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arXiv · 2204.04425

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

Akihiro Goto
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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.

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Akihiro Goto. 2022-04-09. Congruence Relations Connecting Tate-Shafarevich Groups with Bernoulli-Hurwitz Numbers by Elliptic Gauss Sums in Eisenstein Case. https://arxiv.org/abs/2204.04425

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