Cutting out arithmetic Teichmüller curves in genus two via Theta functions

Abstract

We compute the class of arithmetic genus two Teichmüller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants.

The main technical tool is the computation of divisor classes of Hilbert Jacobi forms on the universal abelian surface over the pseudo-Hilbert modular surface.