$p$-adic Gross-Zagier Formula for Heegner Points on Shimura Curves over Totally Real Fields

TL;DR

This paper generalizes Perrin-Riou's p-adic Gross-Zagier formula to Shimura curves over totally real fields, relating derivatives of p-adic L-functions to p-adic heights of Heegner divisors.

Contribution

It extends the p-adic Gross-Zagier formula to Shimura curves over totally real fields, including new constructions of p-adic L-functions and height pairings.

Findings

Constructed a p-adic L-function interpolating special values.

Established the relation between the derivative of the p-adic L-function and Heegner divisor heights.

Showed the contribution of places dividing p is zero, simplifying the formula.

Abstract

The main result of this text is a generalization of Perrin-Riou's p-adic Gross-Zagier formula to the case of Shimura curves over totally real fields. Let $F$ be a totally real field. Let $f$ be a Hilbert modular form over $F$ of parallel weight $2$, which is a new form and is ordinary at $p$. Let $E$ be a totally imaginary quadratic extension of $F$ of discriminant prime to $p$ and to the conductor of $f$. We may construct a $p$-adic $L$ function that interpolates special values of the complex $L$ functions associated to $f$, $E$ and finite order Hecke characters of $E$. The $p$-adic Gross-Zagier formula relates the central derivative of this $p$-adic $L$ function to the $p$-adic height of a Heegner divisor on a certain Shimura curve. The strategy of the proof is close to that of the original work of Perrin-Riou. In the analytic part, we construct the analytic kernel via adelic computations; in the geometric part, we decompose the geometric kernel into two parts: places outside $p$ and places dividing $p$. For places outside $p$, the $p$-adic heights are essentially intersection numbers and are computed in works of S. Zhang, and it turns out that this part is closely related to the analytic kernel. For places dividing $p$, we use the method in the work of J. Nekov\'a\v{r} to show that the contribution of this part is zero.