Let be a simple CM abelian variety over a CM field , a rational prime. Suppose that has potentially ordinary reduction above and is self-dual with root number . Under some further conditions, we prove the generic non-vanishing of (cyclotomic) -adic heights on along anticyclotomic -extensions of . This provides evidence towards Schneider’s conjecture on the non-vanishing of -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 -adic -functions and a Gross–Zagier formula relating the latter to families of rational points on .
Soient une variété abélienne CM simple sur un corps CM , un premier rationnel. On suppose que a une réduction potentiellement ordinaire au dessus de et est auto-duale avec signe . Sous quelques hypothèses supplementaires, on montre la non-annulation générique des hauteurs -adiques (cyclotomiques) sur le long de -extensions anticyclotomiques de . Cela confirme partiellement la conjecture de Schneider sur la non-annulation des hauteurs -adiques. Pour les courbes elliptiques CM sur , 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 -adiques de Katz, et sur une formule de Gross–Zagier qui les relie à des familles de points rationnels sur .
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Keywords: Keywords: $p$-adic heights, Katz $p$-adic $L$-functions, CM abelian varieties
Mot 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 = {http://archive.numdam.org/articles/10.5802/aif.3381/} }
TY - JOUR AU - Burungale, Ashay A. AU - Disegni, Daniel TI - On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions JO - Annales de l'Institut Fourier PY - 2020 SP - 2077 EP - 2101 VL - 70 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3381/ DO - 10.5802/aif.3381 LA - en ID - AIF_2020__70_5_2077_0 ER -
%0 Journal Article %A Burungale, Ashay A. %A Disegni, Daniel %T On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions %J Annales de l'Institut Fourier %D 2020 %P 2077-2101 %V 70 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3381/ %R 10.5802/aif.3381 %G en %F AIF_2020__70_5_2077_0
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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