Geometry of division rings

TL;DR

This paper establishes a geometric criterion for algebraic surfaces to be defined over number fields, linking their definability to coverings of the complex projective plane ramified over three knotted spheres.

Contribution

It proves an analog of Belyi's theorem for algebraic surfaces, extending the classical result from curves to surfaces.

Findings

Any non-singular algebraic surface can be defined over a number field if it covers the complex projective plane with ramification at three knotted spheres.

The theorem provides a geometric characterization of algebraic surfaces over number fields.

The result bridges algebraic geometry and topological properties of surface coverings.

Abstract

We prove an analog of Belyi's theorem for the algebraic surfaces. Namely, any non-singular algebraic surface can be defined over a number field if and only it covers the complex projective plane with ramification at three knotted two-dimensional spheres.