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

Teichmueller curves generated by Weierstrass Prym eigenforms in genus three and genus four

This paper is devoted to the classification of the infinite families of Teichmuller curves generated by Prym eigenforms of genus 3 having a single zero. These curves were discovered by McMullen. The main invariants of our classification is the discriminant D of the corresponding quadratic order, and the generators of this order. It turns out that for D sufficiently large, there are two Teichmueller curves when D=1 modulo 8, only one Teichmueller curve when D=0,4 modulo 8, and no Teichmueller curves when D=5 modulo 8. For small values of D, where this classification is not necessarily true, the…

GL^+(2,R)-orbits in Prym eigenform loci

This paper is devoted to the classification of GL^+(2,R)-orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of quadratic differentials. We show that the following dichotomy holds: an orbit is either closed or dense in a connected component of the Prym eigenform locus. The proof uses several topological properties of Prym eigenforms, which are proved by the authors in a previous work. In particular the tools and the proof are independent of the recent results of Eskin-Mirzakhani-Mohammadi. As an application we obtain a finiteness result for the number…

交集理论与Prym特征形式在$Ω\mathcal{M}_3(2,2)^{\rm odd}$中的Siegel-Veech常数

Intersection theory and Siegel-Veech constants for Prym eigenform loci in $Ω\mathcal{M}_3(2,2)^{\rm odd}$

本文研究了Prym特征形式在特定模空间中的鞍连接的Siegel-Veech常数，涉及交集理论和实二次域的非平方判别式特征。

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. …

$SL(2,\mathbb{Z})$-轨道在$\mathcal{H}(2)$中 origamis 的直径界及$\mathcal{H}(4)$和$\mathcal{H}(6)$中 Prym 区域的轨道界

Diameter bounds for $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2)$ and the Prym loci in $\mathcal{H}(4)$ and $\mathcal{H}(6)$

研究通过算法确定 $SL(2,\mathbb{Z})$-轨道在不同 stratum 中的直径界，包括 $\mathcal{H}(2)$、$\mathcal{H}(4)$ 和 $\mathcal{H}(6)$ 中的 origamis 和 Prym 区域。