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Non-Abelian congruences between special values of \(L\)-functions of elliptic curves: the CM case. (English) Zbl 1279.11107

Summary: We prove congruences between special values of \(L\)-functions of elliptic curves with CM that seem to play a central role in the analytic side of the non-commutative Iwasawa theory. These congruences are the analog for elliptic curves with CM of those proved by Kato, Ritter and Weiss for the Tate motive. The proof is based on the fact that the critical values of elliptic curves with CM, or what amounts to the same, the critical values of Grössencharacters, can be expressed as values of Hilbert-Eisenstein series at CM points. We believe that our strategy can be generalized to provide congruences for a large class of \(L\)-values.

MSC:

11R23 Iwasawa theory
11S40 Zeta functions and \(L\)-functions
11G15 Complex multiplication and moduli of abelian varieties
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