Orbifold points on Prym-Teichm\"{u}ller curves in genus four

TL;DR

This paper classifies orbifold points on Prym-Teichmüller curves in genus four, completing their topological characterization and analyzing their geometric and automorphic properties.

Contribution

It determines the number and types of orbifold points on $W_D(6)$ for all discriminants $D$, extending previous genus 3 results and analyzing intersections with automorphism-rich families.

Findings

Classified orbifold points on $W_D(6)$ for all $D$

Constructed explicit families of genus 4 curves with automorphisms

Described the Prym-Torelli images as products of elliptic curves

Abstract

For each discriminant $D>1$, McMullen constructed the Prym-Teichm\"uller curves $W_D(4)$ and $W_D(6)$ in $\mathcal{M}_{3}$ and $\mathcal{M}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichm\"{u}ller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_D(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau-Nguyen and M\"oller, complete the topological characterisation of all Prym-Teichm\"uller curves and determine their genus. The study of orbifold points relies on the analysis of intersections of $W_D(6)$ with certain families of genus $4$ curves with extra automorphisms. As a side product of this study, we give an explicit construction of such families and describe their Prym-Torelli images, which turn out to be isomorphic to certain products of elliptic curves. We also give a geometric description of the flat surfaces associated to these families and describe the asymptotics of the genus of $W_D(6)$ for large $D$.