Effective Siegel's theorem for modular curves
Yuri Bilu, Marco Illengo
我们证明了在除1/2个模曲线外的所有模曲线上，以及在除1个素幂水平的模曲线外的所有模曲线上，都可以有效地确定整点。
We prove that integral points can be effectively determined on all but finitely many modular curves, and on all but one modular curve of prime power level.
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In order to considering the integrality of nearly holomorphic (vector-valued) Siegel modular forms, we introduce nearly Siegel modular forms and study their integrality. We show that the integrality of nearly Siegel modular forms in terms of their Fourier expansion implies the integrality of their CM values. Furthermore, we show that there exists a one-to-one correspondence between integral nearly Siegel modular forms and integral nearly holomorphic ones. By these results, the integrality of CM values holds for nearly holomorphic Siegel modular forms and for nearly overconvergent p-adic Siegel…
On p-adic properties of Siegel modular forms
证明了水平Γ_0（p^m）的Siegel模形式是p-adic模形式.此外，我们证明了这种Siegel模形式的导数是p-adic的。部分结果对向量值模形式也是有效的。在我们的做法，以p-adic西格尔模形式，我们密切关注塞尔;他的证明，但不推广到西格尔的情况下，或需要一些修改。
On $p$-adic Siegel modular forms of non-real Nebentypus of degree 2
利用Maass提升证明了$Γ_0^{(2)}(p)$的非实Neumypus的所有Siegel模形式都是$p$ -adic Siegel模形式.
We give two applications of Arthur's multiplicity formula to Siegel modular forms. The one is a lifting theorem for vector valued Siegel modular forms, which contains Miyawaki's conjectures and Ibukiyama's conjectures. The other is the strong multiplicity one theorem for Siegel modular forms of scalar weights and level one.
The aim of this paper is to show lifts from pairs of two elliptic modular forms to Siegel modular forms of half-integral weight of even degree under the assumption that the constructed Siegel modular form is not identically zero. The key of the proof is to introduce a certain generalization of the Maass relation for Siegel modular forms of half-integral weight of general degree.
The structure of Siegel modular forms modulo p and U(p) congruences
我们确定Siegel模形式的度g模一个素数p的环结构，延长Nagaoka的结果的情况下，度g=2。我们刻画了Jacobi形式和Siegel模形式的U（p）同余，并惊奇地发现了Siegel模形式在偶数和奇数次上的不同行为.
The goal of this paper is to improve existing bounds for Fourier coefficients of higher genus Siegel modular forms of small weight.
Deligne proved that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g>1 must be integral or half integral. We give a different proof for this. It uses Mennicke's result that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from a paper of Bass-Milnor-Serre.
Ramanujan congruences for Siegel modular forms
我们确定的条件存在和不存在的Ramanujan型同余的Jacobi形式。我们将这些结果扩展到Siegel模形式的程度2和作为一个应用程序，我们建立Ramanujan型同余明确的例子Siegel模形式。
On the existence of Siegel modular forms with extra twists
该研究聚焦带额外扭的Siegel模形式的存在性问题，给出了这类模形式存在额外扭时水平与亏格的必要条件，还构造了不同于复共轭的具体实例，推进了相关数论问题研究。
Explicit matrices for Hecke operators on Siegel modular forms
给出了Hecke算子$T(p)$、$T_j(p^2)$在Siegel模形式上作用的一组显式矩阵.
On the p-adic integration over Igusa towers of Siegel modular varieties
本文提出了一种针对Igusa塔的显式p进积分理论，用于研究模Siegel流形的互反律。
We describe the $p$-divisibility transposition for the Fourier coefficients of Siegel modular forms. This provides a supplement to the result by Wilton for $p$-divisibility satisfied by the Ramanujan $τ$-function.
We deduce a weighted equidistribution theorem of the Satake parameters of Siegel cusp forms on Sp_2({\mathbb Z})with growing even weights.
Galois characterization of Endoscopy for rational Siegel modular forms
对于亏格为2的Siegel尖点形式，我们建立了Galois约化与Endoscopy之间的关系，这些Siegel形式具有有理特征值，且在3处是非分歧的
On the graded ring of Siegel modular forms of degree two with respect to a non-split symplectic group
本文给出了关于非分裂辛群的二次Siegel模形式的分次环。
New bounds for fundamental Fourier coefficients of Siegel modular forms
该研究运用Iwaniec方法得到Jacobi形式Fourier系数的新界，并结合二次Siegel模形式的Fourier-Jacobi展开，推导出Siegel模形式基本Fourier系数的强上界。
Explicit computations of Siegel modular forms of degree two
与经典的模形式不同，目前还没有一种通用的方法来实现任意权、阶和特征的Siegel模形式的计算，即使是二阶的。然而，有一种方法可以以统一的方式来做。在提供了一个调查已知的计算，我们描述了实现一个类建模西格尔模形式的程度在塞奇。特别是，我们描述的算法来计算各种环的西格尔模形式，其中许多是在我们的类实现。多种Siegel模形式（例如，向量值和标量值）都可以通过这个类建模，并且我们通过我们称为形式Siegel模形式的构造来统一这些。我们定义这个概念，并详细讨论它。
Let $E_n$ be the Siegel Eisenstein series of degree $n$ and weight $k$ with a complex parameter $s$. In this paper, using a differential operator $D$ by Ibukiyama which sends a scalar valued Siegel modular form to the tensor product of two vector valued Siegel modular forms, under a certain condition, we give a formula of $DE_{p+q}$ on $H_p\times H_q$, where $H_n$ is the Siegel upper half space of degree $n$. Furthermore, we give some applications of this formula, i.e., analytic properies of standard $L$-functions and the Klingen Eisenstein series and algebraicity results for Siegel modular fo…
On Zeta Functions and Families of Siegel Modular Forms
设$p$是素数，$Γ=\Sp_g(\Z)$是亏格为$g$的Siegel模群.我们研究$p$ -adic族的zeta函数和Siegel模形式.用附在$\Sp_g$上的动机$L$ -函数来描述Siegel模形式的$L$ -函数，并给出它们的解析性质.讨论了旋量$L$ -函数和$p$ -adic构造的临界值.建立了高等亏格的Rankin引理。给出了从$ GSp_{2m} \times GSp_{2m}$提升到$GSp_{4m}$（属$g=4m$）的一般猜想.利用Ikeda-Miyawaki构造给出了Siegel模形式的$p$ -adic族的构造.
We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases g=1,2, which do not extend to the case of general g.
Given a characteristic, we define a character of the Siegel modular group of level 2, the computations of their values are also obtained. By using our theorems, some key theorems of Igusa [1] can be recovered.
Siegel Modular Forms and Theta Series attached to quaternion algebras II
我们继续我们的研究吉田的提升，它与一对自守形式的adelic乘法群的四元数代数西格尔模形式的程度2。我们在这里考虑的情况下，自守形式的四元数代数对应于模形式的任意偶数重量和平方自由水平，特别是我们得到的西格尔模形式的重量3连接到一对椭圆模形式的重量2和4。
Computations of vector-valued Siegel modular forms
我们进行了一些计算的向量值Siegel模形式的程度2，重量（k，2）和水平一。我们的方法是基于Satoh的向量值西格尔模形式的重量（k，2）和明确的描述的Hecke行动上的傅立叶展开的模块的描述。我们强调三个实验结果：（1）我们确定了一个合理的特征形在三维空间的尖点形式形，（2）我们观察到，非尖形的特征形的水平一并不总是合理的和（3）我们验证了一些情况下，关于经典的模形式和Siegel模形式之间的同余关系。
M24-twisted Product Expansions are Siegel Modular Forms
Cheng构造了与M_24月光有关的K3曲面的对称幂的椭圆亏格的扭曲的乘积展开式。我们研究它们中哪些是Siegel模形式。如果预测的水平是非复合的，则它们是模的，并且它们的幂可以表示为重新缩放的Borcherds乘积的乘积。
In this work we obtain algebraicity results on special $L$-values attached to Siegel-Jacobi modular forms. Our method relies on a generalization of the doubling method to the Jacobi group obtained in our previous work, and on introducing a notion of near holomorphy for Siegel-Jacobi modular forms. Some of our results involve also holomorphic projection, which we obtain by using Siegel-Jacobi Poincaré series of exponential type.
Bounds for Siegel Modular Forms of genus 2 modulo $p$
Sturm获得了椭圆模形式$f$的第一傅里叶系数的数目的界限，以确定$f$模素数$p$的消失。本文研究亏格为2的Siegel模形式的Sturm界的类似.我们证明了所得的边界是尖锐的。作为应用，我们研究了亏格为2的Siegel MDoular型的傅里叶系数的包含Atkin $U(p)$ -算子的同余.
Kohnen's limit process for real-analytic Siegel modular forms
Kohnen介绍了一个限制过程西格尔模形式，生产雅可比形式。他问，如果有一个空间的真正分析西格尔模形式，使斜全纯雅可比形式出现通过这一限制过程。本文主要研究调和斜Maass-Jacobi型和调和Siegel-Maass型。我们改进了Maass关于调和Siegel-Maass形式的傅里叶系数的一个结果，这使我们能够与调和斜Maass-Jacobi形式建立联系.特别是，我们肯定地回答Kohnen的问题。
Hecke operators on Hilbert-Siegel modular forms
我们定义了Hilbert-Siegel模形式和作用于它们的Hecke“算子”。与希尔伯特模形式一样，这些线性变换不是线性算子，直到我们考虑模形式（具有不同群）空间的直积，我们可以在某些空间之间进行模自然标识。利用希尔伯特-西格尔形式，我们确定了模形式的某些空间之间的几个自然标识族。我们关联的傅里叶系数的形式在我们的积空间，甚至积分格，独立的基础和选择的系数环。然后，我们确定行动的Hecke运营商对这些傅里叶系数，平行的结果Hafner和Walling西格尔模形式（其中的数域是域的有理数）。
Automatic convergence for Siegel modular forms
本文为 Bruinier-Raum 的“自动收敛”定理的尖点情形提供了一种新证明，并指出该方法同样适用于例外群上的尖点四元数模形式。
Growth and nonvanishing of restricted Siegel modular forms arising as Saito-Kurokawa lifts
本文研究了Siegel模形式对$\mathbb{H} \times \mathbb{H}$的限制在Siegel模形式是Saito-Kurokawa升力的情况下的解析行为。Ichino的一个公式将这种行为与$GL_3 \times GL_2$ $L$ -函数族联系起来。
In this work we study the analytic properties of the standard L-function attached to Siegel-Jacobi modular forms of higher index, generalizing previous results of Arakawa and Murase. Furthermore, we obtain algebraicity results on special L-values in the spirit of Deligne's Period Conjectures.
We construct a ring of meromorphic Siegel modular forms of degree 2 and level 5, with singularities supported on an arrangement of Humbert surfaces, which is generated by four singular theta lifts of weights 1, 1, 2, 2 and their Jacobian. We use this to prove that the ring of holomorphic Siegel modular forms of degree 2 and level $Γ_0(5)$ is minimally generated by eighteen modular forms of weights 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 10, 11, 11, 11, 13, 13, 13, 15.
The supersingular loci and mass formulas on Siegel modular varieties
我们描述了Siegel 3-fold的超奇异轨迹，它具有一个parahoric水平结构。我们还研究了它的高维推广。利用这一对应关系和李和奥尔特的一个深层次的结果，我们评估了西格尔模空间$\calA_{g,1,N}$的超奇异轨迹的不可约分量的数量，对于任意的$g$。
An algorithm to compute upper bounds of dimensions for Siegel Modular Forms of Prime Level and Arbitrary Nebentypus
该研究提出了一种算法，用于计算素数层级和任意特征的塞格尔模形式维数上界，并实现了计算其傅里叶展开的Java程序。
Hecke eigenvalues of Siegel modular forms of "different weights"
给定两个不同权重的西格尔本征形，我们确定明确的Hecke本征值的两种形式，必须是不同的。在二次，并在一些额外的条件下，我们确定明确的傅里叶系数的两种形式，必须是不同的。
Kato explicit reciprocity law for Siegel modular forms of weight $(3, 3)$
本文将 Kato 显式互反律从模曲线推广到两条模曲线的乘积，并利用嵌入 Siegel 三维簇，得到权重 $(3,3)$ 的 Siegel 模形式所附 $p$-进表示的唯一次临界扭转的显式互反律。
Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler--Shimura morphism
本文通过推广完美化方法构建了Siegel模形式的过收敛自守sheaf，其全局截面即为过收敛Siegel模形式，并建立了适用于Siegel模形式的过收敛Eichler—Shimura映射。
On the images of the Galois representations attached to genus 2 Siegel modular forms
我们解决的问题的确定的图像的伽罗瓦表示附属2西格尔尖点形式的水平1具有多重性。这些表示是辛的。我们证明了如果Siegel尖点形式不是Maass特殊形式，则对于几乎所有素数，象都是尽可能大的，并验证了两个易于检验的条件.
An explicit version of Shimura's reciprocity law for Siegel modular functions
本文给出了Siegel模函数奇异值的Shimura互反律的显式版本，并据此构造了比Igusa不变量更小的四次CM域类不变量，同时为Tonghai Yang的定理提供了新证明。
In this work we use the Rankin-Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted $L$-function beyond the usual range of absolute convergence. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.
Hyperbolic distribution problems on Siegel 3-folds and Hilbert modular varieties
本文将W. Duke（Inventiones 1988）在Poincare上半平面中Heegner点和原始正定向闭测地线的等分布上的结果，服从某些次凸性结果。我们还证明了消失的结果与类似的问题的西格尔上半空间的2度的尖形外尔和极限。特别是，这些Weyl和与家庭的亨伯特表面在西格尔3倍和模曲线在这些亨伯特表面。
For positive integers $g$ and $N$, let $\mathcal{F}_N$ be the field of meromorphic Siegel modular functions of genus $g$ and level $N$ whose Fourier coefficients belong to the $N$th cyclotomic field. We present explicit generators of $\mathcal{F}_N$ over $\mathcal{F}_1$ in terms of quotient of theta constants, when $g\geq2$ and $N\geq 3$.
We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $\bmod \ell$, partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto $x$ which do not vanish $\bmod \ell$. We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients $\bmod \ell $ of a Siegel modular form with integral algebraic Fourier coefficients provi…
A Restriction Norm Problem for Siegel Modular Forms
本文建立了2次Siegel尖形式在虚轴上平均$L^2$-范数的渐近公式，误差项有幂次节省，与质量等分布猜想和Lindelöf假设一致，并细致分析了Kitaoka公式。
We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms.
We correct the proof of the theorem in the previous paper presented by the first named author, which concerns Sturm bounds for Siegel modular forms of degree $2$ and of even weights modulo a prime number dividing $2\cdot 3$. We give also Sturm bounds for them of odd weights for any prime numbers, and we prove their sharpness. The results cover the case where Fourier coefficients are algebraic numbers.
Cycle integrals of meromorphic modular forms and Siegel theta functions
本研究探讨正定二元二次型相关亚纯模形式的周期积分，证明其适当线性组合的周期积分为有理值，并借助Hecke不定theta函数算出符号(1,2)偶格对应的Siegel theta函数的周期积分，该副结果也具独立价值。
Sturm type theorem for Siegel modular forms of degree 2
本文试图将Sturm证明的椭圆模形式的一个同余性质推广到Haupttypus的Siegel的2次水平模形式的同余性质。也就是说，我们给出了确定模形式的同余所需的傅里叶系数的显式界。我们给出了Sturm定理在Jacobi形式下的类比，这是证明中所需要的。在Neumypus of Siegel的模形式具有素水平的情况下，为了证明两个模形式之间的全等性，我们证明了只要检查两个傅里叶系数的全等性就足够了.最后，我们给出了我们的主要定理的两个例子。
Upper bound on the number of systems of Hecke eigenvalues for Siegel modular forms (mod p)
本文给出了维数为g、水平N与p互素的Siegel模形式（mod p）的Hecke特征值系统个数的一个显式上界.在椭圆模形式（g=1）的特殊情况下，我们的结果与G.赫里克
A minimal modularity lifting theorem for Siegel modular forms
该研究在亏格为二的西格尔模形式下，证明极小模性提升定理，剩余表示源于稳定吉田提升。基于\(R = \mathbb{T}\)定理，建立通用极小普通伽罗瓦变形环自由性及希达族唯一性。
The Kohnen plus space, roughly speaking, is a space consisting of modular forms of half integral weight with some property in Fourier coefficients. For example, the $n$-th coefficient of a normal modular form of weight $k+1/2$ in the plus space is $0$ unless $(-1)^kn$ is congruent to some square modulo $4$. The concept of plus space was initially introduced by Kohnen in 1980. Eichler and Zagier showed that the plus space is isomorphic to the space of Jacobi forms in the one variable case. Later, Ibukiyama generalized these results to the cases for Siegel modular forms in 1992. Also, Hiraga and…
Runge's method is a tool to figure out integral points on curves effectively in terms of height. This method has been generalised to varieties of any dimension, unfortunately its conditions of application are often too restrictive. In this paper, we provide a further generalisation intended to be more flexible while still effective, and exemplify its applicability by giving finiteness results for integral points on some Siegel modular varieties. As a special case, we obtain a totally explicit finiteness result for integral points on the Siegel modular variety $A_2(2)$.
A Note on Fourier-Jacobi coefficients of Siegel modular forms
设F是权为k亏格n>1的Siegel尖点形式，其Fourier-Jacobi系数为fm.在这篇文章中，我们估计了f_m的Petersson范数的增长性，其中m在等差数列上运行。这一结果改进了Kohnen在[5]中的一个结果。
On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups
证明了关于同余子群的2次Siegel尖点形式的傅里叶系数的Petersson型公式，并作为推论给出了个别傅里叶系数的上界估计.本文中的方法基本上是一个概括的北冈以前的工作，研究了充分的模块化的情况下，但一些修改是必要的，以获得估计是尖锐的水平方面。
This article proves that Tate classes on Siegel modular 3-folds are spanned by the images of Hilbert modular surfaces at degree 2 and by the images of Shimura curves at degree 4. The proof involves a careful study of the period pairing between degree-4 rapidly decreasing cohomological forms and Hilbert modular surfaces.
We investigate Poincaré series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincaré series are almost holomorphic as well. In general this is not the case. The main point of this paper is the study of Siegel-Poincaré series of degree $2$ attached to products of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. …
We prove an equidistribution statement for the Satake parameters of the local representations attached to Siegel cusp forms of degree $2$ of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of $L$-functions attached to these cusp forms.
On certain constructions of p-adic families of Siegel modular forms of even genus
假设p > 5是一个有理数素数。从Hida的p级普通椭圆尖点形式的p-adic解析族出发，通过Ikeda的Duke-Imamoglu提升，构造了与Hida的p-adic解析族相关联的任意偶数亏格的p级全纯Siegel尖点形式的p-adic解析族.此外，我们也给出了具有平凡Neuryypus的偶数亏格的Siegel Eisenstein级数的类似结果。
Cohomological vanishing on Siegel modular varieties and applications to lifting Siegel modular forms
利用Siegel模簇上层上同调的消失结果研究了两个提升问题：（a）Siegel模形式（mod p）何时能提升到特征零？这使用并扩展了Stroh和Lan-Suh以前关于尖点形式的结果。(b)Siegel模形式对模空间边界的限制何时是满射映射？我们研究这个问题的任意特征，推广Weissauer和荒川的分析结果。
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