Integrals on $p$-adic upper half planes and Hida families over totally real fields

TL;DR

This paper generalizes $p$-adic integrals on upper half planes and derives a second derivative formula for two-variable $p$-adic $L$-functions associated with Hilbert modular forms over totally real fields.

Contribution

It extends the theory of $p$-adic integrals to non-$ extbf{Z}$-valued measures and establishes a new formula for the second derivative of $p$-adic $L$-functions for abelian varieties of ${ m GL}(2)$-type.

Findings

Generalized $p$-adic integrals for broader measures

Proved a second derivative formula for $p$-adic $L$-functions

Connected integrals to derivatives of $p$-adic $L$-functions

Abstract

Bertolini-Darmon and Mok proved a formula of the second derivative of the two-variable $p$-adic $L$-function of a modular elliptic curve over a totally real field along the Hida family in terms of the image of a global point by some $p$-adic logarithm map. The theory of $p$-adic indefinite integrals and $p$-adic multiplicative integrals on $p$-adic upper half planes plays an important role in their work. In this paper, we generalize these integrals for $p$-adic measures which are not necessarily $\mathbf{Z}$-valued, and prove a formula of the second derivative of the two-variable $p$-adic $L$-function of an abelian variety of ${\rm GL}(2)$-type associated to a Hilbert modular form of weight 2.