Nontrivial torsion in the Tate--Shafarevich group of elliptic curves via visibility and twists

TL;DR

This paper investigates the existence of nontrivial torsion in the Tate--Shafarevich group of elliptic curves over $Q$ using visibility and twists, providing explicit examples for $ ext{3}$-torsion.

Contribution

It demonstrates how visibility and quadratic twists can produce nontrivial torsion in $ ext{Sha}$, with explicit examples for the prime 3.

Findings

Existence of nontrivial $ ext{ell}$-torsion in $ ext{Sha}(E^D/Q)$ for certain twists.

Pairs of elliptic curves with identical BSD invariants and isomorphic $ ext{Sha}$ groups with nontrivial 3-primary parts.

Explicit construction of elliptic curves with prescribed Tate--Shafarevich group properties.

Abstract

Let $\ell$ be an odd prime. We study the visibility theorem for certain elliptic curves over $\mathbb{Q}$ with additive reduction at $\ell$, and deduce the existence of nontrivial $\ell$-torsion in $\Sha(E^D/\mathbb{Q})$ for suitable quadratic twists $E^D$. As an application for $\ell=3$, we exhibit pairs of non-isomorphic elliptic curves with the same BSD invariants, Kodaira symbols, and minimal discriminants, whose Tate--Shafarevich groups are isomorphic and have nontrivial $3$-primary parts.