Arithmetic Progressions on Conic Sections

The set $1,25,49$ is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set ${(1,1), (5,25), (7,49)}$asa3-termcollectionofrationalpointsontheparabola$y=x^2$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.