On geometric progressions on hyperelliptic curves

Abstract

Let $C$ be a hyperelliptic curve over $\mathbb{Q}$ described by $y^2=a_0{x}^n+a_1{x}^{n-1}+\dots +a_n$, $a_i\in \mathbb{Q}$. The points $P_i=(x_i,y_i)\in C(\mathbb{Q})$, $i=1,2,...,k,$ are said to be in a geometric progression of length $k$ if the rational numbers $x_i$, $i=1,2,...,k,$ form a geometric progression sequence in $\mathbb{Q}$, i.e., $x_i=p{t}^i$ 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.