Weierstrass Prym eigenforms in genus four

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…

Complete periodicity of Prym eigenforms

This paper deals with Prym eigenforms which are introduced previously by McMullen. We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any homological direction is algebraically periodic, and any direction of a regular closed geodesic is a completely periodic direction. As a consequence we draw that the limit set of the Veech group of every Prym eigenform in some Prym loci of genus 3,4, and 5 is either empty, one point, or the full circle at infinity. We also construct new examples of translation…

Prym 特征形式的周期点

Periodic Points of Prym Eigenforms

本文证明亏格 2、3、4 最小层中 Prym 特征形式的周期点恰为 Prym 对合固定点，回答了 Apisa–Wright 的问题，并给出 Möller 亏格 2 分类的几何证明。

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。

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…

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. …