Connected components of Prym eigenform loci in genus three
  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. 
Erwan Lanneau and Duc-Manh Nguyen
math.GT math.AG math.DS