Orbifold points on Prym-Teichmüller curves in genus four

Abstract

For each discriminant $D>1$, McMullen constructed the Prym-Teichmüller 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 Teichmüller 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 Möller, complete the topological characterisation of all Prym-Teichmüller 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 $D$.

38 pages, 5 figures. For PARI files, see http://www.uni-frankfurt.de/50278800/Zachhuber

References in corpus (2)

• Modular embeddings of Teichmueller curves
• Orbifold points on Prym-Teichmüller curves in genus three

Cited by in corpus (1)

• Weierstrass Prym eigenforms in genus four