The derivative formula of $p$-adic $L$-functions for imaginary quadratic fields at trivial zeros

TL;DR

This paper proves an analogue of the Gross conjecture for Katz p-adic L-functions associated with imaginary quadratic fields, using congruences between CM and non-CM forms and the p-adic Rankin-Selberg method.

Contribution

It introduces a novel approach applying the p-adic Rankin-Selberg method to construct congruent Hida families, extending Gross conjecture results to imaginary quadratic fields.

Findings

Established the p-adic Gross conjecture analogue for imaginary quadratic fields.

Constructed a non-CM Hida family congruent to CM forms at specific specializations.

Provided new tools for studying p-adic L-functions via congruences and modular forms.

Abstract

The rank one Gross conjecture for Deligne-Ribet $p$-adic $L$-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. The purpose of this paper is to prove an analogue of the Gross conjecture for the Katz $p$-adic $L$-functions attached to imaginary quadratic fields via the congruences between CM forms and non-CM forms. The new ingredient is to apply the $p$-adic Rankin-Selberg method to construct a non-CM Hida family which is congruent to a Hida family of CM forms at the $1+\varepsilon$ specialization.