Concentration of closed geodesics in the homology of modular curves

TL;DR

This paper demonstrates that homology classes of closed geodesics linked to subgroups of narrow class groups in real quadratic fields tend to concentrate near the Eisenstein line, extending Duke's Theorem and related results.

Contribution

It introduces a new concentration result for homology classes of geodesics in real quadratic fields, analogous to known results in CM-elliptic curves, and explores their implications.

Findings

Homology classes concentrate around the Eisenstein line.

Extension of Duke's Theorem to real quadratic fields.

Applications to group theory and modular forms.

Abstract

We prove that the homology classes of closed geodesics associated to subgroups of narrow class groups of real quadratic fields concentrate around the Eisenstein line. This fits into the framework of Duke's Theorem and can be seen as a real quadratic analogue of results of Michel and Liu--Masri--Young on supersingular reduction of CM-elliptic curves. We also study the level aspect, as well as a homological version of the sup norm problem. Finally we present applications to group theory and modular forms.