Charles H. Conley, Martin Raum
We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.
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Mathieu moonshine attaches a weak Jacobi form of weight zero and index one to each conjugacy class of the largest sporadic simple group of Mathieu. We introduce a modification of this assignment, whereby weak Jacobi forms are replaced by semi-holomorphic Maass-Jacobi forms of weight one and index two. We prove the convergence of some Maass-Jacobi Poincaré series of weight one, and then use these to characterize the semi-holomorphic Maass-Jacobi forms arising from the largest Mathieu group.
Recently Bringmann, Raum and Richter generalised the definition of Jacobi forms and Skoruppa's skew-holomorphic Jacobi forms by intertwining with harmonic Maass forms. We prove the isomorphism of the Kohnen's plus space analogue of harmonic Maass forms of weight $k-1/2$ for $Γ_0(4m)$ and the space of these skew-holomorphic harmonic Maass-Jacobi forms of weight $k$ and index $m$ for $k$ odd and $m=1$ or a prime.
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.
In this paper, considering the Eichler-Shimura cohomology theory for Jacobi forms, we study connections between harmonic Maass-Jacobi forms and Jacobi integrals. As an application we study a pairing between two Jacobi integrals, which is defined by special values of partial $L$-functions of skew-holomorphic Jacobi cusp forms. We obtain connections between this pairing and the Petersson inner product for skew-holomorphic Jacobi cusp forms. This result can be considered as analogue of Haberland formula of elliptic modular forms for Jacobi forms.
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…
Real-analytic Jacobi forms play key roles in different areas of mathematics and physics, but a satisfactory theory of such Jacobi forms has been lacking. In this paper, we fill this gap by introducing a space of harmonic Maass-Jacobi forms with singularities which includes the real-analytic Jacobi forms from Zwegers's PhD thesis. We provide several structure results for the space of such Jacobi forms, and we employ Zwegers's $\widehatμ$-functions to establish a theta-like decomposition.
We define one-parameter "massive" deformations of Maass forms and Jacobi forms. This is inspired by descriptions of plane gravitational waves in string theory. Examples include massive Green's functions (that we write in terms of Kronecker-Eisenstein series) and massive modular graph functions.
Ramanujan's and Lim's Identities and Harmonic Maass--Jacobi Forms
研究Lim对Ramanujan奇数zeta值恒等式的推广，构造Eisenstein级数经典Eichler积分的Jacobi类似物，证明其为奇异调和Maass--Jacobi形式，并建立非全纯部分、反演公式及算子性质。
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.
Period functions for vector-valued Maass cusp forms of real weight, with an application to Jacobi Maass cusp forms
本文定义实权向量值 Maass 尖点形式的周期函数，利用上同调建立其与尖点形式空间的线性同构，并推广到 Jacobi Maass 形式。
Alg{è}bres diff{é}rentielles de formes quasi-Jacobi d'indice nul
本文研究零指标拟Jacobi奇异形式代数中由双重深度区分的子代数在导数及双微分算子序列下的稳定性，这些算子类似于Rankin-Cohen括号或transvectants。
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.
Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular $L$-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivativ…
We construct harmonic weak Maass forms that map to cusp forms of weight $k\geq 2$ with rational coefficients under the $ξ$-operator. This generalizes work of the first author, Griffin, Ono, and Rolen, who constructed distinguished preimages under this differential operator of weight $2$ newforms associated to rational elliptic curves using the classical Weierstrass theory of elliptic functions. We extend this theory and construct a vector-valued Jacobi--Weierstrass $ζ$-function which is a generalization of the classical Weierstrass $ζ$-function.
In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of a meromorphic Jacobi form $ϕ(z;τ)$ are the holomorphic parts of some (vector-valued) almost harmonic Maass forms. We also give a precise description of their completions, which turn out to be uniquely determined by the Laurent coefficients of $ϕ$ at each pole, as well as some well known real analytic functions, that appear for instance in the completion of …
Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between holomorphic Jacobi forms and skew-holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on $Γ^J$ and the space of skew-holomorphic Jacobi cusp forms on $Γ^J$ with the same half-integ…
In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isom…
Polar harmonic Maaß forms and holomorphic projection
本文构造小除数函数生成权重3/2的极调和Maaß形式的全纯部分，通过计算混合调和Maaß形式的全纯投影，并证明p-adic同余及与Hurwitz类数的恒等式。
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.
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 Hecke action on the space of periods of cusp forms, which is compatible with that on the space of cusp forms, was first computed using continued fraction and an explicit algebraic formula of Hecke operators acting on the space of period functions of modular forms was derived by studying the rational period functions. As an application an elementary proof of the Eichler-Selberg trace formula was derived. Similar modification has been applied to period space of Maass cusp forms with spectral parameter $s$. In this paper we study the space of period functions of Jacobi forms by means of Jacobi …
We study the algebras of differential operators invariant with respect to the scalar slash actions of real Jacobi groups of arbitrary rank. These algebras are non-commutative and are generated by their elements of orders 2 and 3. We prove that their centers are polynomial in one variable and are generated by the Casimir operator. For slash actions with invertible indices we also compute the characters of the IDO algebras: in rank exceeding 1 there are two, and in rank 1 there are in general five. In rank 1 we compute in addition all irreducible admissible representations of the IDO algebras.
We find closed-form expressions for the Schur indices of 4d $\mathcal{N}=2^{*}$ super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams associated with spectral zeta functions of an ideal Fermi-gas system. These functions are expressed in terms of the twisted Weierstrass functions, generating functions for quasi-Jacobi forms. The indices lie in the polynomial ring generated by the Kronecker theta function and the Weierstrass functions which contains the polynomial ring of the quasi-Jacobi forms. T…
A Fourier-Jacobi Dirichlet series for cusp forms on orthogonal groups
本文研究正交群上尖点形式的Fourier-Jacobi系数所构成的Dirichlet级数，在Hecke特征形式与Maass提升情形下建立其与标准$L$-函数的联系，并给出具有Euler乘积的显式例子，推广了Kohnen-Skoruppa经典结果。
In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of Jacobi forms of singular and critical weights for root systems using Jacobi theta-series. We describe the Jacobi lifting for Jacobi forms of half-integral indices. In some case it gives additive lifting construction of new reflective modular forms.
We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain Seiberg-Witten curves of type E_6 and E_7 for the E-string theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthmüller.
We extend the dictionary between the BPS spectrum of Heterotic strings and the one of F-/M-theory compactifications on $K3$ fibered Calabi-Yau 3-folds to cases with higher rank non-Abelian gauge groups and in particular to dual pairs between Heterotic CHL orbifolds and compactifications on Calabi-Yau 3-folds with a compatible genus one fibration. We show how to obtain the new supersymmetric index purely from the Calabi-Yau geometry by reconstructing the Noether-Lefschetz generators, which are vector-valued modular forms. There is an isomorphism between the latter objects and vector-valued latt…
False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false t…
Beyond Mock Modularity: Elliptic Corrections for Higher Dyson Ranks
研究高阶戴森秩，发现其单位根特化有隐藏椭圆结构，通过有限\(q\)差分递推及Appell - Lerch修正项得出修正函数，满足扭曲椭圆律，平移后全纯有限部分有有限\(\theta\)分解，揭示其类似物非模拟模性而是特定有限\(\theta\)分解。
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