Kathrin Bringmann, Jonas Kaszian, Antun Milas, Sander Zwegers
In this paper, we study a family of rank two false theta series associated to the root lattice of type $A_2$. We show that these functions appear as Fourier coefficients of a meromorphic Jacobi form of negative definite matrix index. Hypergeometric $q$-series identities are also obtained.
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Let $L$ be a positive definite even lattice. We introduce theta type Jacobi forms and construct three towers of Jacobi forms with a particular easy pullback-structure. We use theta type Jacobi forms to explain the existence of a cusp form of weight 24 with respect to the irreducible root lattice $D_{4}$ which is based on additional symmetries of the Coxeter diagram.
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.
The Petersson norm of the Jacobi theta function
我们计算的Jacobi θ函数的Petersson范数，通过一个简单的应用程序的Rankin-Selberg方法和一个小技巧。
Theta Series Associated with the Weil-Schroedinger Representation
Andre Weil发现的Weil表示在研究θ级数变换性质中起着重要作用。本文定义了Jacobi群的Weil-Schroedinger表示，并证明了与Weil-Schroedinger表示相关的θ级数是关于Jacobi模群的适当算术子群的Jacobi形式.
Linear independence of theta series of positive-definite even unimodular lattices
研究正定单模偶格中theta级数的线性独立性，确定其最小度数范围。
Orbits of Jacobi forms and Theta relations
本文提出Jacobi形式轨道的新方法，用于生成Jacobi theta函数恒等式，简化已知theta关系证明并推广Riemann theta关系。
Liu established an addition formula for the Jacobian theta function by using the theory of elliptic functions. From this addition formula he obtained the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities. In this paper we will give an addition formula for the Jacobian theta function which is equivalent to Liu's addition formula. Based on this formula we deduce some known theta function identities as well as new identities. From these identities we shall establish certain new serie…
On the integral of the fourth Jacobi theta function
我们将Raabe公式推广到$q$ -loggamma函数。由此得到第四Jacobi θ函数在其最小虚零点之间的对数的积分与分拆函数和黎曼ζ函数有关。
We consider a generalization of Jacobi theta series and show that every such function is a quasi-Jacobi form. Under certain conditions we establish transformation laws for these functions with respect to the Jacobi group and prove such functions are Jacobi forms. In establishing these results we construct other functions which are also Jacobi forms. These results are motivated by applications in the theory of vertex operator algebras.
Quasi-Jacobi forms, Appell-Lerch functions, and false theta functions as q-brackets of functions on partitions
研究分拆上类theta函数代数，其生成函数涉及theta函数、拟Jacobi形式、Appell-Lerch和及假theta函数。
Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition
实解析Jacobi型在数学和物理的各个领域都起着重要的作用，但一直缺乏一个令人满意的理论。在本文中，我们填补了这一空白，通过引入一个空间的调和马斯-雅可比形式的奇异性，其中包括真正的分析雅可比形式从Zwegers的博士论文。我们提供了几个结构的结果，这样的Jacobi形式的空间，我们采用Zwegers的$\widehatμ$ -功能，建立一个θ-样分解。
We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting $q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper's $q$-trigonometric identities.
Modular forms of orthogonal type and Jacobi theta-series
本文考虑任意正定格L的半整指标的Jacobi形式（Eichler和Zagier书中的经典Jacobi形式对应于格A_1=<2>）。我们用Jacobi θ-级数给出了根系奇异权和临界权的Jacobi形式的许多例子。我们描述了Jacobi形式的半整指数的Jacobi提升。在某些情况下，给出了新的反射模形式的加法提升构造。
We prove several asymptotic results for partial and false theta functions arising from Jacobi forms, as the modular variable $τ$ tends to $0$ along the imaginary axis, and the elliptic variable $z$ is unrestricted in the complex plane. We observe that these functions exhibit Stokes' phenomenon - the asymptotic behavior of these functions sharply differs depending on where the elliptic variable $z$ is located within the complex plane. We apply our results to study the asymptotic expansions of regularized characters and quantum dimensions of the $(1,p)$-singlet vertex operator algebra coming fro…
We shall answer a question of Mező on the $q$-analogue of the Raabe's integral formula for $0<q<1$ and we shall evaluate an integral involving the first theta function. Moreover, we will reproduce short proofs for some identities of Mező.
Edited 4-Theta embeddings of Jacobians
利用Lefschetz嵌入定理，将主极化$g$维阿贝尔簇簇通过$4^g$半特征θ函数的线性系嵌入到射影空间中.假设我们{ \em edit}这个线性系，通过丢弃所有在原点消失的θ函数，使其阶数大于大于奇偶性要求。我们证明了对于雅可比行列式，编辑后的$4Θ$线性系仍然定义了到射影空间的嵌入。此外，我们证明了投影模型的雅可比矩阵所产生的基本建设的雅可比矩阵最近给出的作者（通过线性外壳后）的副本编辑$4Θ$模型。因此，对于所有紧致黎曼曲面，我们以一种新的方式将代数雅可比行列式和解析雅可比行列式联系在一起。
In this paper, we consider $L$-functions of modular forms of weight 3, which are products of the Jacobi theta series, and express their special values at $s=3$, $4$ in terms of special values of Kampé de Fériet hypergeometric functions. Moreover, via $L$-values, we give some relations between special values of Kampé de Fériet hypergeometric functions and generalized hypergeometric functions.
Hida family of theta lift from U(1) to definite U(2)
本文构造了从代数Hecke特征到确定型酉群U(2)的theta提升的Hida族，并在中心L值模p非消失条件下证明了其本原性。
On Three-Term Linear Relations for Theta Series of Positive-Definite Binary Quadratic Forms
本文扩展Schiemann方法并在多面体锥上开发扩展细化算法，刻画正定整二元二次型theta级数的三项线性关系，证明仅存在一个涉及判别式$-3,-12,-48$的非平凡关系。
We establish the Fourier-Jacobi case of the local Gross-Prasad conjecture for unitary groups, by using local theta correspondence to relate the Fourier-Jacobi case with the Bessel case established by Beuzart-Plessis. To achieve this, we prove two conjectures of D. Prasad on the precise description of the local theta correspondence for (almost) equal rank unitary dual pairs in terms of the local Langlands correspondence.
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…
Generalizing a result of~\cite{Z1991} for modular forms of level~one, we give a closed formula for the sum of all Hecke eigenforms on $Γ_0(N)$, multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level $N$. We also show that for $N=2$,~3 and~5 this formula completely determines the Fourier expansions of all Hecke eigenforms of all weights on $Γ_0(N)$.
We classify the optimal mock Jacobi forms of weight one with rational coefficients. The space they span is thirty-four-dimensional, and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane. We show that their Fourier coefficients can be expressed explicitly in terms of singular moduli, and obtain positivity conditions which distinguish the optimal mock Jacobi forms that appear in umbral moonshine. We find that all of Ramanujan's mock theta functions can be expressed simply in terms of the optimal mock Jacobi forms with rational coefficients.
We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from certain space of such forms, whether scalar-valued or vector-valued, integral-weight or half-integral-weight, of any level, with any character, into a product of finitely many spaces of modular forms. As an application, we investigate linear relations among Jacobi theta series of lattice index. Many linear relations among the second powers of such theta series associated with the $D_4$ lattice and $A_3$ lattice are obtained, along with relations among the third powers of series associated…
We give a new proof for a product formula of Jacobi which turns out to be equivalent to a $q$-trigonometric product which was stated without proof by Gosper. We apply this formula to derive a $q$-analogue for the Gauss multiplication formula for the gamma function. Furthermore, we give explicit formulas for short products of $q$-gamma functions.
Number Field Analogue of Jacobi Theta Relation And Zeros of Dedekind zeta function on Re$(s)=1/2$
本文建立Jacobi theta关系的数域类比，并据此证明任意数域的Dedekind zeta函数在临界线Re$(s)=1/2$上均有无穷多个非平凡零点。
On the Coble quartic and Fourier-Jacobi expansion of theta relations
Q. Ren，S. Sam，G. Schrader，和B. Sturmfels（arXiv：1208.1229），在3属情况下的通用Kummer簇的简化方程。虽然，这些方程中的大多数是从亏格4中θ常数之间的关系的傅里叶-雅可比展开获得的，但更突出的是，Coble的四次曲线是以不同的方式获得的。本文的目的是表明，Coble的四次曲线可以作为亏格4中θ-常数之间关系的Fourier-Jacobi展开式得到。我们还得到一个关系，可以在理想中描述的“普遍库默尔三重”。
We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_1m_2$ with $m_1|N$ and $(N,m_2)=1$ has a non--zero theta component $h_μ$ with either $(μ,2m_1m_2)=1$ or $(μ,2m_1m_2)\nmid 2m_2$. As an application, we prove that a non--zero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin--Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.
We study the Taylor expansion around the point $x=1$ of a classical modular form, the Jacobi theta constant $θ_3$. This leads naturally to a new sequence $(d(n))_{n=0}^\infty=1,1,-1,51,849,-26199,\ldots$ of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of $θ_3$. We prove several results about the numbers $d(n)$ and conjecture that they satisfy the congruence $d(n)\equiv (-1)^{n-1}\ (\textrm{mod }5)$ and other similar congruence relations.
Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant θ_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq 1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all large enough n.
We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta-function and show that they give a powerful new method to construct Jacobi forms and Siegel modular forms, with applications also in lattice theory and algebraic geometry. One of the central questions is when a theta block defines a Jacobi form. It turns out that this seemingly simple question is connected to various deep problems in different fields ranging from Fourier analysis over infinite-dimensional Lie algebras to the theory of moduli spaces in algebraic geometry. We give several answers t…
Recently, Funke and Hofmann constructed a singular theta lift of Borcherds type for the dual reductive pair $U(1,1)\times U(p,q)$, $p,q\geq 1$, the input functions of which are harmonic weak Maass forms of weight $k= 2-p-q$. In the present paper, we give an explicit evaluation of the Fourier-Jacobi expansion of the lift. For this purpose, we adapt a method introduced by Kudla in his paper 'Another product for a Borcherds form'. As an application, in the case $U(p,1)$ we recover a new infinite product expansion associated to a Borcherds form, analogous to the case $O(p,2)$ treated by Kudla.
A theory of theta functions to the quintic base
四个五次θ函数的性质与经典的Jacobi零θ函数的性质并行发展。证明了五次θ函数满足Jacobi的四次θ函数恒等式的类似物和Jacobi的符号公式的重复、减少和改变原理的对应物。由此产生的五次变换公式库是用来描述系列多节的模形式在简单的矩阵运算。这些努力最终在一个正式的技术推导出的各种组合生成函数，包括分拆函数的同余模5的权力。进一步分析的五次θ函数进行探索其模块化的性质和他们的连接到爱森斯坦系列。由此产生的关系导致五次θ函数的微分方程的耦合系统。
Functional Equation for Theta Series
本文给出了Siegel-Jacobi空间上的θ级数在Jacobi模群作用下的变换公式。这个公式推广了Mumford在他的书[第189页，Tata Lectures on Theta I（1983），Birkhauser]中得到的公式（5.1）。
Romik's Conjecture for the Jacobi Theta Function
该论文针对Romik关于Jacobi theta函数Taylor系数模素数周期性的猜想，在p≡1 mod 4情形下给出简洁代数关系，并推导出有效周期。
Truncated Theta Series Related to the Jacobi Triple Product Identity
本文证明了Merca关于雅可比三重积截断恒等式猜想的前三种情况及相关恒等式，提出新组合解释，并用组合方法证明了可简化为已知猜想的恒等式。
We construct many examples of level one Siegel modular forms in the kernel of theta operators mod $p$ by using theta series attached to positive definite quadratic forms.
In Ramanujan's final letter to Hardy, he listed examples of a strange new class of infinite series he called "mock theta functions". It turns out all of these examples are essentially specializations of a so-called universal mock theta function $g_3(z,q)$ of Gordon-McIntosh. Here we show that $g_3$ arises naturally from the reciprocal of the classical Jacobi triple product -- and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms -- under the action of an operator related to statistical physics and partition the…
We discuss an arithmetic approach to some congruence properties of Siegel theta series of even positive definite unimodular quadratic forms.
Some Differential Equations for the Riemann $θ$-Function on Jacobians
本文利用受Arakelov理论启发的Fay公式变体，证明了黎曼曲面雅可比簇上黎曼θ函数满足的若干微分方程。
Positivity and tails of Jacobi theta series
该研究利用基本q级数运算，建立雅可比θ级数尾部正性性质，定义相关级数等式，证明对特定整数范围，系数J_{k,n}(m)恒为正，为该领域研究提供了新结论。
Capparelli conjectured two modular identities for partitions whose parts satisfy certain gap conditions, where were motivated by the calculation of characters for the standard modules of certain affine Lie algebras and by vertex operator theory. These identities were subsequently proved and refined by Andrews, who related them to Jacobi theta functions, and also by Alladi-Andrews-Gordon, Capparelli, and Tamba-Xie. In this paper we prove two new companions to Capparelli's identities, where the evaluations are expressed in terms of Jacobi theta functions and false theta functions.
We obtain infinite product expansions in the sense of Borcherds for theta functions associated with certain positive-definite binary quadratic and binary hermitian forms. Among other things, we show that every weight 1 binary theta function is a Borcherds product. In particular, binary theta functions have zeros only at quadratic irrationalities.
Theta functions for definite signature lattices constitute a rich source of modular forms. A natural question is then their generalization to indefinite signature lattices. One way to ensure a convergent theta series while keeping the holomorphicity property of definite signature theta series is to restrict the sum over lattice points to a proper subset. Although such series do not have the modular properties that a definite signature theta function has, as shown by Zwegers for signature $(1,n-1)$ lattices, they can be completed to a function that has these modular properties by compromising o…
Let $θ_3(τ)=1+2\sum_{ν=1}^{\infty} q^{ν^2}$ with $q=e^{iπτ}$ and $\Im (τ)>0$ denote the Thetanullwert of the Jacobi theta function \[θ(z|τ) \,=\,\sum_{ν=-\infty}^{\infty} e^{πiν^2τ+ 2πiνz} \,.\] Moreover, let $θ_2(τ)=2\sum_{ν=0}^{\infty} q^{{(ν+1/2)}^2}$ and $θ_4(τ)=1+2\sum_{ν=1}^{\infty} {(-1)}^νq^{ν^2}$. For every even integer $n\geq 6$, which is not a power of two, we prove constructively the existence of a nontrivial integer polynomial $Q_n(X,Y)$ such that \[Q_n\Big( \,\frac{θ_3^4(nτ)}{θ_3^4(τ)},\frac{θ_2^4(τ)}{θ_3^4(τ)}\, \Big) \,=\, 0 \] holds for all complex numbers $τ$ from the upper h…
Witten-Reshetikhin-Turaev invariants and indefinite false theta functions for plumbing indefinite H-graphs
本文构造无定假theta函数作为非弱负定管道3-流形的同调块候选，并证明在庞加莱同调球情形与原始同调块一致。
Mock Jacobi forms in basic hypergeometric series
我们证明了一些$q$ -级数，如泛仿θ函数，是θ项与权为1/2的仿Jacobi形式的线性和，当它们被限制在扭点上并乘以$q$的适当幂时，它们成为实解析模形式的全纯部分.证明了某些$q$ -级数的线性和是权为1/2的弱全纯模形式，这是由于模Jacobi形式的湮灭或模Jacobi形式的完备化.作为应用，我们得到了划分的秩与曲柄之间的关系。
Theta Series with Half-integral Characteristics
本文研究奇数正定幺模格上的半整特征Theta级数，在偶数秩（尤其四的倍数）构造非零奇特征Theta级数，并揭示其与整幺模邻格的联系。
We prove the transformation laws of the four Jacobi theta functions using Gordon's proof for the transformation law of the Dedekind eta function.
In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.
In this paper, we consider the Fourier coefficients of a special class of meromorphic Jaocbi forms of negative index. Much recent work has been done on such coefficients in the case of Jacobi forms of positive index, but almost nothing is known for Jacobi forms of negative index. In this paper we show, from two different perspectives, that their Fourier coefficients have a simple decomposition in terms of partial theta functions. The first perspective uses the language of Lie super algebras, and the second applies the theory of elliptic functions. In particular, we find a new infinite family o…
We show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan Delta-function $Δ(τ)=η^{24}(τ)$, $η^{12}(τ)$ and $η^{8}(τ)$ in terms of Cohen's numbers $H(3,N)$ and $H(5,N)$. We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the J…
Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials
我们找到了两个组合恒等式的根格的θ系列的有限维单李代数的$D_{4n}$型和陪集在其积分型，在著名的Essenstein系列$E_4(z)$和Ramanujan系列$\Dlt_{24}(z)$。利用这两个恒等式，我们确定了某些正定偶幺模格的无限族的θ级数，这些无穷族是由$D_{2n}$型有限维单李代数的根格的有限个拷贝胶合而成的.证明了这些θ级数是$E_4(z)$和$\Dlt_{24}(z)$两个固定多项式族的加权对称多项式。
In this paper, we obtain some behaviours of theta sums of higher index for the Schroedinger-Weil representation of the Jacobi group associated with a positive definite symmetric real matrix of degree m.
Clifford algebras and Shimura's lift for Theta-series
本文利用自同构类理论构造了任意三元非退化二次型的二次Z-模到相应Clifford代数的某些子环之间态射的相似性的提升.该结构解释和推广了志村对应的情况下，θ系列的正定二次型。相关的ζ函数之间的关系被认为是。
On Algebraic Conditions for the Non-Vanishing of Linear Forms in Jacobi Theta-Constants
本文研究雅西θ常数线性形式的非零性，在多种参数条件下证明相关线性无关性结论，填补了已有θ常数代数独立/相关性研究的空白，还建立了对应函数域上的线性无关定理。
Two generalizations of Jacobi's derivative formula
本文将著名的Jacobi三重积恒等式推广到更高亏格/维数的情形，其中Jacobi三重积恒等式被认为是θ函数及其导数的特征恒等式.通过应用我们在前一篇论文math.AG/0310085中所发展的结果和方法，得到了Siegel模形式的几个推广。这些推广是特征为零的θ函数及其导数所满足的恒等式。将这些关系的傅里叶展开的系数等于零，得到非平凡的组合恒等式。
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.
We study the reduction of Picard modular surfaces modulo an inert prime, mod p and p-adic modular forms. We construct a theta operator on such modular forms and study its poles and its effect on Fourier-Jacobi expansions.
The theta-block conjecture proposed by Gritsenko--Poor--Yuen in 2013 characterizes Siegel paramodular forms which are simultaneously Borcherds products and additive Jacobi lifts. In this paper, we prove this conjecture for two new infinite series of theta blocks of weights 2 and 3. The proof is based on Scheithauer's classification of reflective modular forms of singular weight.
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