Veech holomorphic families of Riemann surfaces, holomorphic sections,   and Diophantine problems
  In this paper, we construct holomorphic families of Riemann surfaces from Veech groups and characterize their sections by some points of corresponding flat surfaces. The construction gives us concrete solutions for some Diophantine equations over function fields. Moreover, we give upper bounds of the numbers of holomorphic sections of certain holomorphic families of Riemann surfaces. 
Yoshihiko Shinomiya
math.CV math.AG math.GN