@article{AIF_2020__70_5_2077_0,
     author = {Burungale, Ashay A. and Disegni, Daniel},
     title = {On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions},
     journal = {Annales de l'Institut Fourier},
     pages = {2077--2101},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {5},
     year = {2020},
     doi = {10.5802/aif.3381},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3381/}
}
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, Tome 70 (2020) no. 5, pp. 2077-2101. doi : 10.5802/aif.3381. https://aif.centre-mersenne.org/articles/10.5802/aif.3381/

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