Weierstrass Prym eigenforms in genus four

TL;DR

This paper proves the connectedness of Prym eigenform loci in genus four for certain discriminants and classifies primitive square-tiled surfaces within these loci, establishing their structure as Teichmüller curves.

Contribution

It establishes the connectedness of Prym eigenform loci for specific discriminants and classifies primitive square-tiled surfaces in the Prym locus in genus four.

Findings

Prym eigenform loci are connected for discriminants D ≡ 0,1 mod 4, D not in {4,9}.

The projection of these loci in the moduli space forms a single Teichmüller curve.

Primitive square-tiled surfaces in Prym(6) are classified.

Abstract

We prove that for each discriminant $D \equiv 0,1 \mod 4, D \not\in\{4,9\}$, the corresponding Prym eigenform locus discovered by McMullen in the stratum $\mathcal{H}(6)$ is connected. Thus, the projection of any of those loci in the moduli space is a single Teichm\"uller curve. Along the way, we obtain a classification of primitive square-tiled surfaces in the locus $\mathrm{Prym}(6)$ of Prym forms in $\mathcal{H}(6)$.