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

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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modular forms, cycle integrals, markov numbers, j-invariant
Mathematical Subject Classification 2010
Primary: 11F03
Secondary: 11J06
Received: 27 March 2018
Revised: 18 December 2018
Accepted: 8 February 2019
Published: 6 May 2019
Paloma Bengoechea
Department of Mathematics
ETH Zurich
Özlem Imamoglu
Department of Mathematics
ETH Zurich