Periodic Points of Prym Eigenforms

TL;DR

This paper characterizes the periodic points of Prym eigenforms in low-genus translation surfaces, showing they are exactly the fixed points of the Prym involution, and provides a geometric proof of a classification in genus 2.

Contribution

It identifies the periodic points of Prym eigenforms in low genus as fixed points of the Prym involution, answering a previously open question.

Findings

Periodic points are fixed points of the Prym involution.

Provides a geometric proof of M"oller's classification in genus 2.

Extends understanding of periodic points in low-genus translation surfaces.

Abstract

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 M\"oller's classification of periodic points of Veech surfaces in the minimal stratum in genus 2.