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Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterize 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 Weierstra points and the singularities.  Our main tool is the Hodge-theoretic characterization of Teichmller curves. We deduce from it a finiteness result for the Mordell-Weil group of the family of Jacobians over a Teichmller curve. The link to the classification of periodic points is provided by interpreting them as sections of the family of curves over a covering of the Teichmller curve.

Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve