Periodic Points of Prym Eigenforms

A point of a Veech surface is periodic if it has a finite orbit under the surface’s affine automorphism group. We show that the periodic points of Prym eigenforms in the minimal strata of translation surfaces in genera 2, 3 and 4 are the fixed points of the Prym involution. This answers a question of Apisa–Wright and gives a geometric proof of Möller’s classification of periodic points of Veech surfaces in the minimal stratum in genus 2.