$p$-adic heights of generalized Heegner cycles

TL;DR

This paper establishes a relation between the $p$-adic heights of generalized Heegner cycles and the derivatives of associated $p$-adic $L$-functions, extending previous formulas to broader settings involving weight and character variations.

Contribution

It generalizes the $p$-adic Gross-Zagier formula to cases with non-zero infinity type $ ext{(} ext{ell} eq 0 ext{)}$, broadening the scope of height and $L$-function relations.

Findings

Relates $p$-adic heights to derivatives of $p$-adic $L$-functions.

Extends Gross-Zagier formula to new weight and character settings.

Provides explicit formulas connecting cycles and $L$-functions.

Abstract

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 Nekov\'a\u{r} (in higher weight).