Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups: Distribution Part

TL;DR

This paper investigates the distribution of certain congruent elliptic curves with specific prime factor conditions, establishing independence properties and providing results on the structure of their Shafarevich-Tate groups.

Contribution

It proves an independence of residue symbol property for these curves and determines their distribution with respect to the 2-primary part of the Shafarevich-Tate group.

Findings

Distribution of rank zero curves with specific Shafarevich-Tate group structure

Lower bounds on the number of curves with larger Shafarevich-Tate groups

Establishment of independence of residue symbol property

Abstract

We study the distribution of a subclass congruent elliptic curve $E^{(n)}: y^2=x^3-n^2x$, where $n$ is congruent to $1\pmod 8$ with all prime factors congruent to $1\pmod 4$. We prove an independence of residue symbol property. Consequently we get the distribution of rank zero such $E^{(n)}$ with $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z /2\mathbb Z\big)^2$. We also obtain a lower bound of the number of such $E^{(n)}$ with rank zero and $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z /2\mathbb Z\big)^{4}$.