A Note on Maass-Jacobi Forms

Maass-Jacobi Poincaré series and Mathieu Moonshine

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.

The skew-Maass lift I

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…

Massive deformations of Maass forms and Jacobi forms

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.

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…

A $p$-adic Hermitian Maass lift

For $K$ an imaginary quadratic field with discriminant $-D_K$ and associated quadratic Galois character $χ_K$, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level $D_K$ and nebentypus $χ_K$ via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group $U(2,2)$ split over $K$. We generalize this (under certain conditions on $K$ and $p$) to the case of $p$-oldforms of level $pD_K$ and character $χ_K$. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space o…

A note on Maass forms of icosahedral type

Using ideas of Ramakrishnan, we consider the icosahedral analogue of the theorems of Sarnak and Brumley on Hecke-Maass newforms with Fourier coefficients in a quadratic order. Although we are unable to conclude the existence of an associated Galois representation in this case, we show that one can deduce some implications of such an association, including weak automorphy of all symmetric powers and the value distribution of Fourier coefficients predicted by the Chebotarev density theorem.

A note on simultaneous nonvanishing of Dirichlet $L$-functions and twists of Hecke-Maass $L$-functions

In this note, we prove that given a Hecke-Maass cusp form $f$ for $SL_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1q_2$ with $q_j\asymp \sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichlet character $χ$ of conductor $q$ such that $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)\neq 0$. To prove this, we establish asymptotic formulas of $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)$ over the family of even primitive Dirichlet characters $χ$ of conductor $q$ for more general $q$.

Maass lifts of half-integral weight Eisenstein series and theta powers

In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit construction of harmonic weak Maass forms which are Hecke eigenforms and are the preimages of $Θ^k, k \in \{ 3, 5, 7\}$ under the shadow operator, where $Θ$ is the classical Jacobi theta function.

Images of Maass-Poincaré series in the lower half-plane

In this note we extend integral weight harmonic Maass forms to functions defined on the upper and lower half-planes using the method of Poincaré series. This relates to Rademacher's "expansion of zero" principle, which was recently employed by Rhoades to link mock theta functions and partial theta functions.

Hermitian Maass lift for General Level

For an imaginary quadratic field $K$ of discriminant $-D$, let $χ= χ_K$ be the associated quadratic character. We will show that the space of special hermitian Jacobi forms of level $N$ is isomorphic to the space of plus forms of level $DN$ and nebentypus $χ$ (the hermitian analogue of Kohnen's plus space) for any integer $N$ prime to $D$. This generalizes the results of Krieg from $N = 1$ to arbitrary level. Combining this isomorphism with the recent work of Berger and Klosin and a modification of Ikeda's construction we prove the existence of a lift from the space of elliptic modular forms t…

Differential Operators for Siegel-Jacobi forms

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.

调和Maass形式的求和公式

Summation Formulas for Harmonic Maass Forms

本文利用调和Maass形式的L-级数理论，去掉了增长限制，证明了更一般的求和公式，并应用于分拆函数及经典模形式的Riesz和，得到求和公式与渐近性质。

Local and global Maass relations (expanded version)

We characterize the irreducible, admissible, spherical representations of GSp(4,F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito-Kurokawa lifting. We show how the classical Maass relations can be deduced from the local relations in a representation theoretic way, without recourse to the construction of Saito-Kurokawa lifts in term…

On Jacobi--Weierstrass mock modular forms

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.

Koecher-Maass series of a certain half-integral weight modular form related to the Duke-Imamoglu-Ikeda lift

Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k-n/2+1/2, let f be the corresponding primitive form of weight 2k-n for SL(2,Z) under the Shimura correspondence, and I(h) the Duke-Imamoglu-Ikeda lift of h to the space of cusp forms of weight k of genus n. Moreover, let FJ(I(h),1) be the first Fourier-Jacobi coefficient of I(h) and s(FJ(I(h),1)) be the cusp form in the generalized Kohnen plus space of weight k-1/2 corresponding to FJ(I(h),1) under the Ibukiyama isomorphism. We then give an explicit formula for the Koecher-Maass series o…

On the Fourier coefficients of meromorphic Jacobi forms

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 …

Cohomological relation between Jacobi forms and skew-holomorphic Jacobi forms

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…

Rankin-Selberg method for Jacobi forms of integral weight and of half-integral weight on symplectic groups

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…

Heegner divisors, $L$-functions and harmonic weak Maass forms

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…

翻转算子与局部调和 Maass 形式

Flipping operators and locally harmonic Maass forms

本文在适度增长调和 Maass 形式理论中引入翻转算子，并证明该算子作用于双曲型 Poincaré 级数提升得到的局部调和 Maass 形式时，同样具有使指标取反的类似性质。

The Fourier-Jacobi expansion of the singular theta lift

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.

Periods of Jacobi forms and Hecke operator

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 …

来自拉马努金失散笔记的Maass形式周期函数

Period Function of Maass forms from Ramanujan's Lost Notebook

本文证明拉马努金失散笔记第220页的函数$\mathcal{F}_1(x)$满足Maass形式的周期关系，称为拉马努金周期函数，并发现其为Hecke本征函数，推广至参数$s$，连接更广泛的周期函数理论。

向量值弱全纯模形式的L-级数与逆定理

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。

A note on cusp forms as $p$-adic limits

Several authors have recently proved results which express cusp forms as $p$-adic limits of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. The proofs involve techniques from the theory of weak harmonic Maass forms, and in particular a result of Guerzhoy, Kent, and Ono on the $p$-adic coupling of mock modular forms and their shadows. Here we obtain strengthened versions of these results using techniques from the theory of holomorphic modular forms.

Congruences for Ramanujan's f and ω functions via generalized Borcherds products

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan's mock theta functions f and ω in detail.

A note on a fundamental domain for Siegel-Jacobi space

In this paper, we study a fundamental domain for the Siegel-Jacobi space $Sp(g,{\mathbb Z})\ltimes H_{\mathbb Z}^{(g,h)}\backslash {\mathbb H}_g\times {\mathbb C}^{(h,g)}$.

正交群上尖点形式的Fourier-Jacobi Dirichlet级数

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。

A Note on Fourier-Jacobi coefficients of Siegel modular forms

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].

Note on Hermitian Jacobi Forms

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.

A p-adic completion of Zagier's Eisenstein series

This note points out that for any odd prime $p$, Zagier's weight $3/2$ mock Eisenstein series can be completed to a $p$-adic modular form in a way that bears some resemblance to its completion to a harmonic Maass form.

Twists of $GL(3)$ $L$-functions

Let $π$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form, and let $χ$ be a primitive Dirichlet character modulo $M$, which we assume to be prime. In this note we revisit the subconvexity problem addressed in `The circle method and bounds for $L$-functions IV' and establish the following unconditional bound \begin{align*} L\left(\tfrac{1}{2},π\otimesχ\right)\ll M^{3/4-1/308+\varepsilon}. \end{align*}

A note on invariant differential operators on Siegel-Jacobi space

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.

A note on mock automorphic forms and the BPS index

We discuss mock automorphic forms from the point of view of representation theory, that is, obtained from weak harmonic Maass forms give rise to nontrivial $(\mathfrak{g},K)$-cohomology. We consider the possibility of replacing the `holomorphic' condition with `cohomological' when generalizing to general reductive groups. Such a candidate allows for growing Fourier coefficients, in contrast to automorphic forms under the Miatello-Wallach conjecture. In the second part, we provide an overview of the connection with BPS black hole counts as a physical motivation for studying mock automorphic for…

Fourier-Jacobi expansion of cusp forms on $Sp(2,{\Bbb R})$

This paper develops a general theory of the Fourier-Jacobi expansion of cusp forms on the real symplectic group of degree two including generic cusp forms. An explicit description of such expansion is available for cusp forms generating discrete series representations, generalized principal series representations induced from the Jacobi parabolic subgroup and principal series representations, where we note that the latter two cases include non-spherical representations. As the archimedean local ingredients we need Fourier-Jacobi type spherical functions and Whittaker functions, whose explicit …

Remark on Harmonic Analysis on Siegel-Jacobi Space

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 sums, Fermat Jacobians, and ranks of abelian varieties over towers of function fields

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.

On the Space of Slow Growing Weak Jacobi Forms

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 $m$.

关于 Schottky 问题的一个注记

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。

A note on polylogarithms on curves and abelian schemes

In this note we investigate the connection between polylogarithms on curves and abelian schemes. The main result shows that the polylogarithm on the abelian scheme can be obtained as the push-forward of the polylogarithm on a suitable sub-curve. If the abelian scheme is the Jacobian of a smooth projective curve, this push-forward can also be written as a cup-product with the fundamental class of the curve.

Families of Quasimodular Forms and Jacobi Forms: The Crank Statistic for Partitions

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 …