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On arithmetic progressions on genus two curves. (English) Zbl 1243.11069

Summary: 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 \(f_{5}, f_{6}\in\mathbb Q[x]\), \(\deg f_{5}=5\), \(\deg f_{6}=6\) 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 \(f\in\mathbb Q[x]\) and \(\deg f=5\) each containing 11 points in arithmetic progression. We also present an example of \(F\in\mathbb Q[x]\) with \(\deg F=5\) 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 \(g\in\mathbb Q[x]\) and \(\deg g=6\), each containing 16 points in arithmetic progression. Moreover, we present two examples of curves in this form with 18 points in arithmetic progression.

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11B25 Arithmetic progressions
11D41 Higher degree equations; Fermat’s equation
11G05 Elliptic curves over global fields

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References:

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