Connected components of Prym eigenform loci in genus three

TL;DR

This paper classifies the connected components of Prym eigenform loci in genus 3, revealing how their number depends on the discriminant D, and contrasting with known genus 2 results.

Contribution

It provides a complete classification of the connected components of Prym eigenform loci in specific genus 3 strata based on discriminant D.

Findings

For each discriminant D, the locus has one component if D ≡ 0 or 4 mod 8.

The locus has two components if D ≡ 1 mod 8.

The locus is empty otherwise.

Abstract

This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces whose Jacobian variety has a factor admitting real multiplication by some quadratic order Ord_D. It turns out that these subvarieties can be classified by the discriminant D of the corresponding quadratic orders. However there algebraic varieties are not necessarily irreducible. The main result we show is that for each discriminant D the corresponding locus has one component if D is congruent to 0 or 4 mod 8, two components if D is congruent to 1 mod 8, and is empty otherwise. Our result contrasts with the case of Prym eigenform loci in the strata H(1,1) (studied by McMullen) that is connected for every discriminant D.