Continued fractions, modular symbols, and non-commutative geometry

TL;DR

This paper extends the Gauss-Kuzmin theorem for continued fractions, explores their connection to modular symbols and non-commutative geometry, and applies these results to models in cosmology and number theory.

Contribution

It introduces non-commutative geometric methods to study continued fractions, modular symbols, and their applications in physics and number theory.

Findings

Extended Gauss-Kuzmin theorem including congruence properties

Representation of modular forms via continued fractions

Identification of a non-commutative modular curve

Abstract

Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient $PGL(2,\bold{Z})\setminus\bold{P}^1(\bold{R})$ should be considered as non-commutative modular curve, and show that the modular complex can be seen as a sequence of $K_0$-groups of the related crossed-product $C^*$-algebras. This paper is an expanded version of the previous "On the distribution of continued fractions and modular symbols". The main new features are Section 4 on non-commutative geometry and the modular complex and Section 1.2.2 on the Mixmaster Universe.