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arXiv · 2604.13854

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

Kâzım Büyükboduk·Peter Neamti
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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.

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Kâzım Büyükboduk, Peter Neamti. 2026-04-15. A proof of $p$-adic Gross--Zagier theorem via BDP formula. https://arxiv.org/abs/2604.13854

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