A $p$-adic Gross-Zagier formula for twisted triple product $p$-adic $L$-functions attached to finite slope families

TL;DR

This paper generalizes the $p$-adic Gross-Zagier formula to finite slope families of Hilbert modular forms, allowing for inert primes in real quadratic fields, thus broadening the scope of previous results.

Contribution

It extends the $p$-adic Gross-Zagier formula to finite slope families and inert primes, advancing the understanding of $p$-adic $L$-functions in this context.

Findings

Generalized the $p$-adic Gross-Zagier formula to finite slope families.

Allowed the prime $p$ to be inert in the real quadratic field.

Extended previous work to a broader class of Hilbert modular forms.

Abstract

Our main objective in the present paper is to generalise the work of Blanco-Chac\'{o}n and Fornea on the $p$-adic Gross-Zagier formula for twisted triple product $p$-aidc $L$-function. We extend their main result to the case of finite slope families of Hilbert modular forms and also allow the prime $p$ to be inert in the real quadratic field $L$.