An explicit comparison of anticyclotomic $p$-adic $L$-functions for Hida families

TL;DR

This paper compares different constructions of anticyclotomic $p$-adic $L$-functions for modular forms and Hida families, establishing their relationships through specialization and $p$-adic families.

Contribution

It provides a detailed comparison between two-variable anticyclotomic $p$-adic $L$-functions derived from different approaches, clarifying their connections.

Findings

The central critical twist of the two-variable $p$-adic $L$-function matches the one from the three-variable $p$-adic $L$-function under specialization.

The comparison unifies various constructions of $p$-adic $L$-functions for Hida families.

The results enhance understanding of the relationships between different $p$-adic $L$-function frameworks.

Abstract

The aim of this note is to compare several anticyclotomic $p$-adic $L$-functions for modular forms and $p$-adic families of ordinary modular forms, which have been defined and studied from different perspectives by Skinner-Urban, Hida, Perin-Riou, Bertolini-Darmon, Vatsal, Chida-Hsieh, Longo-Vigni, Castella-Longo and Castella-Kim-Longo. The main result of this paper is a comparison between the central critical twist of the two-variable anticyclotomic $p$-adic $L$-function obtained as specialisation of the three-variable $p$-adic $L$-function of Skinner-Urban and the two-variable $p$-adic $L$-function introduced by one of the authors on collaboration with Vigni by means of $p$-adic families of Gross points.