Jae-Hyun Yang
In this paper, we study a fundamental domain for the Siegel-Jacobi space $Sp(g,{\mathbb Z})\ltimes H_{\mathbb Z}^{(g,h)}\backslash {\mathbb H}_g\times {\mathbb C}^{(h,g)}$.
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For any positive integers $n$ and $m$, $\mathbb{H}_{n,m}:=\mathbb{H}_n\times\mathbb{C}^{(m,n)}$ is called the Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are defined on this space. In this article we compute the Chern connection of the Siegel-Jacobi space and use it to obtain derivations of Jacobi forms. Using these results, we constructed a series of invariant differential operators for Siegel-Jacobi forms. Also two kinds of Maass-Shimura type differential operators for $\mathbb{H}_{n,m}$ are obtained.
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.
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.
In this paper, we compute Riemannian metrics on the Siegel-Jacobi space which are invariant under the natural action of the Jacobi group explicitly and also provide the Laplacians of these invariant metrics. These are expressed in terms of the trace form.
Let ${\mathbb D}_n $ be the generalized unit disk of degree $n$. In this paper, we find Riemannian metrics on the Siegel-Jacobi disk ${\mathbb D}_n \times {\mathbb C}^{(m,n)}$ which are invariant under the natural action of the Jacobi group explicitly and also compute the Laplacians of these invariant metrics explicitly. These are expressed in terms of the trace form. We give a brief remark on the theory of harmonic analysis on the Siegel-Jacobi disk ${\mathbb D}_n \times {\mathbb C}^{(m,n)}$.
Let F be a Siegel cusp form of weight k and genus n>1 with Fourier-Jacobi coefficients f_m. In this article, we estimate the growth of the Petersson norms of f_m, where m runs over an arithmetic progression. This result sharpens a recent result of Kohnen in [5].
Let ${\mathbb H}_g$ and ${\mathbb D}_g$ be the Siegel upper half plane and the generalized unit disk of degree $g$ respectively. Let ${\mathbb C}^{(h,g)}$ be the Euclidean space of all $h\times g$ complex matrices. We present a partial Cayley transform of the Siegel-Jacobi disk ${\mathbb D}_g\times {\mathbb C}^{(h,g)}$ onto the Siegel-Jacobi space ${\mathbb H}_g\times {\mathbb C}^{(h,g)}$ which gives a partial bounded realization of ${\mathbb H}_g\times {\mathbb C}^{(h,g)}$ by ${\mathbb D}_g\times {\mathbb C}^{(h,g)}$. We prove that the natural actions of the Jacobi group on ${\mathbb D}_g\tim…
Kohnen introduced a limit process for Siegel modular forms that produces Jacobi forms. He asked if there is a space of real-analytic Siegel modular forms such that skew-holomorphic Jacobi forms arise via this limit process. In this paper, we initiate the study of harmonic skew-Maass-Jacobi forms and harmonic Siegel-Maass forms. We improve a result of Maass on the Fourier coefficients of harmonic Siegel-Maass forms, which allows us to establish a connection to harmonic skew-Maass-Jacobi forms. In particular, we answer Kohnen's question in the affirmative.
For two positive integers $m$ and $n$, we let ${\mathbb H}_n$ be the Siegel upper half plane of degree $n$ and let ${\mathbb C}^{(m,n)}$ be the set of all $m\times n$ complex matrices. In this article, we study differential operators on the Siegel-Jacobi space ${\mathbb H}_n\times {\mathbb C}^{(m,n)}$ that are invariant under the natural action of the Jacobi group $Sp(n,{\mathbb R}\ltimes H_{\mathbb R}^{(n,m)}$ on ${\mathbb H}_n\times {\mathbb C}^{(m,n)}$, where $H_{\mathbb R}^{(n,m)}$ denotes the Heisenberg group. We give some explicit invariant differential operators. We present important pr…
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。
The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space.
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.
Problems in the Geometry of the Siegel-Jacobi Space
本文针对非对称齐次空间 Siegel-Jacobi 空间，提出其几何中的基本问题，强调其在几何与算术中的重要性。
An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms
该研究针对Sp₄的傅里叶-雅可比周期，在新分歧情形下计算相关局部积分，构建了关联二次西格尔尖型形式与半整数权形式Petersson范数的显式猜想恒等式，并指出其在范数增长等方面的推论。
In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential operators. We present problems which are natural. We give some new partial solutions for these natural problems.
$L^{\infty}$-norm bounds for Siegel--Jacobi cusp forms
该研究针对任意亏格的Siegel模群，借助高亏格Eichler--Zagier theta分解与已有Siegel尖点形式范数界，首次得到整权、指数参数下L²归一化Siegel--Jacobi尖点形式的显式一致L^∞范数上界。
We generalize the work of Fei, Bhowmik and Halupczok, and Jia relating the Goldbach conjecture to real zeros of Dirichlet $L$-functions.
Rings of Siegel-Jacobi forms of bounded relative index are not finitely generated
本文利用环面 b-除子与凸几何证明固定次数、权指比固定或有界的 Siegel–Jacobi 形式环非有限生成，并证明 Kramer 猜想、恢复 Tai 的渐近维数公式。
Modules of Jacobi forms of degree two of small levels
本文通过合并已知结果，为小水平2、3、4的二阶雅可比形式的模结构提供简单结构定理。
This is the lecture note of my invited lecture given at the International Conference on Number Theory at Harish-Chandra Research Institute in Allahabad (quite near the River Ganges), India on December 5, 2006. I gave an invited lecture on the same subject at Department of Mathematics, Kyoto University, Japan on June 23, 2009. The new lecture note contains more new contents and references.
Jacobi-Forms can be decomposed as a linear combination of Thetafunctions with modular forms as coefficients. It is shown that the space of these coefficient modular forms of Fourier-Jacobi-Forms, which come from Siegel cusp forms, has full rank in every point of the Satake boundary, if the index is 1, the weight is sufficiently large and the Satake boundary point has trivial stabilizer in $Γ_{n-1}$. This yields a local automorphic embedding of the Siegel modular variety. Klingen-Poincare series are the main tool. Despite of this richness it is proved that there are more Jacobi index 1 cusp for…
Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $ρ$ with respect to some congruence subgroup of $\mathrm{Sp}_{2n}(\mathbb Q)$. In this note, we prove that the function on $\mathrm{Sp}_{2n}(\mathbb R)$ obtained by lifting $F$ has the moderate growth (or "slowly increasing") property. This is a consequence of the following bound that we prove: $\|ρ(Y^{1/2})F(Z) \| \ll \prod_{i=1}^n (μ_i(Y)^{λ_1/2} + μ_i(Y)^{-λ_1/2})$ where $ λ_1 \ge \ldots \ge λ_n$ is the highest weight of $ρ$ and $μ_i(Y)$ are the eigenvalues of the matrix $Y$.
We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on $\mathbf H_2$ and those on $\mathbf{H}_1 \times \mathbf{C}$ (here $\mathbf H_n$ denotes the Siegel upper half space of degree $n$). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadrat…
We investigate the relation between Klingen's decomposition of the space of Siegel modular forms and Dulinski's analogous decomposition of the space of Jacobi forms.
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。
We compute the space $S_2(K(N))$ of weight $2$ Siegel paramodular cusp forms of squarefree level $N<300$. In conformance with the paramodular conjecture of A. Brumer and K. Kramer, the space is only the additive (Gritsenko) lift space of the Jacobi cusp form space $J_{2,N}^{\text{cusp}}$ except for $N=249,295$, when it further contains one nonlift newform. For these two values of $N$, the Hasse-Weil $p$-Euler factors of a relevant abelian surface match the spin $p$-Euler factors of the nonlift newform for the first two primes $p\nmid N$.
Notes on Mitsui's Prime Number Theorem with Siegel zeros
本文通过纳入Hecke L-函数的Siegel零点，改进了三井1957年的素数定理，使模的范数能以伪多项式速率增长，为后续研究素元素线性模式提供关键支持。
We prove that formal Fourier Jacobi expansions of degree 2 are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension 2, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree 2. Combining both results enables us to compute relations of special cycles in the second Chow group.
Dans cette note, nous montrons que certaines formes modulaires de Siegel de caractéristique p et de genre 2 ou 3 se relèvent en caractéristique 0. Ce résultat généralise un théorème classique obtenu par Katz pour les formes de genre 1. Nous utilisons des résultats de Shepherd-Barron et de Hulek et Sankaran, ainsi que des théorèmes d'annulation de la cohomologie cohérente dûs à Deligne, Illusie et Raynaud et à Esnault et Viehweg. ----- In this note, we show that cuspidal Siegel modular forms of characteristic p and genus 2 or 3 can be lifted to characteristic 0. This result extends a classical …
In this note, we announce results on integral points on some modular varieties, based on a generalisation of Runge's method in higher dimensions which will be explained beforehand. In particular, we obtain an explicit result in the case of the Siegel modular variety $A_2(2)$.
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。
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…
Certain Siegel Cusp Forms with Level are Determined by their Fundamental Fourier Coefficients
本文证明具有特定nebentypus的向量值Siegel尖点形式由与水平互素的基本Fourier系数决定，并在亏格3时加强至四元数极大序情形，同时推广到Jacobi形式。
This article is an expository paper. We first survey developments over the past three decades in the theory of harmonic analysis on reductive symmetric spaces. Next we deal with the particular homogeneous space of non-reductive type, the so called Siegel-Jacobi space that is important arithmetically and geometrically. We present some new results on the Siegel-Jacobi space.
For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an explicit expression for the $L$-functions of these curves in terms of Jacobi sums. Moreover, we deduce from this calculation that the curves $E_d$ satisfy an analogue of the Brauer-Siegel theorem. More precisely, we estimate the asymptotic growth of the product of the order of the Tate-Shafarevich group of $E_d$ (which is known to be finite) by its Néron-Tate regulator, in …
We investigate the growth of Fourier coefficients of Siegel paramodular forms built by exponentially lifting weak Jacobi forms, focusing on terms with large negative discriminant. To this end we implement a method based on deforming contours that expresses the coefficients of all such terms as residues. We find that there are two types of weak Jacobi forms, leading to two different growth behaviors: the more common type leads to fast, exponential growth, whereas a second type leads to slower growth, akin to the growth seen in ratios of theta functions. We give a simple criterion to distinguish…
This paper is motivated by a 2001 paper of Choie and Kim and a 2006 paper of Bump and Choie. The paper of Choie and Kim extends an earlier result of Bol for elliptic modular forms to the setting of Siegel and Jacobi forms. The paper of Bump and Choie provides a representation theoretic interpretation of the phenomenon, and shows how a natural generalization of Choie and Kim's result on Siegel modular forms follows from a natural conjecture regarding (g,K)-modules. In this paper, it is shown that the conjecture of Bump and Choie follows from work of Boe. A second proof which is along the lines …
In this paper, we introduce the notion of Maass-Jacobi forms and investigate some properties of these new automorphic forms. We also characterize these automorphic forms in several ways.
Weak Jacobi forms of weight $0$ and index $m$ can be exponentially lifted to meromorphic Siegel paramodular forms. It was recently observed that the Fourier coefficients of such lifts are then either fast growing or slow growing. In this note we investigate the space of weak Jacobi forms that lead to slow growth. We provide analytic and numerical evidence for the conjecture that there are such slow growing forms for any index $m$.
Note on the asymptotic of the auxiliary function
本文改进了辅助函数 $\mathop{\mathcal R}(s)$ 的近似式，将误差项从不趋于零的 $U$ 提升为 $o(t)$，并精确化 Siegel 的结果，从而为其论文中最后定理的证明奠定基础。
We compare the spaces of Hermitian Jacobi forms (HJF) of weight $k$ and indices $1,2$ with classical Jacobi forms (JF) of weight $k$ and indices $1,2,4$. Using the embedding into JF, upper bounds for the order of vanishing of HJF at the origin is obtained. We compute the rank of HJF as a module over elliptic modular forms and prove the algebraic independence of the generators in case of index 1. Some related questions are discussed.
Between his arrival in Frankfurt in $1922$ and and his proof of his famous finiteness theorem for integral points in $1929$, Siegel had no publications. He did, however, write a letter to Mordell in $1926$ in which he explained a proof of the finiteness of integral points on hyperelliptic curves. Recognizing the importance of this argument (and Siegel's views on publication), Mordell sent the relevant extract to be published under the pseudonym "X". The purpose of this note is to explain how to optimize Siegel's $1926$ technique to obtain the following bound. Let $K$ be a number field, $S$ a f…
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。
Siegel modular forms associated to Weil representations
该研究利用2-上同调分析与Weil表示相关的Siegel模形式的八次单位根，将研究拓展至完全Siegel群，得到两个矩阵值Siegel模形式并明确其对应的有限维表示。
A note on the Schottky problem
本文综述了 Schottky 问题的最新进展，并探讨了 Andr{é}-Oort 猜想、Okounkov 凸体、Coleman 猜想等若干数学概念与该问题之间的关联。
Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.
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