Fabien Cléry, Gerard van der Geer, Samuel Grushevsky
We study vector-valued Siegel modular forms of genus 2 and level 2. We describe the structure of certain modules of vector-valued modular forms over rings of scalar-valued modular forms.
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We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.
The Siegel modular forms of genus 2 with the simplest divisor
证明了亏格为2的仿模群的Hecke型同余子群沿Siegel上半平面的对角线精确为零的Siegel模形式存在8个.这是在“黑洞、黑环和模形式“会议（ENS，巴黎，2007年8月）期间提出的一个问题的解决方案。这些模形式推广了经典的Igusa形式和Gritsenko和Nikulin在1998年构造的形式。
We extend Igusa's description of the relation between invariants of binary sextics and Siegel modular forms of degree two to a relation between covariants and vector-valued Siegel modular forms of degree two. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree two.
We define a family of arithmetic zero cycles in the arithmetic Chow group of a modular curve X_0(N), for N>3 odd and squarefree, and identify the arithmetic degrees of these cycles as q-coefficients of the central derivative of a Siegel Eisenstein series of genus two. This parallels work of Kudla-Rapoport-Yang for Shimura curves.
We use vanishing results for sheaf cohomology on Siegel modular varieties to study two lifting problems: (a) When can Siegel modular forms (mod p) be lifted to characteristic zero? This uses and extends previous results for cusp forms by Stroh and Lan-Suh. (b) When is the restriction of Siegel modular forms to the boundary of the moduli space a surjective map? We investigate this question in arbitrary characteristic, generalising analytic results of Weissauer and Arakawa.
On the cohomological dimension of Siegel modular varieties and the modularity of formal Siegel modular forms
本文证明亏格g≥2时Siegel模簇的凝聚上同调维数至多为g(g+1)/2-2，由此推出其紧化边界满足Grothendieck-Lefschetz条件，从而形式Siegel模形式自动成为经典模形式，推广了Bruinier和Raum的工作。
Computing genus 2 Hilbert-Siegel modular forms over $\Q(\sqrt{5})$ via the Jacquet-Langlands correspondence
本文给出了计算窄类数为1的实二次域上Hilbert-Siegel尖点形式的Hecke本征系的一个算法。我们使用二次域$\Q(\sqrt{5})$给出一些说明性的例子。在这些例子中，我们确定希尔伯特-西格尔本征形是希尔伯特本征形的可能提升。
On the supersingular locus of the Siegel modular variety of genus 3 or 4
本文研究亏格3或4的Siegel模簇的超奇异轨迹，在perfection后将其分解为Coxeter型Deligne-Lusztig簇与有限维仿射空间乘积的不交并。
This paper gives a simple method for constructing vector-valued Siegel modular forms from scalar-valued ones. The method is efficient in producing the siblings of Delta, the smallest weight cusp forms that appear in low degrees. It also shows the strong relations between these modular forms of different genera. We illustrate this by a number of examples.
Siegel Modular Forms
讲座西格尔模形式在Nordfjordeid。除了调查西格尔模形式它提出了联合工作与卡雷尔费伯向量值西格尔模形式的属2，它提供了证据的猜想哈德之间的同余g=1和g=2的形式。
Theory of the Siegel Modular Variety
本文从算术和几何两个方面讨论了Siegel模簇的理论。本文涵盖了理论的西格尔模形式，赫克理论，解除椭圆尖点形式，几何性质的西格尔模品种，（假设）动机重视西格尔模形式和上同调的西格尔模品种。
We construct one parameter families of overconvergent Siegel-Hilbert modular forms. In particular, for any classical Siegel-Hilbert modular eigenform one can find a rigid analytic disc centered at this point, on which an infinite family of classical points with varying weights accumulates at the center.
In this paper we will describe all vector-valued Siegel modular forms of degree 2 and weight ${\rm Sym}^6({\rm St}) \otimes \det^{k}({\rm St})$ with $k$ odd. These vector-valued forms constitute a module over the ring of classical Siegel modular forms of degree 2 and even weight and this module turns out to be free. In order to find generators, we generalize certain Rankin-Cohen differential operators on triples of classical Siegel modular forms that were first considered by Ibukiyama and we find a Rankin-Cohen bracket on vector-valued Siegel modular forms.
The geometry of Siegel modular varieties
这是一篇关于复数上Siegel模簇的综述文章。它主要是从阿贝尔簇的模的角度来写的，尤其是曲面。我们讨论了Siegel模簇的紧化，紧化簇的Kodaira维数分类等。交换曲面的模，特别是Jacobi形式到模形式的提升的应用;一些特殊Siegel模3-折叠的投影模型;非主极化交换曲面;以及构造退化的阿贝尔簇族。
This is a survey based on the construction of Siegel modular forms of degree 2 and 3 using invariant theory in joint work with Fabien Cléry and Carel Faber.
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。
In this paper we extend some results of Norman and Oort and of de Jong, and give an explicit description of the geometry of the Siegel modular threefold with paramodular level structure. We also discuss advantages and restrictions of three standard methods for studying moduli spaces of abelian varieties.
Computing the Satake p-parameters of Siegel modular forms
经典模形式的Hecke特征值经常编码大量的算术数据。的佐竹$p$ -参数的西格尔模形式发挥的作用类似于一个发挥赫克特征值的表征经典模形式。本文给出了一个算法，如果给定$F$关于局部Hecke代数的生成元的Hecke特征值，就可以计算Siegel模形式$F$的Satake $p$ -参数.我们计算明确的例子，并与我们的计算的Ramanujan-Petersson猜想。
Let p be an odd prime and g an integer greater or equal to 2. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this result relies on the construction of a family of sheaves of locally analytic overconvergent modular forms.
Siegel modular forms arising from higher Chow cycles
本文证明了一般阿贝尔簇上高阶Chow环的无穷小不变量可构造出具有有界奇点的亚纯Siegel模形式，且该构造与秩1退化下的Siegel算子具有函子性。
Siegel modular forms (mod p) and algebraic modular forms
在给泰特的信中，塞尔证明了系统的Hecke特征值给出的模形式（mod p）是相同的本地常数函数的一个adelic双陪集空间构造的自同态代数的超奇椭圆曲线。我们将这一结果推广到Siegel模形式（mod p），证明了从它们得到的Hecke特征值系统与Gross定义的四元数酉群上的代数模形式（mod p）给出的Hecke特征值系统相同。通过限制到阿贝尔簇的模空间的超特殊轨迹，得到了对应关系。
We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichmüller cusp forms on \overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases.
We prove that the ring of Siegel modular forms of weight divisible by g+n+1 is isomorphic to the ring of (log) pluricanonical forms on the n-fold Kuga family of abelian varieties and its certain compactifications, for every arithmetic group for a symplectic form of rank 2g>2. We also give applications to the Kodaira dimension of the Kuga variety. In most cases, the Kuga variety has canonical singularities.
Siegel modular forms of degree three and invariants of ternary quartics
本研究确定了三次Siegel模形式分次环的结构，明确其由19个模形式生成并找到7个权的齐次参数系，同时建立了三元四次式的Dixmier-Ohno不变量与这些生成元的完整对应。
Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms
在给泰特的一封信（1996年发表在《以色列数学杂志》上）中，J. P.Serre证明了在由超奇椭圆曲线的自同态代数构造的椭圆双陪集空间上，由模形式（mod p）给出的Hecke特征值系与由局部常数函数给出的Hecke特征值系是相同的.我们将这个结果推广到Siegel模形式，证明了Siegel模形式（mod p）给出的Hecke特征值系统与Gross在Israel J. Math.于1999年定义的四元数酉群上的代数模形式（mod p）给出的Hecke特征值系统相同。通过限制到阿贝尔簇的模空间的超特殊轨迹，得到了对应关系。
We develop two structure theorems for vector valued Siegel modular forms for Igusa's subgroup Γ_2[2,4], the multiplier system induced by the theta constants and the representation Sym^2. In the proof, we identify some of these modular forms with rational tensors with easily handleable poles on P^3\C. It follows that the observed modules of modular forms are generated by the Rankin-Cohen brackets of the four theta series of the second kind.
The number of Ekedahl-Oort strata intersecting the supersingular locus in the Siegel modular variety
本文证明了 Siegel 模簇中与超奇异轨迹相交的 Ekedahl-Oort 层数 $N_g$ 在 $g$ 趋于无穷时渐近等于 $2^{g-1}$。
Genus Two Quasi-Siegel Modular Forms and Gromov-Witten Theory of Toric Calabi-Yau Threefolds
本文发展了亏格二拟Siegel模形式与拟Siegel Jacobi形式的微分环理论，并将其应用于带膜的局部Calabi-Yau三流形的拓扑递归，证明了开闭Gromov-Witten势本质上是亏格二拟Siegel模形式。
In this article we prove several level lowering results for cuspidal automorphic representations occurring in the cohomology of the Siegel modular threefold with paramodular level structure by adapting a method of Ribet in his proof of the Serre's epsilon conjecture. The proof is purely geometric and relies on the description of the supersingular locus of certain quaternionic unitary Shimura variety and an arithmetic level raising result on this Shimura variety. The heart of the proof is a comparison of the dimension of the space of vanishing cycles on the paramodular Siegel modular threefold …
We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.
Siegel modular forms of genus 2 attached to elliptic curves
本文的目的是构造一类有算术意义的亏格为2的全纯Siegel尖点形式F，它们既不是Saito-Kurokawa型，也不是Yoshida型，在Saito-Kurokawa型的情况下，4次旋量L-函数L（s，F）可被交换L-函数整除，在Yoshida型的情况下，L（s，F）是一对椭圆尖点形式的L-级数的乘积.两个关键的例子是由有理数上的非CM椭圆曲线E的对称立方定义的形式F，以及由这样的E到虚二次域K的基变化的反分圆扭曲（偶非零权重）定义的形式F。其中一个主要成分是GL（4）上的尖点形式π到GSp（4）的转移，尽管是间接的，但具有许多预期的性质，这可能是独立的兴趣。当π是正则的并且与一个\ell -adic伽罗瓦表示相关联时，转移信息是完整的.我们还呼吁（独立）的结果Laumon和Weissauer的zeta函数的西格尔模块化的三倍。
On the transcendence degree of the differential field generated by Siegel modular forms
椭圆模函数满足一个三阶代数微分方程，且不满足低阶方程，这是一个经典事实。我们展示了如何推广到西格尔模函数的任意程度。其关键思想是，它们所满足的偏微分方程是由高斯-马宁联络所支配的，其单值群是众所周知的。模θ函数对我们的结果提供了一个具体的解释，并且我们在2次的情况下详细研究了它们的微分性质。
Dimension formulas for spaces of vector-valued Siegel modular forms of degree two and level two
本文结合局部系统上同调描述与 Euler 特征计算，求得了二阶二级向量值 Siegel 模形式空间在 6 个字母对称群下的同型分解。
In this paper we investigate the (classical) weights of mod $p$ Siegel modular forms of degree 2 toward studying Serre's conjecture for $GSp_4$. We first construct various theta operators on the space of such forms a la Katz and define the theta cycles for the specific theta operators. Secondly we study the partial Hasse invariants on each Ekedahl-Oort strata and their local behaviors. This enable us to obtain a kind of weight reduction theorem for mod $p$ Siegel modular forms of degree 2 without increasing level.
We study over rings of scalar valued Siegel modular forms. modules of vector valued modular forms of degree two. For the two simplest representations, standard and Sym^2, appears rather natural consider the cases of the group $Γ[4,8] $ and $Γ[2,4].$ In these case we give the complete structure of the modules . The main tools are Theta functions and their derivatives evaluated at $z=0$
The Kodaira dimension of Siegel modular varieties of genus 3 or higher
我们考虑具有互质极化和满能级$n$结构的（非主）极化阿贝尔簇属$g\geq3$的模空间$A_{pol}(n)$.基于对math/0405321中Tits构造的分析，给出了$n$的一个显式下界，当满足一个进一步的显式条件时，该下界足以使紧化模空间是一般型的.
Embeddings of Certain Exceptional Shimura Varieties into Siegel Modular Varieties
本文定义了一类包含某些例外群局部Shimura变种的局部Shimura变种，并为该类构建了从(G, μ)-显示到p-可除群的函子。作为应用，证明了对于该类，局部Shimura变种在无穷级可代表且完美oid。考虑该类的全局对应物，将某些例外Shimura变种嵌入到Siegel模变种中。特别是，证明它们在无穷级是完美oid。
Siegel modular forms mod p
证明了特征p不等于2的域上亏格g>1的具有水平结构N的主极化阿贝尔簇模叠加的一个形式的消灭定理.这用于计算这些堆栈的Picard群。作为应用，我们得到了Siegel积分模形式之间的一个同余关系.
We state conjectures on the asymptotic behavior of the Masur-Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel-Veech constants as the genus tends to infinity.
We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A_3 of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower genera. The conjecture is based on extensive counts of curves of genus three and abelian threefolds over finite fields. It provides a lot of new information about vector-valued Siegel modular forms of d…
Let $\mathcal Q_D$ be the set of moduli points on Siegel's modular threefold whose corresponding principally polarized abelian surfaces have quaternionic multiplication by a maximal order $\mathcal O$ in an indefinite quaternion algebra of discriminant $D$ over $\mathbb Q$ such that the Rosati involution coincides with a positive involution of the form $α\mapstoμ^{-1}\overlineαμ$ on $\mathcal O$ for some $μ\in\mathcal O$ with $μ^2+D=0$. In this paper, we first give a formula for the number of irreducible components in $\mathcal Q_D$, strengthening an earlier result of Rotger. Then for each irr…
Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures
本文研究了具有2级结构的主极化交换曲面的模空间上的某些局部系统的上同调。Frobenius在这些局部系统的平展上同调群的交错和上的迹可以通过在给定的有限域上计算亏格为2的点曲线的个数来计算。这种上同调与向量值Siegel模形式密切相关。在[FvdG]中进行了水平1中的相应方案。在这里，我们将其扩展到出现新现象的级别2。我们确定了Eisenstein上同调及其S_6作用量对全二阶结构的贡献，并在此基础上对内窥镜贡献作了精确的计算。我们还预测了齐藤-黑川升力的矢量值模拟的存在。假设这些假设是基于充分的数值证据，我们得到的Heck算子T（p）的p < 41的“真正的”西格尔模形式的剩余空间上的痕迹。我们提出了一些例子，1维空间的特征形，这些痕迹与赫克特征值相吻合。我们希望，专家在解除和内窥镜检查将能够证明我们的知识。
All Siegel Hecke eigensystems (mod p) are cuspidal
我们证明了在具有固定维数g、固定水平N和变权的Siegel模形式（mod p）空间中出现的Hecke特征值系统与在具有相同参数和变权的Siegel尖点形式空间中出现的Hecke特征值系统是相同的。特别地，在g=1的情况下，这表明来自经典模形式的Hecke特征系统（mod p）与来自尖点形式的Hecke特征系统相同。证明使用了对超特殊轨迹的限制。我们也给出了比较尖点形式的佐竹紧化与环形紧化的西格尔模品种。
Arithmetic Siegel--Weil formula on the product of modular curves
该研究证明了两条模曲线乘积上带任意奇异系数的完整生成级数算术Siegel–Weil公式，对应Kudla纲领的GSpin(2,2)情形。
For $g=8,12,16$ and $24$, there is a nonzero alternating $g$-multilinear form on the ${\rm Leech}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of ${\rm Leech}$. The harmonic Siegel theta series built from these alternating forms are Siegel modular cuspforms of weight $13$ for ${\rm Sp}_{2g}(\mathbb{Z})$. We prove that they are nonzero eigenforms, determine one of their Fourier coefficients, and give informations about their standard ${\rm L}$-functions. These forms are interesting since, by a recent work of the authors, they are the only nonzero Siegel modular fo…
There is no stable Siegel modular form that vanishes on the trigonal locus in every moduli space of curves.
Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms
本文研究了两类关系：一类是Calabi-Yau流形的椭圆亏格与Jacobi模形式之间的关系，另一类是第二量子化椭圆亏格、Siegel模形式与Lorentzian Kac-Moody李代数之间的关系。我们还确定了具有整傅里叶系数的弱Jacobi形式的分次环的结构。它给了我们一些应用理论的椭圆亏格和第二个量化的椭圆亏格。
We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case $g$ is even.
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 $\mathcal C_{α, β}$. …
Picard groups of Siegel modular threefolds and theta lifting
证明了Humbert曲面有理生成Siegel模三重的Picard群.
Siegel modular forms of level (4,8) and weight two
本文研究了权为2的Siegel模形式空间，证明了当亏格 $g\ge 8$ 时，空间 $[Γ_g[4,8],2]$ 由4个theta nullwerte的乘积生成，补充了已有研究。
The Kudla-Millson lift of Siegel cusp forms
研究亏格2 Siegel尖点形式的Kudla-Millson提升的 injectivity，证明在偶格分裂出两个双曲平面且秩足够大时该提升是 injective 的，并引入向量值不定Siegel theta函数推广了Borcherds的结果。
Siegel operators for holomorphic differential forms
本文利用环面紧化与全纯 Leray 滤过，几何解释 Siegel 模簇上微分形式的 Siegel 算子，并建立多种边界消失概念的等价性。
The Satake compactification of the moduli space of principally polarized abelian surfaces with a level two structure has a degree 8 endomorphism. The aim of this paper is to show that this result can be extended to other modular threefolds. The main tools are Siegel modular forms and Satake compactifications of arithmetic quotients of the Siegel upper-half space. Indeed, the construction of the degree 8 endomorphism on suitable modular threefolds is done via an isomorphism of graded rings of modular forms. By studying the action of the Fricke involution one gets a further extension of the prev…
We study the cone of moving divisors on the moduli space ${\mathcal A}_g$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin-Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on ${\mathcal A}_g$ for $g\leq 4$, and gives an explicit upper bound for the moving slope of ${\mathcal A}_5$ and a conjectural upper bound for the moving slope of ${\mathcal A}_6$.
The EKOR-stratification on the Siegel modular variety with parahoric level structure
本文研究具有抛物水平结构的Siegel模簇模p约化的算术几何，通过构造到参数化均匀极化截断显示链的代数栈的光滑态射，实现了该簇上的EKOR-分层。
Quasimodular forms were first studied in the context of counting torus coverings. Here we show that a weighted version of these coverings with Siegel-Veech weights also provides quasimodular forms. We apply this to prove conjectures of Eskin and Zorich on the large genus limits of Masur-Veech volumes and of Siegel-Veech constants. In Part I we connect the geometric definition of Siegel-Veech constants both with a combinatorial counting problem and with intersection numbers on Hurwitz spaces. We introduce modified Siegel-Veech weights whose generating functions will later be shown to be quasimo…
The goal of the present paper is two-fold. First, we present a classification of algebraic K3 surfaces polarized by the lattice H+E_8+E_7. Key ingredients for this classification are: a normal form for these lattice polarized K3 surfaces, a coarse moduli space and an explicit description of the inverse period map in terms of Siegel modular forms. Second, we give explicit formulas for a Hodge correspondence that relates these K3 surfaces to principally polarized abelian surfaces. The Hodge correspondence in question underlies a geometric two-isogeny of K3 surfaces.
Igusa's modular form and the classification of Siegel modular threefolds
本文证明了关于Siegel模三重分类的两个结果。设A_{1,d}（n）是具有（1，d）-极化和全n阶结构的阿贝尔曲面的模空间，A_{1,d} ^{lev}（n）是其中固定了一个额外的正则水平结构的空间.证明了当（d，n）=1且n至少为4时，A_{1,d}（n）是一般型的.这是对所有d同时可以证明的最好的可能结果。设p是一个奇素数，并假设（p，n）=1。证明了A_{1,p} ^{lev}（n）的Voronoi紧化是光滑的且有充分的典范丛当且仅当n大于或等于5.
We study the hyperbolicity of compactifications of quotients of bounded symmetric domains by arithmetic groups. We prove that, up to an étale cover, they are Kobayashi hyperbolic modulo the boundary. Applying our techniques to Siegel modular varieties, we improve some former results of Nadel on the non-existence of certain level structures on abelian varieties over complex function fields.
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