Lower bounds for ranks using Pell equations

TL;DR

This paper investigates the ranks of specific algebraic curves linked to Pell equations, demonstrating many have moderately large ranks and establishing conditions for minimum rank guarantees.

Contribution

It introduces a method to derive curves with guaranteed rank at least three from solutions to Pell equations, expanding understanding of their rank distribution.

Findings

Existence of many curves with large rank

Proof that these curves have rank at least three under mild conditions

Connection between Pell solutions and algebraic curve ranks

Abstract

We examine the ranks of a subfamily of curves in a previous article, which are derived from the existence of solutions to certain Pell equations. We exhibit an abundance of curves of moderately large rank, and prove under mild conditions that these curves have rank at least three.