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

Bengoechea, Paloma; Imamoglu, Özlem

doi:10.2140/ant.2019.13.943

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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

DOI: 10.2140/ant.2019.13.943

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

Mathematical Subject Classification 2010

Primary: 11F03

Secondary: 11J06

Milestones

Received: 27 March 2018

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

Vol. 13, No. 4, 2019