The p-adic Gross-Zagier formula on Shimura curves

TL;DR

This paper establishes a general p-adic Gross-Zagier formula on Shimura curves, linking p-adic heights of Heegner points to derivatives of p-adic L-functions, extending previous results and exploring applications in Iwasawa theory.

Contribution

It introduces a new formula for p-adic heights of Heegner points on Shimura curves, generalizing prior work and constructing interpolating functions in anticyclotomic variables.

Findings

Derived a formula relating p-adic heights to derivatives of p-adic L-functions.

Constructed analytic functions interpolating Heegner points in anticyclotomic variables.

Connected the formula to implications in p-adic BSD conjecture in anticyclotomic families.

Abstract

We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a Rankin-Selberg $p$-adic $L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author, to the context of the work of Yuan-Zhang-Zhang on the archimedean Gross-Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is $+1$ rather than $-1$, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross-Zagier formula implies one divisibility in a $p$-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.