Recherches autour de la theorie de Markoff

TL;DR

This paper explores generalizations of the Markoff equation, providing methods for their complete resolution, and connecting algebraic, geometric, and analytical perspectives, with applications in physics and number theory.

Contribution

It introduces new generalized Markoff equations, offers a comprehensive algebraic and geometric resolution method, and links these to Teichmuller theory and Riemann surfaces.

Findings

Complete resolution method for generalized Markoff equations

Classification of punctured toruses via Teichmuller theory

Applications to physics, 1/f noise, and zeta functions

Abstract

The text deals with generalizations of the Markoff equation in number theory, arising from continued fractions. It gives the method for the complete resolution of such new equations, and their interpretation in algebra and algebraic geometry. This algebraic approach is completed with an analytical development concerning fuchsian groups. The link with the Teichmuller theory for punctured toruses is completely described, giving their classification with a reduction theory. More general considerations about Riemann surfaces, geodesics and their hamiltonian study are quoted, together with applications in physics, 1/f noise and zeta function. Ideas about important conjectures are presented. Reasons why the Markoff theory appears in different geometrical contexts are given, thanks to decomposition results in the group GL(2,Z).