On arithmetic progressions on genus two curves

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dc.creatorUlas, Maciej
dc.date2007-05-21
dc.date.accessioned2026-07-07T08:02:36Z
dc.date.available2026-07-07T08:02:36Z
dc.descriptionWe 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.
dc.description7 pages, to appear in Rocky Mountain Journal of Mathematics
dc.identifierhttps://arxiv.org/abs/0705.2919
dc.identifierhttp://arxiv.org/abs/0705.2919
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/129275
dc.subjectNumber Theory
dc.subject11B25; 11D41
dc.titleOn arithmetic progressions on genus two curves
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