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There are classical congruences between the class number of an imaginaryquadratic field and a Bernoulli number or an Euler number. Under the BSDconjecture, Onishi obtained an elliptic generalization of these congruences,which gives congruences between the order of the Tate-Shafarevich group ofcertain elliptic curves with CM by the Gauss integers ring and Mordell-Weilrank 0, and a coefficient of power series expansion of an elliptic functionassociated to Gauss integers ring. In this paper, we provide Onishi's typecongruences for the Eisenstein integers case.

Congruence Relations Connecting Tate-Shafarevich Groups with Bernoulli-Hurwitz Numbers by Elliptic Gauss Sums in Eisenstein Case