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

TL;DR

This paper establishes congruences between special values of CM elliptic curves' L-functions, extending known results to the CM case and linking them to non-commutative Iwasawa theory.

Contribution

It proves new congruences for L-values of CM elliptic curves, using Hilbert-Eisenstein series, and suggests potential generalizations to broader classes of L-values.

Findings

Proved congruences between special L-values of CM elliptic curves.

Expressed critical values as Hilbert-Eisenstein series at CM points.

Indicated possible extensions to other L-value classes.

Abstract

In this work we prove congruences between special values 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 analogue 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 Gr\"{o}ssencharacters, 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.