CM congruence and trivial zeros of the Katz $p$-adic $L$-functions for CM fields

TL;DR

This paper explores the trivial zeros of Katz $p$-adic $L$-functions in CM fields, proving their existence and deriving a derivative formula using CM congruence and $p$-adic limit formulas.

Contribution

It establishes the existence of trivial zeros for Katz $p$-adic $L$-functions in general CM fields and derives a first derivative formula at these zeros under Leopoldt hypothesis.

Findings

Existence of trivial zeros for Katz $p$-adic $L$-functions in CM fields

First derivative formula at trivial zeros under Leopoldt hypothesis

Connection between CM congruence and trivial zeros

Abstract

The aim of this paper is to investigate the trivial zeros of the Katz $p$-adic $L$-functions by the CM congruence. We prove the existence of trivial zeros of the Katz $p$-adic $L$-functions for general CM fields and establish a first derivative formula of the cyclotomic $p$-adic $L$-functions at trivial zeros under some Leopoldt hypothesis. The crucial ingredients in our proof are a special case of $p$-adic Kronecker limit formula for CM fields and a leading term formula of anticyclotomic $p$-adic $L$-functions at trivial zeros via the explicit congruences between CM and non-CM Hida families of Hilbert cusp forms.