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Shimura–Teichmüller curves in genus 5

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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  • 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.

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

    Citation:

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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

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

    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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