Real geometric transcendence for uniformization maps of algebraic Riemann surfaces

2025/08/20 by Arshay Sheth, Matteo Tamiozzo, Sheth, Arshay +1
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Mathematical functions and polynomials #Polynomial and algebraic computation #math.NT

paper · pdf · doi:10.48550/arxiv.2508.14645

openalex publication_date 2025/08/20 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/31

Abstract

Let X be a smooth connected complex algebraic curve that is not simply connected, and let X be the universal cover of X. We study the set of irreducible real algebraic curves in X (seen as a real algebraic surface) containing the image of an arc of a real algebraic curve in X. In particular, we give necessary and sufficient conditions on X in order for this set to be non-empty or infinite, and we describe the set explicitly when X is projective of genus one, or X is hyperbolic and its fundamental group is arithmetic.

Citations

• Special curves in modular surfaces
• A quaternionic construction of p-adic singular moduli
• Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds
• Special subvarieties of non-arithmetic ball quotients and Hodge Theory
• Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian\n groups
• Algebraicity of analytic maps to a hyperbolic variety
• Some structure theorems for algebraic groups
• Introduction To Arithmetic Groups

Related

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• Uniform Hyperbolicity of the Graphs of Curves