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

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.