Cutting out arithmetic Teichmueller curves in genus two via Theta functions

TL;DR

This paper computes the classes of specific genus two Teichmueller curves using theta functions and Hilbert Jacobi forms, providing new insights into their geometric and arithmetic properties.

Contribution

It introduces a method to compute divisor classes of Teichmueller curves in genus two via theta functions and Hilbert Jacobi forms, linking geometry and arithmetic.

Findings

Class of arithmetic genus two Teichmueller curves computed

Number of genus two square-tiled surfaces with given invariants determined

Divisor classes of Hilbert Jacobi forms on universal abelian surfaces calculated

Abstract

We compute the class of arithmetic genus two Teichmueller 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.