Skip to main content
    • Aa
    • Aa


  • David Torres-Teigell (a1) and Jonathan Zachhuber (a2)

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 $D$ .

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

M. Bainbridge , Euler characteristics of Teichmüller curves in genus two, Geom. Topol. 11 (2007), 18872073.

C. Birkenhake and H. Lange , Complex Abelian Varieties, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, (Springer, Berlin, 2004).

S. A. Broughton , Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69(3) (1991), 233270.

E. Bujalance and M. Conder , On cyclic groups of automorphisms of Riemann surfaces, J. Lond. Math. Soc. (2) 59(2) (1999), 573584.

H. Cohen , A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics, Volume 138 (Springer, Berlin, 1993).

E. Lanneau and D.-M. Nguyen , Teichmüller curves generated by Weierstrass Prym eigenforms in genus 3 and genus 4, J. Topol. 7(2) (2014), 475522.

J. L. Lehman , Levels of positive definite ternary quadratic forms, Math. Comp. 58(197) (1992), 399417.

C. T. McMullen , Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16(4) (2003), 857885.

C. T. McMullen , Prym varieties and Teichmüller curves, Duke Math. J. 133(3) (2006), 569590.

C. T. McMullen , R. E. Mukamel and A. Wright , Cubic curves and totally geodesic subvarieties of moduli space, Ann. of Math. (2) 185(3) (2017), 957990.

M. Möller and D. Zagier , Modular embeddings of Teichmüller curves, Compos. Math. 152 (2016), 22692349.

R. E. Mukamel , Orbifold points on Teichmüller curves and Jacobians with complex multiplication, Geom. Topol. 18(2) (2014), 779829.

J. Schiller , Moduli for special Riemann surfaces of genus 2, Trans. Amer. Math. Soc. 144 (1969), 95113.

G. Shimura , On modular forms of half-integral weight, Ann. of Math. (2) 97(2) (1973), 440481.

D. Torres-Teigell and J. Zachhuber , Orbifold points on Prym–Teichmüller curves in genus three, Int. Math. Res. Not. IMRN (2016), doi:10.1093/imrn/rnw277, to appear.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 8 *
Loading metrics...

Abstract views

Total abstract views: 71 *
Loading metrics...

* Views captured on Cambridge Core between 22nd May 2017 - 21st September 2017. This data will be updated every 24 hours.