Square-tiled surfaces and curves over number fields

TL;DR

This paper establishes a correspondence between square-tiled surfaces and algebraic curves defined over number fields, extending classical results about Riemann surfaces and conformal structures.

Contribution

It generalizes a known result relating conformal structures and algebraic definitions from Riemann surfaces to square-tiled surfaces.

Findings

Square-tiled surfaces correspond to algebraic curves over number fields.

The result extends classical conformal structure characterization to square-tiled surfaces.

Provides a new link between geometric structures and arithmetic properties of algebraic curves.

Abstract

The paper presents an analog of the old result by the author and V. Voevodsky, according to which a Riemann surface admits a conformal structure, defined by an equilateral triangulation, if and only if the corresponding algebraic curve can be defined over the field of the algebraic numbers; the similar result is obtained for the square-tiled surfaces.