Non-Existence and Finiteness Results for Teichmüller Curves in Prym Loci

Abstract

The minimal stratum in Prym loci have been the first source of infinitely many primitive, but not algebraically primitive Teichmüller curves. We show that the stratum Prym(1, 2) contains no such Teichmüller curve and the stratum Prym(2) at most 92 such Teichmüller curves. This complements the recent progress establishing general – but non-effective – methods to prove finiteness results for Teichmüller curves and serves as proof of concept how to use the torsion condition in the non-algebraically primitive case.

Keywords:

• Veech surfaces
• translation surfaces
• Teichmüller curves

Acknowledgments

Computational assistance was provided by SAGE and a program written by Alex Eskin for computing cylinder decompositions. The authors are indebted to the programmers for the help provided. We would like to thank the two anonymous referee for the suggestions and comments.

Notes

1 Note that so far we have not been specifying the order of real multiplication, just the field and we specify it by a square-free integer D₀.

Additional information

Funding

The first author is partially supported by the Labex Persyval and the Project GeoSpec. The second author is partially supported by the DFG-project MO 1884/2-1 “Classification of Teichmüller curves.”