Rational points in arithmetic progression on $y^2=x^n+k$

TL;DR

This paper demonstrates the existence of polynomials and parameters for which hyperelliptic and elliptic curves contain multiple rational points in arithmetic progression, extending previous results and showing infinite families of such curves.

Contribution

It constructs explicit examples of polynomials and parameters where hyperelliptic and elliptic curves have multiple rational points in arithmetic progression, generalizing earlier work.

Findings

Existence of a polynomial k(t) with four independent rational points in arithmetic progression on y^2=x^3+k(t).

Infinite families of curves y^2=x^n+k with four or six rational points in arithmetic progression for odd and even n.

Extension of previous results by Lee and Vélz on rational points in arithmetic progression.

Abstract

Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$, where $f\in\Z[x]$ and $f$ hasn't multiple roots. We say that points $P_{i}=(x_{i}, y_{i})\in C(\Q)$ for $i=1,2,..., n$ are in arithmetic progression if the numbers $x_{i}$ for $i=1,2,..., n$ are in arithmetic progression. In this paper we show that there exists a polynomial $k\in\Z[t]$ with such a property that on the elliptic curve $\cal{E}: y^2=x^3+k(t)$ (defined over the field $\Q(t)$) we can find four points in arithmetic progression which are independent in the group of all $\Q(t)$-rational points on the curve $\cal{E}$. In particular this result generalizes some earlier results of Lee and V\'{e}lez from \cite{LeeVel}. We also show that if $n\in\N$ is odd then there are infinitely many $k$'s with such a property that on the 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 the curves $y^2=x^n+k$ there are six rational points in arithmetic progression.