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# Cambridge Journal of Mathematics

## Volume 3 (2015)

### Number 3

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

Pages: 355 – 438

DOI: http://dx.doi.org/10.4310/CJM.2015.v3.n3.a3

#### Author

#### 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.