A proof of $p$-adic Gross--Zagier theorem via BDP formula

TL;DR

This paper offers a novel proof of the $p$-adic Gross--Zagier formula for $p$-adic $L$-functions associated with modular forms over imaginary quadratic fields, using a wall-crossing approach centered on BDP formulas.

Contribution

It introduces a new proof method employing wall-crossing and Beilinson--Flach elements, extending results to non-ordinary cases and higher weights.

Findings

Proves the $p$-adic Gross--Zagier formula in broader settings.

Includes cases with $k > 2$ and positive $p$-adic valuation of $a_p(f)$.

Employs a wall-crossing strategy based on BDP formulas.

Abstract

This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements.