On genus one curves violating the local-global principle

TL;DR

This paper constructs explicit examples of elliptic curves over certain number fields that have nontrivial Shafarevich-Tate groups, demonstrating violations of the local-global principle.

Contribution

It provides explicit constructions of elliptic curves with nontrivial Shafarevich-Tate groups over number fields not containing , showcasing violations of the local-global principle.

Findings

Existence of elliptic curves with nontrivial Shafarevich-Tate groups over specified fields

Explicit construction method for such curves

Demonstration of violations of the local-global principle

Abstract

For any number field not containing $\QQ(i),$ we give an explicit construction to prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group is nontrivial.