Heisenberg Groups, Theta Functions and the Weil Representation

Classical Theta Lifts for Higher Metaplectic Covering Group

The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issu…

On the Siegel-Weil formula for classical groups over function fields

We establish a Siegel-Weil formula for classical groups over a function field with odd characteristic, which asserts in many cases that the Siegel Eisenstein series is equal to an integral of a theta function. This is a function-field analogue of the classical result proved by A. Weil in his 1965 Acta Math. paper. We also give a convergence criterion for the theta integral by using Harder's reduction theory over function fields.

On the Whittaker range of the generalized metaplectic theta lift

The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to automorphic representations on special orthogonal groups. It is of interest to vary the orthogonal group and describe the behavior in this theta tower (the Rallis tower). In prior work, the authors obtained an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups that is based on the tensor product of the Weil representation with another small representation. …

酉群Weil表示的环面周期

On tori periods of Weil representations of unitary groups

本文确定了酉群Weil表示到极大环面的限制，证明了局部相容特征与全局环面周期非零的充要条件，并利用跷跷板论证和theta提升结果为构造G2的Arthur包提供了基础。

扩展Weil表示：实数域情形

Extended Weil representations: the real field case

本文在实数域辛向量空间上，通过扭曲作用将两种Weil表示重组为覆盖群$\widetilde{Sp}^{\pm}(W)$的表示，并研究其中的对偶对及二维情形下的theta级数。

An exceptional Siegel-Weil formula and poles of the Spin L-function of $PGSp_6$

We show a Siegel-Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin-Selberg integral for the Spin L-function of $PGSp_6$ discovered by A. Pollack, we prove that a cuspidal representation of $PGSp_6$ is a (weak) functorial lift from the exceptional group $G_2$ if its (partial) Spin L-function has a pole at $s=1$.

Theta Correspondence for U(1,1) and U(2)

In this paper, we parametrize certain irreducible supercuspidal representations of U(1,1) and U(2) via explicit induction data. The parametrization depends on traceless elements of negative valuation in a quadratic extension of base field. We use the lattice model of the Weil representation to determine which traceless elements are involved in the theta correspondence for reductive dual pair U(1,1) and U(2).

关于Weil表示的格模型

On the lattice model of the Weil representation

本文通过引入有理Metaplectic群和Metaplectic群，将基于非自对偶格的Heisenberg-Weil表示扩展至Jacobi子群与Jacobi群，并推广至非阿基米德局部域等情形。

A weak second term identity of the regularized Siegel-Weil formula for unitary groups

Following W.T.Gan and S.Takeda, we obtain a weak second term identity of the regularized Siegel-Weil formula for the unitary dual pair $(U(n,n),U(V))$, where $V$ is a split hermitian space of dimension $2r$ with $r+1 \leq n \leq 2r-1$. As an application, we obtain a Rallis inner product formula for theta lifts from $U(W)$ to $U(V)$ for a skew-hermitian space $W$ of dimension $n$.

Remarks on the theta decomposition of vector-valued Jacobi forms

We study the theta decomposition of Jacobi forms of nonintegral lattice index for a representation that arises in the theory of Weil representations associated to even lattices, and suggest possible applications.

非伽罗瓦四次CM域上\(\mathrm{GL}_2\)的扭曲西格尔 - 韦伊公式

Twisted Siegel-Weil formulas for $\mathrm{GL}_2$ over non-Galois quartic CM fields

研究非伽罗瓦四次CM域上\(\mathrm{GL}_2\)的扭曲西格尔 - 韦伊公式，通过联系扭曲西塔积分与赫克积分的土居 - 永野提升，证明雅克比 - 朗兰兹对应基变换是同构，还得出博切尔兹形式扭曲CM值相关结论。

The Weil Representations of the Jacobi Group

The Jacobi group is the semi-direct product of the symplectic group and the Heisenberg group. The Jacobi group is an important object in the framework of quantum mechanics, geometric quantization and optics. In this paper, we study the Weil representations of the Jacobi group and their properties. We also provide their applications to the theory of automorphic forms on the Jacobi group and representation theory of the Jacobi group.

Arithmetic Siegel-Weil formula on $X_{0}(N)$

In this paper, we proved an arithmetic Siegel-Weil formula and the modularity of some arithmetic theta function on the modular curve $X_0(N)$ when $N$ is square free. In the process, we also construct some generalized Delta function for $Γ_0(N)$ and proved some explicit Kronecker limit formula for Eisenstein series on $X_0(N)$.

A construction of antisymmetric modular forms for Weil representations

We give coefficient formulas for antisymmetric vector-valued cusp forms with rational Fourier coefficients for the Weil representation associated to a finite quadratic module. The forms we construct always span all cusp forms in weight at least three. These formulas are useful for computing explicitly with theta lifts.

Vector valued theta functions associated with binary quadratic forms

We study the space of vector valued theta functions for the Weil representation of a positive definite even lattice of rank two with fundamental discriminant. We work out the relation of this space to the corresponding scalar valued theta functions of weight one and determine an orthogonal basis with respect to the Petersson inner product. Moreover, we give an explicit formula for the Petersson norms of the elements of this basis.

An infinite family of vector-valued mock theta functions

We exhibit an infinite family of vector-valued mock theta functions indexed by positive integers coprime to $6$. These are built from specializations of Dyson's rank generating function and related functions studied by Watson, Gordon, and McIntosh. The associated completed harmonic Maass forms transform according to the Weil representation attached to a rank one lattice. This strengthens a 2010 result of Bringmann and Ono and a 2019 result of Garvan.

Jacobi Forms of Critical Weight and Weil Representations

Jacobi forms can be considered as vector valued modular forms, and Jacobi forms of critical weight correspond to vector valued modular forms of weight $\frac12$. Since the only modular forms of weight $\frac12$ on congruence subgroups of $\SL$ are theta series the theory of Jacobi forms of critical weight is intimately related to the theory of Weil representations of finite quadratic modules. This article explains this relation in detail, gives an account of various facts about Weil representations which are useful in this context, and it gives some applications of the theory developed herein …

Godement--Jacquet L-函数与同调 theta 提升

Godement--Jacquet L-function and homological theta lifting

本文结合同调与解析方法，研究非阿基米德局部域上 II 型对偶对的 theta 提升，完全确定了临界点处 L-函数全纯的不可约表示的大 theta 提升，计算了所有特征标的大 theta 提升，并给出了 Weil 表示射影性的充要条件。

围绕Yu构造的无特征方法

Characteristic-free approaches around Yu's construction

本文提出无特征构造扭曲Heisenberg-Weil表示，扩展Yu构造至剩余特征2，并给出几何实现及正深度Deligne-Lusztig诱导的显式描述。

Weil表示的模形式基问题

The basis problem for modular forms for the Weil representation

本文证明了正定偶格的向量值theta级数生成Weil表示的尖点形式空间，从而肯定回答了Eichler基问题，并由此导出Waldspurger的结果及Borcherds提升满射性的新证明。

与Weil表示相关的Siegel模形式：SL₂(ℝ)与GL₂(ℝ)情况

Siegel modular forms associated to Weil representations: $\operatorname{SL}_2(\mathbb{R}) \& \operatorname{GL}_2(\mathbb{R})$ cases

本文研究通过Rao、Kudla、Perrin、Lion-Vergne和Satake-Takase的2-余弦得到的经典Weil表示所生成的重量为1/2和3/2的经典、减法和费米子theta级数，并通过张量归纳重新组织这些形式，进而扩展研究到相似群GL₂(ℝ)。

与Weil表示相关的Siegel模形式

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。

H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series

It was shown in previous work that the one-variable $\widehatμ$-function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature $(r\!+\!1,1)$ are both Heisenberg harmonic Maaß-Jacobi forms. We extend the concept of Heisenberg harmonicity to Maaß-Jacobi forms of arbitrary many elliptic variables, and produce indefinite theta series of "product type" for non-degenerate lattices of signature $(r\!+\!s,s)$. We thus obtain a clean generalization of $\widehatμ$ to these negative definite lattices. From restrictions to torsion points of Heisenberg harmonic M…

Twisted arithmetic Siegel Weil formula on X0(N)

In this paper, we study twisted arithmetic divisors on the modular curve X_0(N) with N square-free. For each pair (Δ, r) where Δ>0 and Δ\equiv r^2 \mod 4N, we constructed a twisted arithmetic theta function ϕ_{Δ, r}(τ) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of ϕ_{Δ, r}(τ), along the way, we also identify the arithmetic pairing \langle ϕ_{Δ, r}(τ),\widehatω_N \rangle with special value of some Eisenstein series, where \widehatω_N is a normalized metric Hodge line bundle.

Shintani theta lifts of harmonic Maass forms

We define a regularized Shintani theta lift which maps weight $2k+2$ ($k \in \Z, k \geq 0$) harmonic Maass forms for congruence subgroups to (sesqui-)harmonic Maass forms of weight $3/2+k$ for the Weil representation of an even lattice of signature $(1,2)$. We show that its Fourier coefficients are given by traces of CM values and regularized cycle integrals of the input harmonic Maass form. Further, the Shintani theta lift is related via the $ξ$-operator to the Millson theta lift studied in our earlier work. We use this connection to construct $ξ$-preimages of Zagier's weight $1/2$ generating…

$\mathrm{SL}_2(\mathbb{Z})$ 的 Weil 表示的不变量

The invariants of the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$

本文研究了正定偶格的向量值 theta 函数在 $\mathrm{Mp}_2(\mathbb{Z})$ 下由 Weil 表示描述的变换行为，证明其不变量由 $5$ 个基本不变量诱导，并以此给出奇异权 Jacobi 形式的简单生成集。

The transformation laws of algebraic theta functions

We give a comprehensive treatment of the transformation laws of theta functions from an algebro-geometric perspective, that is, in terms of moduli of abelian schemes. This is accomplished by introducing geometric notions of theta-descent structures, metaplectic stacks, and bundles of half-forms, whose analytic incarnations underlie different aspects of the classical transformation laws. As an application, we lay the foundations for a geometric theory of modular forms of half-integral weight and, more generally, for modular forms taking values in the Weil representation. We discuss further appl…

Theta函数、特征形式的四阶矩与上确界问题 I

Theta functions, fourth moments of eigenforms, and the sup-norm problem I

本文通过构造基于Bergman核的全纯theta核，给出了Eichler序对应的算术双曲曲面上模形式四阶矩在权方面的逐点精确上界，加强了Xia的结果并推广至余紧格，同时为符号(2,2)的整二次型产生了新的基本theta级数。

A $p$-adic Approach to the Weil Representation of Discriminant Forms Arising from Even Lattices

Suppose that $M$ is an even lattice with dual $M^{*}$ and level $N$. Then the group $Mp_{2}(\mathbb{Z})$, which is the unique non-trivial double cover of $SL_{2}(\mathbb{Z})$, admits a representation $ρ_{M}$, called the Weil representation, on the space $\mathbb{C}[M^{*}/M]$. The main aim of this paper is to show how the formulae for the $ρ_{M}$-action of a general element of $Mp_{2}(\mathbb{Z})$ can be obtained by a direct evaluation which does not depend on ``external objects'' such as theta functions. We decompose the Weil representation $ρ_{M}$ into $p$-parts, in which each $p$-part can be…

Some remarks on K-lattices and the Adelic Heisenberg Group for CM curves

We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic $1$-dimensional $\mathbb{K}$-lattices (here and subsequently, $\mathbb{K}$ denotes a quadratic imaginary number field) and define theta functions on it. We also embed the groupoid of commensurability modulo dilations into the union of adelic Heisenbe…

On two arithmetic theta lifts

Our aim is to clarify the relationship between Kudla's and Bruinier's Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type. These functions play a key role in the arithmetic geometry of the special cycles in the context of Kudla's program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a modular form with trivial holomorphic projection. Along the way, we construct a section of the Maass lowering operator for moderate growth modular forms valued in the Weil representation us…

拉马努金伪θ函数系数的超越公式

Transcendental formulas for the coefficients of Ramanujan's mock theta functions

本研究证明了拉马努金最后一封信中全部17个伪θ函数的精确系数公式，同时给出了奇秩格相伴韦伊表示的克洛斯特曼和之和的界定理，推进了伪θ函数系数的精确表达研究。

紧 $\mathrm{Spin}_8$ 的例外 Siegel–Weil 定理

Exceptional Siegel-Weil theorems for compact $\mathrm{Spin}_8$

本文研究四元数 $E_8$ 中 $G_E\times S_E$ 对偶对，利用极小表示计算平凡表示的提升，证明其对应退化 Eisenstein 级数，并用于 $G_2$ 模形式算术性。

On a Critical Case of Rallis Inner Product Formula

Let $π$ be a genuine cuspidal representation of the metaplectic group of rank $n$. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension $2n+1$. We show a case of regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands $L$-function of $π$ twisted by a character. The bulk of this article focuses on proving a case of regularised Siegel-Weil formula, on which the Rallis inner product formula is based and whose proof is missing in the literature.

与ADE李代数相关的Epstein向量ζ函数

Epstein vector zeta functions related to the ADE Lie algebras

本文研究了与ADE型李代数根格相关的Epstein向量ζ函数，通过构造Weil表示的Metaplectic群Mp(2,Z)的向量值theta级数，并利用其等变性质推导出Riemann型矩阵函数方程，最终对满足该方程的格和不变子空间进行了完整分类。

二次Gauss和 over Z^n/cZ^n：显式公式，对偶定理，以及应用于Weil表示和立方超曲面 over F_p

Quadratic Gauss sums over $\mathbb{Z}^n/c\mathbb{Z}^n$: explicit formulas, a duality theorem, and applications to Weil representations and cubic hypersurfaces over $\mathbb{F}_p$

本文研究二次Gauss和 over Z^n/cZ^n的显式公式，推导对偶定理，并应用于Weil表示、二次同余方程和立方超曲面的解数计算。

van der Geer--van der Vlugt曲线在特征2下的L多项式

The $L$-polynomials of van der Geer--van der Vlugt curves in characteristic $2$

本文在特征2下推导了van der Geer--van der Vlugt曲线的L多项式公式，并构造了达到Hasse--Weil界限的曲线示例。

Faltings heights of big CM cycles and derivatives of L-functions

We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d,2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Mass forms, and it involves the Siegel-Weil formula and the central derivatives of incoherent Eisenstein series for totally real fields. In the case of a weakly holomorphic form, the formula is an explicit combination of quantities obtain…

迈向类 SIC 现象的理论：正则花束与广义海森堡群

Towards a Theory of SIC-like Phenomena: Regular Bouquets and Generalised Heisenberg Groups

本文通过广义海森堡群和正则花束构建了涵盖 SIC 的广泛代数理论，以阐明其与 Stark 单位的联系，并引入源自数域分式理想的算术例子。