Computing Periodic Points on Veech Surfaces

TL;DR

This paper introduces an algorithm to compute periodic points on Veech surfaces, demonstrating its application to Prym eigenforms in genus 3 and revealing the absence of non-trivial periodic points in certain cases.

Contribution

The paper presents a novel algorithm for identifying periodic points on non-square-tiled Veech surfaces and applies it to specific Prym eigenforms, providing new insights into their structure.

Findings

Finitely many periodic points on non-square-tiled Veech surfaces.

In low discriminant, Prym eigenforms lack periodic points except fixed points.

Algorithm effectively identifies periodic points on these surfaces.

Abstract

A non-square-tiled Veech surface has finitely many periodic points, i.e., points with finite orbit under the affine automorphism group. We present an algorithm that inputs a non-square-tiled Veech surface and outputs its set of periodic points. We apply our algorithm to Prym eigenforms in the minimal stratum in genus 3, proving that in low discriminant these surfaces do not have periodic points, except for the fixed points of the Prym involution.