Continued fractions, modular symbols, and noncommutative geometry

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, Z) \ P ¹(R) should be considered as noncommutative 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.