On arithmetic progressions on Edwards curves

TL;DR

This paper investigates the structure of rational numbers forming arithmetic progressions on Edwards curves, parametrizing these sets via rational points on algebraic curves to understand their properties.

Contribution

It introduces a method to parametrize the set of rational arithmetic progressions on Edwards curves using algebraic curves, providing a new approach to analyze such progressions.

Findings

Characterization of AP_m(a,q) through algebraic curves

Parametrization of rational progressions on Edwards curves

Insights into the structure of rational solutions on these curves

Abstract

Let m be a positive integer and a,q two rational numbers. Denote by AP_m(a,q) the set of rational numbers d such that a,a+q,...,a+(m-1)q form an arithmetic progression in the Edwards curve E_d:x^2+y^2=1+d x^2 y^2. We study the set AP_m(a,q) and we parametrize it by the rational points of an algebraic curve.