Invariants of complex and p-adic origami-curves

Origamis (also known as square-tiled surfaces) are Riemann surfaces which are constructed by glueing together finitely many unit squares. By varying the complex structure of these squares one obtains easily accessible examples of Teichmüller curves in the moduli space of Riemann surfaces.
Different Teichmüller curves can be distinguished by several invariants, which are explicitly computed. The results are then compared to a p-adic analogue where Riemann surfaces are replaced by Mumford curves.