Shimura–Teichmüller curves in genus 5

Abstract

We prove that there are no Shimura–Teichmuller curves generated by genus five translation surfaces, thereby completing the classification of Shimura–Teichmuller curves in general. This was conjectured by Moller in his original work introducing Shimura–Teichmuller curves. Moreover, the property of being a Shimura–Teichmuller 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.