STNB2020(34th edition)

Periods of modular functions and diophantine approximation


Paloma Bengoechea


For a real quadratic irrationality w and the classical Klein's modular invariant j, the "value" j(w) has been recently defined using the period of j along the closed geodesic associated to w in the hyperbolic plane. Works of Duke, Imamoglu, Toth, and Masri establish analogies between these values and singular moduli when they are both gathered in traces. However, the arithmetic properties of the individual values j(w) remain inaccessible. In this talk, we will address conjectures of Kaneko on bounds for these values as well as a specific behaviour of j on Markov quadratics. Our strategy consists in studying the values j(w) according to the diophantine properties of w. This is joint work with O. Imamoglu.


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