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

There are classical congruences between the class number of the imaginary quadratic field ℚ(√−p) for a rational prime p > 3 and a Bernoulli number or an Euler number. Under the BSD conjecture on the 2-parts of the leading term, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with complex multiplication by the Gaussian integers ℤ[√−1] and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to ℤ[√−1]. In this paper, we provide Onishi's type congruences for the Eisenstein integers ℤ[-1+√−3/2]