Teichmueller curves on Hilbert modular surfaces

Teichmuller curves are rather exotic objects that have received a lot of attention in recent years but are still only very partially understood.
A Teichmuller curve is an algebraic curve in the moduli space of curves of genus $g$ which is totally geodesic for the Teichmuller metric. For large $g$ it is not even known whether there are infinitely many such curves, but for $g=2$ a great deal is known, thanks to the work of McMullen and others: there are countably many such curves, each lying on some (unique) Hilbert modular surface, and conversely one or two Teichmuller curves on each Hilbert modular surface. In recent work with M. Moller we have found much more explicit descriptions of these curves than were previously known, as the zero-loci of explicit Hilbert modular forms; this casts new light on the number theory and algebraic geometry of the curves and also on the properties of the associated Picard–Fuchs differential equations.