On geometric progressions on hyperelliptic curves

Alaa, Mohamed ; Sadek, Mohammad

Journal of integer sequences, Tome 19 (2016) no. 6

Résumé

Let $C$ be a hyperelliptic curve over ${\mathbb Q}$ described by $y^2=a_0x^n+a_1x^{n-1}+\cdots+a_n, a_i\in{\mathbb Q}$. The points $P_{i}=(x_{i},y_{i})\in C(\mathbb{Q} ), i=1,2,\ldots,k$, are said to be in a geometric progression of length $k$ if the rational numbers $x_{i}, i=1,2,\ldots,k$, form a geometric progression sequence in ${\mathbb Q}$, i.e., $x_{i} = pt^{i}$ for some $p,t\in{\mathbb Q}$. In this paper we prove the existence of an infinite family of hyperelliptic curves on which there is a sequence of rational points in a geometric progression of length at least eight.

Détail

Classification : 14G05, 11B83
Keywords: geometric progression, hyperelliptic curve, rational point