On geometric progressions on hyperelliptic curves

TL;DR

This paper proves the existence of infinitely many hyperelliptic curves that contain rational points forming a geometric progression of length at least eight.

Contribution

It establishes the existence of an infinite family of hyperelliptic curves with rational points in a geometric progression of length at least eight, a new result in the study of rational points.

Findings

Existence of infinite hyperelliptic curves with long geometric progressions

Construction of rational points in geometric progression on these curves

Progression length of at least eight points

Abstract

Let $C$ be a hyperelliptic curve over $\mathbb Q$ described by $y^2=a_0x^n+a_1x^{n-1}+\ldots+a_n$, $a_i\in\mathbb Q$. 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., $x_i=pt^{i}$ 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.