Shimura-Teichm\"uller Curves in Genus 5
Abstract
We prove that there are no Shimura-Teichm\"uller curves generated by genus five translation surfaces, thereby completing the classification of Shimura-Teichm\"uller curves in general. This was conjectured by M\"oller in his original work introducing Shimura-Teichm\"uller curves. Moreover, the property of being a Shimura-Teichm\"uller 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.
Cite
@article{arxiv.1904.01625,
title = {Shimura-Teichm\"uller Curves in Genus 5},
author = {David Aulicino and Chaya Norton},
journal= {arXiv preprint arXiv:1904.01625},
year = {2020}
}
Comments
37 pages, 2 figures. An error was found in Lemma 2.1 and corrected here. The correction in Lemma 2.1 was then applied to the proofs in Section 4. The main technical theorem Thm. 4.5 and the main result of the paper remain the same