Arithmetic Progressions on Conic Sections

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 $Y^2+4XY+4kY=X^3+k{X}^2$ classifying those curves possessing a rational 4-torsion point.

17 pages, submitted for publication