Arithmetic Progressions on Conic Sections

TL;DR

This paper generalizes the concept of arithmetic progressions of rational points on conic sections, linking them to elliptic curves with rational 4-torsion points, expanding understanding of rational point configurations.

Contribution

It introduces a framework for 3-term arithmetic progressions on arbitrary conics via linear rational maps, connecting these to rational points on specific elliptic curves.

Findings

Established a general method for arithmetic progressions on conics

Connected progressions to rational points on elliptic curves with 4-torsion

Provided insights into the structure of rational points on conics and elliptic curves

Abstract

The set ${1, 25, 49}$ is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set ${(1,1), (5,25), (7,49)}$ as a 3-term collection of rational points on the parabola $y=x^2$ whose $y$-coordinates form an arithmetic progression. In this exposition, we provide a generalization to 3-term arithmetic progressions on arbitrary conic sections $\mathcal C$ with respect to a linear rational map $\ell: \mathcal C \to \mathbb P^1$. We explain how this construction is related to rational points on the universal elliptic curve $Y^2 + 4XY + 4kY = X^3 + kX^2$ classifying those curves possessing a rational 4-torsion point.