  Periodic points are points on Veech surfaces, whose orbit under the group of
affine diffeomorphisms is finite. We characterise those points as being torsion
points if the Veech surfaces is suitably mapped to its Jacobian or an
appropriate factor thereof. For a primitive Veech surface in genus two we show
that the only periodic points are the Weierstrass points and the singularities.
  Our main tool is the Hodge-theoretic characterisation of Teichmueller curves.
We deduce from it a finiteness result for the Mordell-Weil group of the family
of Jacobians over a Teichmueller curve.
