Kathrin Bringmann, Sander Zwegers
In this paper we show how Rank-Crank type PDE's (first found by Atkin and Garvan) occur naturally in the framework of non-holomorphic Jacobiforms and find an infinite family of such differential equations. As an application we show an infinite family of congruences for odd Durfee symbols, a partition statistic introduced by George Andrews.
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Families of quasimodular forms arise naturally in many situations such as curve counting on Abelian surfaces and counting ramified covers of orbifolds. In many cases the family of quasimodular forms naturally arises as the coefficients of a Taylor expansion of a Jacobi form. In this note we give examples of such expansions that arise in the study of partition statistics. The crank partition statistic has gathered much interest recently. For instance, Atkin and Garvan showed that the generating functions for the moments of the crank statistic are quasimodular forms. The two variable generating …
Crank equidistribution and $(k,j)$-overlined partitions
本文给出$(k,j)$-着色划线分拆数的渐近公式，证明Wagner构造的crank在算术级数上渐近等分布，推进了分拆理论中拉马努金型同余式相关研究。
We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of $q$. And we prove that certain linear sums of $q$-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.
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 paper, we prove modularity results of Taylor coefficients of certain non-holomorphic Jacobi forms. It is well-known that Taylor coefficients of holomorphic Jacobi forms are quasimoular forms. However recently there has been a wide interest for Taylor coefficients of non-holomorphic Jacobi forms for example arising in combinatorics. In this paper, we show that such coefficients still inherit modular properties. We then work out the precise spaces in which these coefficients lie for two examples.
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.
Jacobi Forms of Affine Weight in Higher Cogenus and Nearly Holomorphic Functions
本文通过微分几何方法研究高余数下雅可比形式的结构，并揭示了近全纯函数的性质。
We define Jacobi forms with complex multiplication. Analogous to modular forms with complex multiplication, they are constructed from Hecke characters of the associated imaginary quadratic field. From this construction we obtain a Jacobi form which specializes to $η(τ)^{26}$ which we present to highlight an open question of Dyson and Serre. We give other examples and applications of Jacobi forms with complex multiplication including constructing theta blocks associated to elliptic curves with complex multiplication and new families of congruences and cranks for certain partition functions.
For every point on the Jacobian of the modular curve $X_0(l)$ we define and study certain twisted holomorphic Eisenstein series. These are particular cases of a more general notion of twisted modular forms which correspond to sections on the modular curve $X_1(l)$ of the degree zero twists of line bundles of usual modular forms. We conjecture that a point on the Jacobian is rational if and only if the ratios of these twisted Eisenstein series of the same weights have rational coefficients.
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.
In this paper, we consider the Fourier coefficients of meromorphic Jacobi forms of negative index. This extends recent work of Creutzig and the first two authors for the special case of Kac-Wakimoto characters which occur naturally in Lie theory, and yields, as easy corollaries, many important PDEs arising in combinatorics such as the famous rank-crank PDE of Atkin and Garvan. Moreover, we discuss the relation of our results to partial theta functions and quantum modular forms as introducted by Zagier, which together with previous work on positive index meromorphic Jacobi forms illuminates the…
We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows to generate a Jacobi eigenform directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.
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.
We analyze holomorphic Jacobi forms of weight one with level. One such form plays an important role in umbral moonshine, leading to simplifications of the statements of the umbral moonshine conjectures. We prove that non-zero holomorphic Jacobi forms of weight one do not exist for many combinations of index and level, and use this to establish a characterization of the McKay--Thompson series of umbral moonshine in terms of Rademacher sums.
The classical Maass lift is a map from holomorphic Jacobi forms to holomorphic scalar-valued Siegel modular forms. Automorphic representation theory predicts a non-holomorphic and vector-valued analogue for Hecke eigenforms. This paper is the first part of a series of papers. In this series of papers, we provide an explicit construction of the non-holomorphic Maass lift that is linear and also applies to non-eigenforms. In this first part, we develop new techniques to study Fourier series expansions of Siegel modular forms, which allow us to construct a Maass lift from harmonic Maass-Jacobi fo…
By modifying a slash operator of index zero we define \textit{modified Jacobi forms} of \textit{index zero}. Such forms play a role of generating nearly holomorphic modular forms of integral weight. Furthermore, by observing a relation between the coefficients of Fourier development of a modified Jacobi form we construct a family of finite-dimensional subspaces.
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…
L-Series for Vector-Valued Weakly Holomorphic Modular Forms and Converse Theorems
本文利用拉普拉斯变换引入弱全触模形式的L-级数并给出函数方程，进而确定了向量值调和弱Maass形式、Jacobi形式及半整数权椭圆模形式的逆定理。
In this paper we study Jacobi forms associated with the Leech lattice $Λ$ which are invariant under the Conway group $\mathrm{Co}_0$. We determine and construct generators of modules of both weak and holomorphic Jacobi forms of integral weight and fixed index $t\leq 3$. As applications, (1) we find the modular linear differential equations satisfied by the holomorphic generators; (2) we determine the decomposition of many products of orbits of Leech vectors; (3) we calculate the intersection between orbits and Leech vectors; (4) we derive some conjugate relations among orbits modulo $tΛ$.
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 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…
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 …
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.
Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$
本文研究了$D_{2n+1}$指数雅可比形式与水平为2的椭圆模形式之间的同构关系，并给出了雅可比-艾森斯坦级数的傅里叶系数公式，以及从Zagier-Eisenstein级数构造半正定模形式的方法。
We study various properties of quasimodular forms by using their connections with Jacobi-like forms and pseudodifferential operators. Such connections are made by identifying quasimodular forms for a discrete subgroup $\G$ of $SL(2, \bR)$ with certain polynomials over the ring of holomorphic functions of the Poincaré upper half plane that are $\G$-invariant. We consider a surjective map from Jacobi-like forms to quasimodular forms and prove that it has a right inverse, which may be regarded as a lifting from quasimodular forms to Jacobi-like forms. We use such liftings to study Lie brackets an…
In 1992 Wirthmüller showed that for any irreducible root system not of type $E_8$ the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for $E_8$ the ring is not a polynomial algebra. Weyl invariant $E_8$ Jacobi forms have many applications in string theory and it is an open problem to describe such forms. The scaled refined free energies of $E$-strings with certain $η$-function factors are conjectured to be Weyl invariant $E_8$ quasi holomorphic Jacobi forms. It is further observed that the scaled refined free energies up t…
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.
We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of $(0,2)$-model in string theory. We also determine the structure of the graded ring of all weakly holomorphic Jacobi forms of weight zero and integral index with integral Fourier coefficients. These forms are the data for Borcherds products for the Siegel paramodular groups.
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。
Arithmetic special cycles and Jacobi forms
本文研究了通过增强特殊循环与Green流形，将生成函数与非全纯希伯特-雅可比模形式的傅里叶展开关联起来，验证了Kudla猜想的算术类比。
Pseudodifferential Jacobi forms and Geometric Rankin-Cohen Brackets
通过Jacobi群在伪微分算子上的作用，构造了Jacobi形式空间与不变伪微分算子空间的同构，并导出了带复参数的几何Rankin-Cohen括号族。
Quasi-modular forms for the orthogonal group and Gromov-Witten theory of Enriques surfaces
本文建立符号 $(2,n)$ 正交群的近全纯与拟模形式理论及 theta 提升性质，并据此猜想 Enriques、双椭圆曲面的 Gromov–Witten 势满足全纯异常方程。
We show that the meromorphic Jacobi form that counts the quarter-BPS states in N=4 string theories can be canonically decomposed as a sum of a mock Jacobi form and an Appell-Lerch sum. The quantum degeneracies of single-centered black holes are Fourier coefficients of this mock Jacobi form, while the Appell-Lerch sum captures the degeneracies of multi-centered black holes which decay upon wall-crossing. The completion of the mock Jacobi form restores the modular symmetries expected from $AdS_3/CFT_2$ holography but has a holomorphic anomaly reflecting the non-compactness of the microscopic CFT…
Let $λ$ be an integer, and $f(z)=\sum_{n\gg-\infty} a(n)q^n$ be a weakly holomorphic modular form of weight $λ+\frac 12$ on $Γ_0(4)$ with integral coefficients. Let $\ell\geq 5$ be a prime. Assume that the constant term $a(0)$ is not zero modulo $\ell$. Further, assume that, for some positive integer $m$, the Fourier expansion of $(f|U_{\ell^m})(z) = \sum_{n=0}^\infty b(n)q^n$ has the form \[ (f|U_{\ell^m})(z) \equiv b(0) + \sum_{i=1}^{t}\sum_{n=1}^{\infty} b(d_i n^2) q^{d_i n^2} \pmod{\ell}, \] where $d_1, \ldots, d_t$ are square-free positive integers, and the operator $U_\ell$ on formal pow…
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。
This work considers aspects of almost holomorphic and meromorphic Siegel modular forms from the perspective of physics and mathematics. The first part is concerned with (refined) topological string theory and the direct integration of the holomorphic anomaly equations. Here, a central object to compute higher genus amplitudes, which serve as the generating functions of various enumerative invariants, is provided by the so-called propagator. We derive a universal expression for the propagator for geometries that have mirror curves of genus two which is given by the derivative of the logarithm o…
Singular-weight Conway-invariant Jacobi forms of index four
本文解决了指标4的奇异重量Co₀不变雅可比形式空间的精确维数问题，证明其维数为6，并构造了该空间的一组自然基。
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