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arXiv · 2508.14645

Real geometric transcendence for uniformization maps of algebraic Riemann surfaces

Arshay Sheth·Matteo Tamiozzo
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Abstract

Let $X$ be a smooth connected complex algebraic curve that is not simply connected, and let $\tilde{X}$ 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 $\tilde{X}$. 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.

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Arshay Sheth, Matteo Tamiozzo. 2025-08-20. Real geometric transcendence for uniformization maps of algebraic Riemann surfaces. https://arxiv.org/abs/2508.14645

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