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In this work we prove congruences between special values of elliptic curveswith CM that seem to play a central role in the analytic side of thenon-commutative Iwasawa theory. These congruences are the analogue for ellipticcurves 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 withCM, or what amounts to the same, the critical values of Gr\"{o}ssencharacters,can be expressed as values of Hilbert-Eisenstein series at CM points. Webelieve that our strategy can be generalized to provide congruences for a largeclass of $L$-values.

Non-abelian congruences between special values of $L$-functions of elliptic curves; the CM case