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Rational Points in Arithmetic Progressions on y2 = xn + k

Published online by Cambridge University Press:  20 November 2018

Maciej Ulas
Maciej Ulas*
Affiliation:
Jagiellonian University, Institute of Mathematics, 30 - 348 Kraków, Poland e-mail: maciej.ulas@gmail.com
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Abstract

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Let $C$ be a hyperelliptic curve given by the equation ${{y}^{2}}\,=\,f(x)$ for $f\,\in \,\mathbb{Z}[x]$ without multiple roots. We say that points ${{P}_{i}}\,=\,({{x}_{i}},\,{{y}_{i}})\,\in \,C(\mathbb{Q})$ for $i\,=\,1,\,2,\,\ldots ,\,m$ are in arithmetic progression if the numbers ${{x}_{i}}$ for $i\,=\,1,\,2,\,\ldots ,\,m$ are in arithmetic progression.

In this paper we show that there exists a polynomial $k\,\in \,\mathbb{Z}[t]$ with the property that on the elliptic curve $\varepsilon \prime :{{y}^{2}}={{x}^{3}}+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four points in arithmetic progression that are independent in the group of all $\mathbb{Q}(t)$-rational points on the curve $\varepsilon \prime $. In particular this result generalizes earlier results of Lee and Vélez. We also show that if $n\,\in \,\mathbb{N}$ is odd, then there are infinitely many $k$'s with the property that on curves ${{y}^{2}}\,=\,{{x}^{n}}\,+\,k$ there are four rational points in arithmetic progressions. In the case when $n$ is even we can find infinitely many $k$'s such that on curves ${{y}^{2}}\,=\,{{x}^{n}}\,+\,k$ there are six rational points in arithmetic progression.

Keywords

11G05arithmetic progressionselliptic curvesrational points on hyperelliptic curves
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 1 , 01 March 2012 , pp. 193 - 207
Copyright
Copyright © Canadian Mathematical Society 2012

References

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