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 K1​-group of the completed localized Iwasawa
algebra.Comment: 59 page