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Computing periodic points on Veech surfaces

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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. Our algorithm serves as a new proof of the finiteness of periodic points for non-square-tiled Veech surfaces. 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.

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Data availability statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

We thank the Institute for Computational and Experimental Research in Mathematics (ICERM) for running the Summer@ICERM 2021 REU where this work took place. We also thank Curt McMullen for comments and suggestions on an early draft of this paper. Finally, we are grateful to Paul Apisa for proposing the problem and providing invaluable guidance and support.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, USA

    Zawad Chowdhury

  2. University of Chicago, Chicago, USA

    Samuel Everett

  3. Brown University, Providence, USA

    Sam Freedman

  4. Columbia University, New York, USA

    Destine Lee

Authors
  1. Zawad Chowdhury
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  2. Samuel Everett
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  3. Sam Freedman
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  4. Destine Lee
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Correspondence to Sam Freedman.

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Chowdhury, Z., Everett, S., Freedman, S. et al. Computing periodic points on Veech surfaces. Geom Dedicata 217, 66 (2023). https://doi.org/10.1007/s10711-023-00804-z

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