Biquadrates and Elliptic Curves

TL;DR

This paper investigates the rank of elliptic curves defined by specific biquadratic forms, establishing lower bounds based on representations of numbers as sums of two fourth powers, with implications for the structure of these curves over rational numbers.

Contribution

It proves that the rank of certain elliptic curves is at least 2, and at least 4 when the parameter can be expressed in two distinct ways as a sum of two fourth powers.

Findings

Rank at least 2 for the elliptic curve y^2= x^3-Nx with N=m^4+n^4.

Rank at least 4 when N has two different representations as sum of two fourth powers.

Demonstrates a link between number representations and elliptic curve rank.

Abstract

The elliptic curve y^2= x^3-Nx where N=m^4+n^4 has rank at least 2 over Q(m,n). When N can be written in two different ways as sum of two fourth powers, then we prove that the rank is at least 4.