August 1, 2025

Periodic Points in Genus Two: Holomorphic Sections Over Hilbert Modular Varieties, Teichmüller Dynamics, and Billiards

Key Points

All equivariant point markings arise from pairs of points exchanged by hyperelliptic involution, Weierstrass points, or golden points.
The classification of holomorphically varying families of points aids in understanding orbifold covers of genus two surfaces.
The study solves the finite blocking problem specifically related to genus two translation surfaces, indicating complexity in dynamics.
Insights from Teichmüller dynamics can further influence the understanding of billiards in the context of geometry.

Abstract

Abstract We show that all GL (2, R) equivariant point markings over orbit closures of primitive genus two translation surfaces arise from pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points on E₅. As corollaries, we classify holomorphically varying families of points over orbifold covers of genus two Hilbert modular surfaces and solve the finite blocking problem on genus two translation surfaces.

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Cite This Study

Paul Apisa (Fri,) studied this question.

synapsesocial.com/papers/68af454cad7bf08b1ead325b https://doi.org/https://doi.org/10.1093/imrn/rnaf254

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