# Shimura-Teichm\"uller Curves in Genus 5

## TL;DR

This paper proves that no Shimura-Teichmüller curves exist in genus five, completing their classification and linking their property to the degeneracy of the Kontsevich-Zorich spectrum, using advanced calculations and computer searches.

## Contribution

It completes the classification of Shimura-Teichmüller curves by proving their non-existence in genus five, confirming a conjecture by Möller.

## Key findings

- No Shimura-Teichmüller curves in genus five
- Equivalence between Shimura-Teichmüller property and degenerate Kontsevich-Zorich spectrum
- Successful computational exclusion of remaining cases

## 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.