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

Abstract We show that all $\operatorname{GL}(2, \mathbb{R})$ 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 $\Omega E_{5}$. 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.