A continuity of cycle integrals of modular functions

TL;DR

This paper investigates the nuanced continuity properties of modular functions at real quadratic numbers, focusing on their cycle integrals along geodesics and revealing a finer structure related to continued fractions.

Contribution

It introduces a detailed analysis of the continuity of modular function values at quadratic irrationals, highlighting differences from Euclidean topology.

Findings

Continuity of cycle integrals depends on continued fraction expansions.

Finer structure of continuity differs from Euclidean topology.

Provides new insights into modular functions at quadratic irrationals.

Abstract

In this paper we study a continuity of the "values" of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer structure of the continuity of these values with respect to continued fraction expansions and it turns out that it is different from the continuity with respect to Euclidean topology.