# Non-trivial Shafarevich-Tate Groups of Elliptic Curves

## TL;DR

This paper characterizes specific quadratic twists of elliptic curves with particular Mordell-Weil and Shafarevich-Tate group structures, and studies their distribution, advancing understanding of their arithmetic properties.

## Contribution

It provides a detailed characterization of quadratic twists with isomorphic Mordell-Weil and Shafarevich-Tate groups, and analyzes their distribution patterns.

## Key findings

- Identifies conditions for quadratic twists with isomorphic groups.
- Establishes distribution results for these elliptic curves.
- Enhances understanding of the structure of Shafarevich-Tate groups.

## Abstract

We characterize quadratic twists of $y^2=x(x-a^2)(x+b^2)$ with Mordell-Weil groups and $2$-primary part of Shafarevich-Tate groups being isomorphic to $(\mathb Z/2\mathbb Z)^2$ under certain conditions. We also obtain the distribution result of these elliptic curves.