Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups

TL;DR

This paper investigates a specific subclass of congruent elliptic curves with particular properties of their Shafarevich-Tate groups, providing characterizations for cases with rank zero and certain 2-primary parts.

Contribution

It characterizes congruent elliptic curves with rank zero and specific 2-primary Shafarevich-Tate group structures based on prime factor conditions.

Findings

Identifies conditions for rank zero elliptic curves in the subclass.

Describes the structure of the 2-primary part of Shafarevich-Tate groups.

Provides criteria for the size of the 2-primary Shafarevich-Tate group.

Abstract

We study a subclass of congruent elliptic curves $E^{(n)}: y^2=x^3-n^2x$, where $n$ is a positive integer congruent to $1\pmod 8$ with all prime factors congruent to $1\pmod 4$. We characterize such $E^{(n)}$ with Mordell-Weil rank zero and $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z/2\mathbb Z \big)^2$. We also discuss such $E^{(n)}$ with 2-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z/2\mathbb Z \big)^{2k}$ with $k\ge2$.