# The $p$-adic Gross-Zagier formula on Shimura curves, II: nonsplit primes

## TL;DR

This paper extends the $p$-adic Gross-Zagier formula to nonsplit primes, relating $p$-adic heights of Heegner points to derivatives of $p$-adic $L$-functions for Hilbert-modular abelian varieties, removing previous splitting assumptions.

## Contribution

It generalizes the $p$-adic Gross-Zagier formula to cases where $p$ is nonsplit, broadening its applicability to Hilbert-modular abelian varieties.

## Key findings

- Proves the $p$-adic Gross-Zagier formula for nonsplit primes.
- Establishes the relation between $p$-adic heights and derivatives of $p$-adic $L$-functions in a more general setting.
- Removes the splitting assumption previously required in the formula.

## Abstract

The formula of the title relates $p$-adic heights of Heegner points and derivatives of $p$-adic $L$-functions. It was originally proved by Perrin-Riou for $p$-ordinary elliptic curves over the rationals, under the assumption that $p$ splits in the relevant quadratic extension. We remove this assumption, in the more general setting of Hilbert-modular abelian varieties.