Orbifold points on Prym-Teichmüller curves in genus three

Abstract

Prym-Teichmüller curves $W_D(4)$ constitute the main examples of known primitive Teichmüller curves in the moduli space ${\mathcal{M}}_3$. We determine, for each non-square discriminant $D>1$, the number and type of orbifold points in $W_D(4)$. These results, together with the formulas of Lanneau-Nguyen and Möller for the number of cusps and the Euler characteristic, complete the topological characterisation of Prym-Teichmüller curves in genus 3. Crucial for the determination of the orbifold points is the analysis of families of genus 3 cyclic covers of degree $4$ and $6$, branched over four points of ${\mathbb{P}}^1$. As a side product of our study, we provide an explicit description of the Jacobians and the Prym-Torelli images of these two families, together with a description of the corresponding flat surfaces.

41 pages, 16 figures; for PARI files, see http://www.uni-frankfurt.de/50278800/Zachhuber. Rewrote introduction, updated references, fixed typos, improved exposition and added a section on flat geometry

References in corpus (1)

• Rank two affine submanifolds in $\mathcal{H}(2,2)$ and $\mathcal{H}(3,1)$

Cited by in corpus (2)

• Weierstrass Prym eigenforms in genus four
• Orbifold points on Prym-Teichmüller curves in genus four