Markoff Triples and Strong Approximation

TL;DR

This paper studies the symmetry actions on solutions to the Markoff equation and related surfaces, revealing how these symmetries influence the distribution of integer points and enabling effective sieving methods.

Contribution

It analyzes the transitivity of Vieta involution groups on Markoff-type surfaces and applies these findings to strong approximation and sieving techniques.

Findings

Finite orbits of the group actions can be effectively determined.

Results establish strong approximation properties for integer points.

Applications include improved sieving methods on these surfaces.

Abstract

We investigate the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated by the finite $\bar{\mathbb Q}$ orbits of these actions and these can be determined effectively. The results are applied to give forms of strong approximation for integer points, and to sieving, on these surfaces