On Bounds and Diophantine Properties of Elliptic Curves

TL;DR

This paper investigates the solutions of Mordell equations, providing explicit bounds, classifying equations with a specific number of solutions, and improving lower bounds on integral solutions by analyzing elliptic curves and their ranks.

Contribution

It classifies Mordell equations with a given number of solutions and establishes explicit bounds, enhancing understanding of elliptic curves and their Diophantine properties.

Findings

Classified all Mordell equations with exactly |k| solutions.

Established explicit bounds for families of elliptic curves.

Improved lower bounds for the number of integral solutions using high-rank curves.

Abstract

Mordell equations are celebrated equations within number theory and are named after Louis Mordell, an American-born British mathematician, known for his pioneering research in number theory. In this paper, we discover all Mordell equations of the form $y^2 = x^3 + k$, where $k \in \mathbb Z$, with exactly $|k|$ integral solutions. We also discover explicit bounds for Mordell equations, parameterized families of elliptic curves and twists on elliptic curves. Using the connection between Mordell curves and binary cubic forms, we improve the lower bound for the number of integral solutions of a Mordell curve by looking at a pair of curves with unusually high rank.