On arithmetic progressions on genus two curves

Abstract

We study arithmetic progression in the $x$-coordinate of rational points on genus two curves. As we know, there are two models for the curve $C$ of genus two: $C: y^2=f_{5}(x)$ or $C: y^2=f_{6}(x)$, where $f_{5}, f_{6}\in\Q[x]$, $\operatorname{deg}f_{5}=5, \operatorname{deg}f_{6}=6$ and the polynomials $f_{5}, f_{6}$ do not have multiple roots. First we prove that there exists an infinite family of curves of the form $y^2=f(x)$, where $f\in\Q[x]$ and $\operatorname{deg}f=5$ each containing 11 points in arithmetic progression. We also present an example of $F\in\Q[x]$ with $\operatorname{deg}F=5$ such that on the curve $y^2=F(x)$ twelve points lie in arithmetic progression. Next, we show that there exist infinitely many curves of the form $y^2=g(x)$ where $g\in\Q[x]$ and $\operatorname{deg}g=6$, each containing 16 points in arithmetic progression. Moreover, we present two examples of curves in this form with 18 points in arithmetic progression.