STNB2018 (32nd edition)

Periods of modular functions around Markov geodesics


Paloma Bengoechea


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. They have been the object of various recent works of Duke, Imamoglu and Toth and, when f is the well-known j-function, have been viewed as the analog of singular moduli for real quadratic fields. Kaneko conjectured, based on numerical evidence, 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. In this talk we will address Kaneko's conjectures.


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