Periodic Points in Genus Two: Holomorphic Sections over Hilbert Modular Varieties, Teichmuller Dynamics, and Billiards

TL;DR

This paper classifies special point markings on genus two translation surfaces, linking them to hyperelliptic involutions and eigenform loci, and applies these results to problems in Hilbert modular surfaces, billiards, and orbit closures.

Contribution

It provides a classification of GL(2, R)-equivariant point markings on genus two surfaces and explores their implications for modular varieties and dynamical systems.

Findings

Classified point markings arising from hyperelliptic involutions and eigenform loci.

Solved the finite blocking problem for genus two translation surfaces.

Established the uniqueness of nonarithmetic rank two orbit closure in genus four.

Abstract

We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varying families of points over orbifold covers of genus two Hilbert modular surfaces, solve the finite blocking problem on genus two translation surfaces, and show that there is at most one nonarithmetic rank two orbit closure in the minimal stratum in genus four.