Periods of modular functions around Markov geodesics

The periods of a modular function f are integrals of f along geodesics in the hyperbolic plane joining a real irrational quadratic number with its Galois conjugate. When f is the well-known j-function, its periods have been the object of various recent works of Duke, Imamoglu and Toth, and have been viewed as analogs of singular moduli for real quadratic fields. In this talk we address two conjectures of Kaneko that predict some specific behaviours of the periods of j around geodesics that correspond to Markov quadratics. Markov quadratics are those which can be worse approximated by rationals; they give the beginning of the Lagrange spectrum in Diophantine approximation. This is joint work with O. Imamoglu.