Shimura–Teichmüller curves in genus 5

David Aulicino; Chaya Norton

doi:10.3934/jmd.2020009

Article Contents

2020, Volume 16: 255-288. Doi: 10.3934/jmd.2020009

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Shimura–Teichmüller curves in genus 5

David Aulicino^1, and
Chaya Norton^2,

1.
Department of Mathematics, Brooklyn College and CUNY Graduate Center, 2900 Bedford Avenue, Brooklyn, NY 11210-2889, USA
2.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Received: November 17, 2019

Revised: April 28, 2020

Published: September 2020

Partially supported by the National Science Foundation under Award Nos. DMS - 1738381, DMS - 1600360, and PSC-CUNY Grant # 61639-00 49

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Abstract

We prove that there are no Shimura–Teichmüller curves generated by genus five translation surfaces, thereby completing the classification of Shimura–Teichmüller curves in general. This was conjectured by Möller in his original work introducing Shimura–Teichmüller curves. Moreover, the property of being a Shimura–Teichmüller curve is equivalent to having completely degenerate Kontsevich–Zorich spectrum.

The main new ingredient comes from the work of Hu and the second named author, which facilitates calculations of higher order terms in the period matrix with respect to plumbing coordinates. A large computer search is implemented to exclude the remaining cases, which must be performed in a very specific way to be computationally feasible.

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Mathematics Subject Classification: 32G20, 37D40; Secondary: 14H15, 14G35, 37D25.

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Figure 1. Schematic of the surface coordinates used in the algorithm: Circle vertices represent zeros and square vertices could represent zeros or regular points

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Figure 2. Proof of Corollary 5.10: The shaded region on the top of $ C_2 $ represents the admissible locations of $ \tau_0 $

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Table 1. List of Strata Possibly Containing ST-Curves and Values of $ d_{opt} $: Dashed lines separate genus

Stratum	$ d_{opt} $
$ \mathscr{H}(1, 1, 1, 1) $	$ 2 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1) $	$ 4 $
$ \mathscr{H}(2, 2, 2) $	$ 3 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1,2) $	$ 36 $
$ \mathscr{H}(1, 1, 1, 1, 2,2) $	$ 18 $
$ \mathscr{H}(1, 1, 2, 2,2) $	$ 12 $
$ \mathscr{H}(2, 2, 2,2) $	$ 9 $
$ \mathscr{H}(1, 1, 1, 1, 1,3) $	$ 16 $
$ \mathscr{H}(1, 1, 3,3) $	$ 8 $
$ \mathscr{H}(1, 1, 1, 1,4) $	$ 10 $

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