On the ranks of elliptic curves with isogenies

TL;DR

This paper investigates the ranks of elliptic curves over  with specific algebraic structures, especially those with isogenies, using both existing and new computational techniques to find high-rank examples.

Contribution

It introduces a new computational method based on Mazur's observation to efficiently find high-rank elliptic curves with isogenies, complementing existing techniques.

Findings

Identified elliptic curves over  with large ranks and specific isogenies.

Developed a more feasible computational approach for large-height elliptic curves.

Provided data supporting the unboundedness of ranks in certain families.

Abstract

In recent years, the question of whether the ranks of elliptic curves defined over $\mathbb{Q}$ are unbounded has garnered much attention. One can create refined versions of this question by restricting one's attention to elliptic curves over $\mathbb{Q}$ with a certain algebraic structure, e.g., with a rational point of a given order. In an attempt to gather data about such questions, we look for examples of elliptic curves over $\mathbb{Q}$ with an $n$-isogeny and rank as large as possible. To do this, we use existing techniques due to Rogers, Rubin, Silverberg, and Nagao and develop a new technique (based on an observation made by Mazur) that is more computationally feasible when the naive heights of the elliptic curves are large.