On the transfer congruence between \(p\)-adic Hecke \(L\)-functions. (English) Zbl 1333.11111
Author’s 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.
Reviewer: Andrzej Dąbrowski (Szczecin)
MSC:
11S40 | Zeta functions and \(L\)-functions |
11R23 | Iwasawa theory |
11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |