$p$-adic heights of generalized Heegner cycles

Shnidman, Ariel

Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1117-1174.

Nous relions les hauteurs $p$-adiques des cycles de Heegner généralisés à la dérivée d’une fonction $L$ $p$-adique attachée à une paire $(f,\chi )$, où $f$ est une forme modulaire ordinaire de poids $2r$ et $\chi$ est un caractère de Hecke non-ramifé de type $(\ell ,0)$, pour $0<\ell <2r$. Ceci généralise la formule de Perrin-Riou (en poids deux) and Nekovář (poids plus élevé).

We relate the $p$-adic heights of generalized Heegner cycles to the derivative of a $p$-adic $L$-function attached to a pair $(f,\chi )$, where $f$ is an ordinary weight $2r$ newform and $\chi$ is an unramified imaginary quadratic Hecke character of infinity type $(\ell ,0)$, with $0<\ell <2r$. This generalizes the $p$-adic Gross-Zagier formula in the case $\ell =0$ due to Perrin-Riou (in weight two) and Nekovář (in higher weight).

Classification : 11G40, 11G18
Keywords: algebraic cycles, modular forms, $p$-adic $L$-functions
Mot clés : cycles algébriques, formes modulaires, fonctions $L$ $p$-adiques

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