Tate-Shafarevich groups of elliptic curves with nontrivial 2-torsion subgroups

TL;DR

This paper proves that over any number field, there exists an elliptic curve with a nontrivial 2-torsion subgroup in its Shafarevich-Tate group, highlighting new instances of such structures.

Contribution

It establishes the existence of elliptic curves with nontrivial 2-torsion in their Shafarevich-Tate groups over arbitrary number fields, a novel result in the field.

Findings

Existence of elliptic curves with nontrivial 2-torsion in Shafarevich-Tate groups over any number field

Construction methods for such elliptic curves

Implications for the structure of Shafarevich-Tate groups

Abstract

For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup.