Special curves in modular surfaces

TL;DR

This paper characterizes special algebraic curves in modular surfaces, showing that certain geodesics related to tori are uniquely associated with algebraic curves under the $j$-invariant.

Contribution

It identifies and classifies special geodesic curves in modular surfaces that correspond to algebraic curves via the $j$-invariant, revealing their uniqueness.

Findings

Geodesics attached to split and quadratic tori are the only irreducible algebraic curves with algebraic $j$-image.

Characterization of special curves in modular surfaces.

Connection between geodesic dynamics and algebraic geometry in modular contexts.

Abstract

We show that geodesics in the upper half-plane attached to a maximal split torus or a real quadratic torus in $GL_{2, \mathbf{Q}}$ are the only irreducible algebraic curves whose image via the $j$-invariant is contained in an algebraic curve.