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

TL;DR

This work extends the $p$-adic Gross--Zagier formula for twisted triple product $p$-adic $L$-functions to finite slope Hilbert modular form families, including the inert case for $p$ in a real quadratic field. It constructs a three-variable $p$-adic $L$-function $\\mathscr{L}_p(\\omega_{\\mathbf{g}}, \\omega_{\\mathbf{f}})$ over weight spaces using nearly overconvergent Hilbert modular forms and the Gauss--Manin connection, and proves an interpolation formula incorporating explicit Euler factors. The core results express special values of $\\mathscr{L}_p$ at balanced classical weights as syntomic Abel--Jacobi images of generalized Hirzebruch--Zagier cycles, computed via finite polynomial cohomology and $p$-adic differential operators. The paper also develops the split and inert local computations, enabling applications to Selmer-group dimension results and automorphic-motivic $L$-function comparisons in the finite slope setting, with potential implications for BSD-type conjectures in non-ordinary contexts.

Abstract

Our main objective in the present paper is to generalise the work of Blanco-Chacón 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 $L$.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• proof