Teichmüller curves and Hilbert modular surfaces

The theory of Teichmüller curves, which combines elements of complex algebraic geometry and dynamical systems theory, has seen spectacular development in recent years, thanks to the work of C. McMullen and others. In particular, in the case of genus 2, there is a complete classification of these curves, due to K. Calta and McMullen. This classification implies that Teichmüller curves in genus 2 are always contained in Hilbert modular surfaces, which in principle allows an approach using methods from number theory and modular form theory. Such an approach was developed by M. Möller and myself and was the subject of this year's lecture, some of the results having already been presented in the 2008-2009 course. In particular, we obtain a description of Teichmüller curves on a given Hilbert modular surface as the set of zeros of the derivative of a certain theta function. This explicit description makes it much easier than before to understand and demonstrate the main results of the theory obtained in the work of McMullen and M. Bainbridge.