Linear relations among holomorphic quadratic differentials and induced Siegel's metric on M_g

Hodge Bundles on Smooth Compactifications of Siegel Varieties and Applications

We study Hodge bundles on Siegel varieties and their various extensions to smooth toroidal compactifications. Precisely, we construct a canonical Hodge bundle on an arbitrary Siegel variety so that the holomorphic tangent bundle can be embedded into the Hodge bundle, and we observe that the Bergman metric on the Siegel variety is compatible with the induced Hodge metric. Therefore we obtain the asymptotic estimate of the Bergman metric explicitly. Depending on these properties and the uniformitarian of Kähler-Einstein manifold, we study extensions of the tangent bundle over any smooth toroidal…

算术子群的形式西格尔模形式

Formal Siegel modular forms for arithmetic subgroups

本文研究算术子群的形式西格尔模形式，证明在仿射覆盖数有上界时其均可由经典全纯西格尔模形式的傅里叶-雅可比展开给出，并验证属2至4时该条件恒成立，进而应用于属2无平方水平参数模群。

局部对称簇上全纯微分形式的平方可积性

Square-integrability of holomorphic differential forms on locally symmetric varieties

该研究证明了具有余维数大于1的Baily-Borel边界的局部对称簇上非顶层全纯微分形式的平方可积性，推广了Weissauer的Siegel模情形定理。

A Universal Genus-Two Curve from Siegel Modular Forms

Let $\mathfrak p$ be any point in the moduli space of genus-two curves $\mathcal M_2$ and $K$ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal C_{α, β}$ defined over $K(α, β)$, corresponding to $\mathfrak p$, where $α$ and $β$ satisfy a quadratic $α^2+ b β^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{α, β}$ is defined over the field of moduli $K$ if and only if the quadratic has a $K$-rational point $(α, β)$. We discover some interesting symmetries of the Weierstrass equation of $\mathcal C_{α, β}$. …

Shimura 曲线上的特殊循环交集与Siegel Maass形式

Intersections of special cycles on Shimura curves and Siegel Maass forms

本文研究Shimura曲线上的特殊循环交集与Siegel Maass形式的关系，通过非全纯模形式和theta提升方法，推广了Rickards的计数结果，并给出了傅里叶系数的几何解释。

某些曲线上有效 Brauer-西格尔定理

Effective Brauer-Siegel on some curves in $Y(1)^n$

本文通过 Yves André 的 G 函数方法，在某些 $Y(1)^n$ 曲线上建立了二次虚数域类数的有效 Siegel 下界。

Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, II: the singularity of the invariant (with an Appendix by Ken-Ichi Yoshikawa)

We study the boundary behavior of the invariant of $K3^{[2]}$-type manifolds with antisymplectic involution, which we obtained using equivariant analytic torsion. We show the algebraicity of the singularity of the invariant by using the asymptotic of equivariant Quillen metrics and equivariant $L^2$-metrics. We prove that, in some cases, the invariant coincides with Yoshikawa's invariant for 2-elementary K3 surfaces. Hence, in these cases, our invariant is expressed as the Petersson norm of a Borcherds product and a Siegel modular form.

Pillowcase covers: Counting Feynman-like graphs associated with quadratic differentials

We prove the quasimodularity of generating functions for counting pillowcase covers, with and without Siegel-Veech weight. Similar to prior work on torus covers, the proof is based on analyzing decompositions of half-translation surfaces into horizontal cylinders. It provides an alternative proof of the quasimodularity results of Eskin-Okounkov and a practical method to compute area Siegel-Veech constants. A main new technical tool is a quasi-polynomiality result for 2-orbifold Hurwitz numbers with completed cycles.

Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems and the Siegel--Veech Constants

A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. The similar phenomenon is valid for the families of parallel closed geodesics. We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geo…

Siegel modular forms of genus 2 attached to elliptic curves

The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F) is divisible by an abelian L-function, nor of Yoshida type, in which case L(s,F) is a product of L-series of a pair of elliptic cusp forms. Two key examples are the forms F defined by the symmetric cube of a non-CM elliptic curve E over the rationals, and those defined by anticyclotomic twists (of even non-zero weight) of the base change of such an E to an imaginary qu…

On some Siegel threefold related to the tangent cone of the Fermat quartic surface

Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{\rm sa}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential $(2,0)$-form on $Z$. The variety $Z$ is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of $Z$. One of these, denoted by $W$, is a kind of Igusa compactification such…

Hodge环和CM阿贝尔簇上解析周期之间的二次关系

Hodge cycles and quadratic relations between holomorphic periods on CM abelian varieties

本文研究了CM阿贝尔簇上解析周期的Hodge关系，证明通过单调二次关系可生成所有非平凡关系，并展示了构造的B在特定域上的分裂性质。

On the subring of special cycles

For a totally real field F of degree d>1 and a quadratic space V of signature (m,2)^{d_+} x (m+2,0)^{d-d_+} with associated Shimura variety Sh(V), we consider the subring of cohomology generated by the classes of weighted special cycles. We assume that d_+<d. We take the quotient SC(V) of this ring by the radical of the restriction of the intersection pairing to it. We show that the inner products of classes in SC(V) are determined by Fourier coefficients of pullbacks of Hilbert-Siegel Eisenstein series of genus m to products of smaller Siegel spaces and that the products of classes in SC(V) a…