Rational Points in Arithmetic Progressions on y 2 = xn + k

Ulas, Maciej

Canadian mathematical bulletin, Tome 55 (2012) no. 1, pp. 193-207

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-058-1

Mots-clés : 11G05, arithmetic progressions, elliptic curves, rational points on hyperelliptic curves

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