Arithmetic Veech sublattices of $\SL(2,\Z)$

TL;DR

This paper proves that every algebraic curve over the algebraic closure of rationals is birationally equivalent over complex numbers to a Teichmuller curve, linking algebraic geometry and Teichmuller theory.

Contribution

It establishes a universal birational correspondence between algebraic curves over  and Teichmuller curves, revealing a deep connection between these areas.

Findings

Every algebraic curve over  is birational over f to a Teichmuller curve

Bridges algebraic geometry and Teichmuller theory

Provides a new perspective on the structure of algebraic curves

Abstract

We prove that every algebraic curve X defined over the algebraic closure of the rationals is birational over the complex numbers to a Teichmuller curve.