Teichmüller curves: where to find them and how to cut them out

Teichmüller curves are totally geodesic curves inside the moduli space of Riemann surfaces. They can always be seen, via certain Prym-Torelli map, as Kobayashi geodesics inside a Hilbert modular variety parametrizing abelian varieties with real multiplication. Our main objective is to cut Teichmüller curves out as the vanishing locus of a Hilbert modular form. The building blocks of these modular forms turn out to be certain theta functions and their derivatives, which can be made very precise. This calculation allows us to compute the Euler characteristics of the Teichmüller curves and the Masur-Veech volume of the affine invariant manifold where they live. In this talk we will give an overview of the subject and focus on the case of gothic Teichmüller curves.