A classical family of elliptic curves having rank one and the $2$-primary part of their Tate-Shafarevich group non-trivial

TL;DR

This paper investigates specific elliptic curves defined by cubic equations related to primes congruent to 2 or 5 modulo 9, establishing cases of the Birch-Swinnerton-Dyer conjecture and linking their Selmer groups to class groups, revealing rank one curves with non-trivial Tate-Shafarevich groups.

Contribution

It demonstrates the 3-part of the BSD conjecture for these curves and connects their 2-Selmer groups to class group ranks, providing explicit examples with rank one and non-trivial Tate-Shafarevich groups.

Findings

3-part of BSD conjecture verified for these curves

Link between 2-Selmer groups and class group ranks established

Examples of rank one elliptic curves with non-trivial 2-part of Tate-Shafarevich group

Abstract

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their $2$-Selmer group to the $2$-rank of the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ to obtain some examples of elliptic curves with rank one and non-trivial $2$-part of the Tate-Shafarevich group.