Vol. 13, No. 4, 2019

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Cycle integrals of modular functions, Markov geodesics and a conjecture of Kaneko

Paloma Bengoechea and Özlem Imamoglu

Vol. 13 (2019), No. 4, 943–962
Abstract

In this paper we study the values of modular functions at the Markov quadratics which are defined in terms of their cycle integrals along the associated closed geodesics. These numbers are shown to satisfy two properties that were conjectured by Kaneko. More precisely we show that the values of a modular function $f$, along any branch $B$ of the Markov tree, converge to the value of $f$ at the Markov number which is the predecessor of the tip of $B$. We also prove an interlacing property for these values.

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Keywords
modular forms, cycle integrals, markov numbers, j-invariant
Primary: 11F03
Secondary: 11J06
Milestones
Revised: 18 December 2018
Accepted: 8 February 2019
Published: 6 May 2019
Authors
 Paloma Bengoechea Department of Mathematics ETH Zurich Switzerland Özlem Imamoglu Department of Mathematics ETH Zurich Switzerland