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

TL;DR

This paper establishes new congruences connecting Tate-Shafarevich groups and Bernoulli-Hurwitz numbers via elliptic Gauss sums in the Eisenstein case, extending classical results under the BSD conjecture.

Contribution

It generalizes Onishi's elliptic congruences to the Eisenstein integers, linking Tate-Shafarevich groups with special number-theoretic coefficients.

Findings

Proves congruences between Tate-Shafarevich groups and Bernoulli-Hurwitz numbers.

Extends classical congruences to Eisenstein integers case.

Provides explicit formulas connecting elliptic functions and algebraic groups.

Abstract

There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.