Asymptotic bounds for cycle integrals of the j-function on Markov geodesics

TL;DR

This paper establishes asymptotic bounds for the cycle integrals of the modular j-function along Markov geodesics, advancing understanding of their behavior in number theory.

Contribution

It provides the first asymptotic bounds for these cycle integrals specifically on Markov geodesics, linking modular functions with Markov irrationalities.

Findings

Derived asymptotic upper bounds for cycle integrals.

Derived asymptotic lower bounds for cycle integrals.

Enhanced understanding of the behavior of the j-function on special geodesics.

Abstract

We give asymptotic upper and lower bounds for the real and imaginary parts of cycle integrals of the classical modular j-function along geodesics that correspond to Markov irrationalities.