Non-existence and finiteness results for Teichmueller curves in Prym loci

TL;DR

This paper proves the non-existence of certain Teichmueller curves in Prym loci and establishes an upper bound for their number in specific strata, advancing understanding of their finiteness properties.

Contribution

It demonstrates the non-existence of primitive Teichmueller curves in Prym(2,1,1) and bounds their number in Prym(2,2), introducing a new approach using torsion conditions.

Findings

No primitive Teichmueller curves in Prym(2,1,1)

At most 92 such curves in Prym(2,2)

Uses torsion conditions to establish finiteness

Abstract

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