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

TL;DR

This paper investigates the values of modular functions at Markov quadratic points, proving convergence and interlacing properties that confirm conjectures by Kaneko, linking number theory, geometry, and dynamical systems.

Contribution

It establishes convergence and interlacing properties of modular function values at Markov quadratics, confirming Kaneko's conjectures and deepening understanding of their number-theoretic and geometric structure.

Findings

Values of modular functions at Markov quadratics converge to specific limits.

An interlacing property of these values is proven.

The results confirm conjectures proposed by Kaneko.

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.