On the transfer congruence between $p$-adic Hecke $L$-functions

TL;DR

This paper establishes a transfer congruence for $p$-adic Hecke $L$-functions in CM fields, extending classical results to a non-abelian setting and applying it to special values and non-commutative $p$-adic $L$-functions.

Contribution

It introduces a non-abelian transfer congruence for $p$-adic Hecke $L$-functions and constructs a non-commutative $p$-adic $L$-function in Iwasawa theory.

Findings

Proved transfer congruence between $p$-adic Hecke $L$-functions for CM fields.

Established explicit congruences for special values of $L$-functions of CM elliptic curves.

Constructed a non-commutative $p$-adic $L$-function in algebraic $K_1$-group.

Abstract

We prove the transfer congruence between $p$-adic Hecke $L$-functions for CM fields over cyclotomic extensions, which is a non-abelian generalization of the Kummer's congruence. The ingredients of the proof include the comparison between Hilbert modular varieties, the $q$-expansion principle, and some modification of Hsieh's Whittaker model for Katz' Eisenstein series. As a first application, we prove explicit congruence between special values of Hasse-Weil $L$-function of a CM elliptic curve twisted by Artin representations. As a second application, we prove the existence of a non-commutative $p$-adic $L$-function in the algebraic $K_1$-group of the completed localized Iwasawa algebra.