Poincaré series and modular functions for U(n, 1)

Almost holomorphic Poincare series corresponding to products of harmonic Siegel-Maass forms

We investigate Poincaré series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincaré series are almost holomorphic as well. In general this is not the case. The main point of this paper is the study of Siegel-Poincaré series of degree $2$ attached to products of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. …

On the Non-Vanishing of Poincaré Series on the Metaplectic Group

In this paper, we study the $ K $-finite matrix coefficients of integrable representations of the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $ and give a result on the non-vanishing of their Poincaré series. We do this by adapting the techniques developed for $ \mathrm{SL}_2(\mathbb R) $ by Muić to the case of the metaplectic group.

On hyperderivatives of single-cuspidal Drinfeld modular forms with A-expansions

We show that the Drinfeld modular forms with $A$-expansion that have been constructed by the author are precisely the hyperderivatives of the subfamily of single-cuspidal Drinfeld modular forms with $A$-expansions that remain modular after hyperdifferentiation. In addition, we show that Drinfeld-Poincaré series display a similar behavior with respect to hyperdifferentiation, giving indirect evidence that the Drinfeld modular forms with $A$-expansions are Drinfeld-Poincaré series. The Drinfeld-Poincaré series that we consider generalize previous examples of such series by Gekeler, and Gerritzen…

Zero mode of the Fourier series of some modular graphs from Poincare series

We consider specific linear combinations of two loop modular graph functions on the toroidal worldsheet with $2s$ links for $s=2, 3$ and $4$. In each case, it satisfies an eigenvalue equation with source terms involving $E_{2s}$ and $E_s^2$ only. On removing certain combinations of $E_{2s}$ and $E_s^2$ from it, we express the resulting expression as an absolutely convergent Poincare series. This is used to calculate the power behaved terms in the asymptotic expansion of the zero mode of the Fourier expansion of these graphs in a simple manner.

$D^6R^4$ curvature corrections, modular graph functions and Poincaré series

In this note we study the U-duality invariant coefficient functions of higher curvature corrections to the four-graviton scattering amplitude in type IIB string theory compactified on a torus. The main focus is on the $D^6R^4$ term that is known to satisfy an inhomogeneous Laplace equation. We exhibit a novel method for solving this equation in terms of a Poincaré series ansatz and recover known results in $D=10$ dimensions and find new results in $D<10$ dimensions. We also apply the method to modular graph functions as they arise from closed superstring one-loop amplitudes.

Faber polynomials and Poincaré series

In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished set of generators $f_{m,2-k}$. Such weakly holomorphic modular forms have been classified in terms of finitely many Eisenstein series, the unique weight 12 newform $Δ$, and certain Faber polynomials in the modular invariant $j(z)$, the Hauptmodul for $\SL_2(\Z)$. We employ the theory of harmonic weak Maass forms and (non-holomorphic) Maass-Poincaré series in…

Modular graph functions and asymptotic expansions of Poincaré series

In this note we study $SL(2,\mathbb{Z})$-invariant functions such as modular graph functions or coefficient functions of higher derivative corrections in type IIB string theory. The functions solve inhomogeneous Laplace equations and we choose to represent them as Poincaré series. In this way we can combine different methods for asymptotic expansions and obtain the perturbative and non-perturbative contributions to their zero Fourier modes. In the case of the higher derivative corrections, these terms have an interpretation in terms of perturbative string loop effects and pairs of instantons/a…

不可约有界对称域上庞加莱级数的非零性研究

On the non-vanishing of Poincaré series on irreducible bounded symmetric domains

本文将Muić积分非零准则推广至向量值情形，研究不可约有界对称域上多项式型庞加莱级数对应的全纯自守形式非零性，并在SU(p,q)群情形下给出应用示例。

Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms

We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenst…

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.

Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series

Modular graph functions are $SL(2,{\mathbb Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $τ$. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincaré series with respect to $Γ_\infty \backslash PSL(2,{\mathbb Z})$. The Fourier and Poincaré series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg…

On Poincaré series of half-integral weight

We use Poincaré series of $ K $-finite matrix coefficients of genuine integrable representations of the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $ to construct a spanning set for the space of cusp forms $ S_m(Γ,χ) $, where $ Γ$ is a discrete subgroup of finite covolume in the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $, $ χ$ is a character of $ Γ$ of finite order, and $ m\in\frac52+\mathbb Z_{\geq0} $. We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any $ f\in S_m(Γ,χ) $. Using this last result, we construct a Poin…

一族向量值Siegel Poincaré级数的构造与非消失性

Construction and non-vanishing of a family of vector-valued Siegel Poincaré series

本文利用可积反全纯离散系列表示的矩阵系数构造了一族向量值Siegel Poincaré级数，为Siegel尖点形式空间提供生成集，并借助Muić积分判据的变体证明了这些级数的非消失性。

A Poincaré series on hyperbolic space

Let $L$ be the unique even self-dual lattice of signature $(25,1)$. The automorphism group $\operatorname{Aut}(L)$ acts on the hyperbolic space $\mathcal{H}^{25}$. We study a Poincaré series $E(z,s)$ defined for $z$ in $\mathcal{H}^{25}$, convergent for $\operatorname{Re}(s) > 25$, invariant under $\operatorname{Aut}(L)$ and having singularities along the mirrors of the reflection group of $L$. We compute the Fourier expansion of $E(z,s)$ at a "Leech cusp" and prove that it can be meromorphically continued to $\operatorname{Re}(s) > 25/2$. Analytic continuation of Kloosterman sum zeta function…

Hilbert Poincaré series and kernels for products of $L$-functions

We study Hilbert Poincaré series associated to general seed functions and construct Cohen's kernels and double Eisenstein series as series of Hilbert Poincaré series. Then we calculate the Rankin-Cohen brackets of Hilbert Poincaré series and Hilbert modular forms and extend Zagier's kernel formula to totally real number fields. Finally, we show that the Rankin-Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant.

Poincaré 级数的正交关系

Orthogonality relations for Poincaré series

本文证明了具有有限中心的连通半单 Lie 群上由 Poincaré 级数给出的尖点自守形式的内积公式，并以此推广了上半平面尖点形式的相关结果。

Non-vanishing of vector-valued Poincaré series

We prove a vector-valued version of Muić's integral non-vanishing criterion for Poincaré series on the upper half-plane $ \mathcal H $. Moreover, we give an accompanying result on the construction of vector-valued modular forms in the form of Poincaré series. As an application of these results, we construct and study the non-vanishing of the classical and elliptic vector-valued Poincaré series.

Non-vanishing of Jacobi Poincaré series

We prove that under suitable conditions, the Jacobi Poincaré series of exponential type of integer weight and matrix index does not vanish identically. For classical Jacobi forms, we construct a basis consisting of the "first" few Poincaré series and also give conditions both dependent and independent of the weight, which ensures non-vanishing of classical Jacobi Poincaré series. Equality of certain Kloosterman-type sums is proved. Also, a result on the non-vanishing of Jacobi Poincaré series is obtained when an odd prime divides the index.

Rankin-Cohen brackets and Serre derivatives as Poincaré series

We give expressions for the Serre derivatives of Eisenstein and Poincaré series as well as their Rankin-Cohen brackets with arbitrary modular forms in terms of the Poincaré averaging construction, and derive several identities for the Ramanujan tau function as applications.

Poincaré square series of small weight

We extend the author's earlier computation and give coefficient formulas for the (quasimodular) Poincaré square series of weight 3/2 and weight 2 for the dual Weil representation for an even lattice.

Non-vanishing of Hilbert Poincaré series

We prove some non-vanishing results of Hilbert Poincaré series. We derive these results, by showing that the Fourier coefficients of Hilbert Poincaré series satisfy some nice orthogonality relations for sufficiently large weight as well as for sufficiently large level. To prove later results, we generalize a method of E. Kowalski et. al.

Parabolic, hyperbolic and elliptic Poincaré series

Following Petersson, we study the parabolic, hyperbolic and elliptic expansions of holomorphic cusp forms and the associated Poincaré series. We show how these ideas extend to the space of second-order cusp forms.

A note on Fourier coefficients of Poincaré series

We give a short and "soft" proof of the asymptotic orthogonality of Fourier coefficients of Poincaré series for classical modular forms as well as for Siegel cusp forms, in a qualitative form.

Hyperbolic Fourier coefficients of Poincaré series

Poincaré in 1911 and Petersson in 1932 gave the now classical expression for the parabolic Fourier coefficients of holomorphic Poincaré series in terms of Bessel functions and Kloosterman sums. Later, in 1941, Petersson introduced hyperbolic and elliptic Fourier expansions of modular forms and the associated hyperbolic and elliptic Poincaré series. In this paper we express the hyperbolic Fourier coefficients of Poincaré series, of both parabolic and hyperbolic type, in terms of hypergeometric series and Good's generalized Kloosterman sums. In an explicit example for the modular group, we see t…

Poincaré and Eisenstein series for Jacobi forms of lattice index

Poincaré and Eisenstein series are building blocks for every type of modular forms. We define Poincaré series for Jacobi forms of lattice index and state some of their basic properties. We compute the Fourier expansions of Poincaré and Eisenstein series and give an explicit formula for the Fourier coefficients of the trivial Eisenstein series. For even weight and fixed index, finite linear combinations of Fourier coefficients of non-trivial Eisenstein series are equal to finite linear combinations of Fourier coefficients of the trivial one.

Linear Relations of Siegel Poincare Series and Nonvanshing of the Central Value of Spinor L-functions

In this paper, we will first investigate the linear relations of a one parameter family of Siegel Poincaré series. Then we give the applications to the non-vanishing of Fourier coefficients of Siegel cusp eigenforms and the central values.

关于平均意义下的庞加莱级数非零性

Non-vanishing of Poincaré Series on Average

本文研究了在平均意义下庞加莱级数非零性的条件，证明了当权重或索引在特定范围内变化时，适合参数的庞加莱级数几乎不为零。

Ramanujan and coefficients of meromorphic modular forms

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincaré series introduced by Petersson and a new family of Poincaré series which we construct here and which are of independent interest. In addition we est…

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.

On the zeros of certain Poincaré series for $Γ_0^*(2)$ and $Γ_0^*(3)$

We locate all of the zeros of certain Poincare series associated with the Fricke groups $Γ_0^*(2)$ and $Γ_0^*(3)$ in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (``On the zeros of Eisenstein series'', 1970).

同余子群的 Poincaré 级数的消失

Vanishing of Poincaré series for congruence subgroups

本文研究同余子群 $Γ_0(N)$ 的 Poincaré 级数消失问题，证明在若干权 $k$、指标 $m$ 和级 $N$ 的参数选择下 $P_{k,m,N}$ 非零，并改进 Rankin 与 Mozzochi 的既有结果。

Niebur-Poincaré Series and Traces of Singular Moduli

We compute the Fourier coefficients of analogues of Kohnen and Zagier's modular forms $f_{k,D}$ of weight $2$ and negative discriminant. These functions can also be written as twisted traces of certain weight $2$ Poincaré series with evaluations of Niebur-Poincaré series as Fourier coefficients. This allows us to study twisted traces of singular moduli in an integral weight setting. In particular, we recover explicit series expressions for twisted traces of singular moduli and extend algebraicity results by Bengoechea to the weight $2$ case. We also compute regularized inner products of these …

关于 Hilbert Poincaré 级数的非零性

On the non-vanishing of Hilbert Poincaré series

本文研究 Hilbert Poincaré 级数的非零性，证明小范数条件下其不恒为零，并在数域上建立 Kloosterman 和的 Selberg 恒等式，进而在阿代尔框架中用 Hecke 算子加以解释。

非阿基米德Poincaré级数与Bruhat-Tits树上的测地线

Non-Archimedean Poincaré series and geodesics on the Bruhat-Tits tree

本文在Drinfeld半平面上构造非阿基米德Poincaré级数，证明其收敛性与非零性，并用于生成模形式空间，推广了Kurihara的方法。

Uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums

This thesis contains some new results on the uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums

Non-vanishing of Taylor coefficients and Poincaré series

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan's Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincaré series. At a "generic" point all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.

Linear Relations Among Poincare Series via Harmonic Weak Maass Forms

We discuss the problem of the vanishing of Poincaré series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan's mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincaré series and the connection to Ramanujan's mock theta functions.

辛 Kloosterman 和与 Poincaré 级数

Symplectic Kloosterman Sums and Poincaré Series

本文通过分层论证和 p-adic 驻相方法，证明了 Sp(4) 上一般 Kloosterman 和的幂节省界，并将其与 Poincaré 级数的 Fourier 系数相关联。

Eisenstein-Kronecker series via the Poincaré bundle

A classical construction of Katz gives a purely algebraic construction of Eisenstein--Kronecker series using the Gauß--Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein--Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable $p$-adic Eisenstein measure through $p$-adic theta functions of the Poincaré b…

关于Niebur-Poincaré级数的生成函数

On a generating function of Niebur-Poincaré series

本文研究了与i∞相关的Niebur-Poincaré级数生成函数在s=1处的延拓，将其与解析核和非全纯艾森斯坦级数联系，并通过多项式对数和点对不变量的周期化进行扩展。

Twisted correlations of the divisor function via discrete averages of $\operatorname{SL}_2(\mathbb{R})$ Poincaré series

We prove a theorem that allows one to count solutions to determinant equations twisted by a periodic weight with high uniformity in the modulus. It is obtained by using spectral methods of $\operatorname{SL}_2(\mathbb{R})$ automorphic forms to study Poincaré series over congruence subgroups. By keeping track of interactions between multiple orbits we get advantages over the widely used sums of Kloosterman sums techniques. We showcase this with applications to correlations of the divisor functions twisted by periodic functions and the fourth moment of Dirichlet $L$-functions on the critical lin…

On a family of Siegel Poincaré series

Let $ Γ$ be a congruence subgroup of $ \mathrm{Sp}_{2n}(\mathbb Z) $. Using Poincaré series of $ K $-finite matrix coefficients of integrable discrete series representations of $ \mathrm{Sp}_{2n}(\mathbb R) $, we construct a spanning set for the space $ S_m(Γ) $ of Siegel cusp forms of weight $ m\in\mathbb Z_{>2n} $. We prove the non-vanishing of certain elements of this spanning set using Muić's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of $ \mathrm{Sp}_{2n}(\mathbb R) $, we study the Petersson inner pro…

Flag Hilbert-Poincaré series of hyperplane arrangements and their Igusa zeta functions

We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called flag Hilbert-Poincaré series. These series are intimately connected with Igusa local zeta functions of products of linear polynomials, and their motivic and topological relatives. Our main results include a self-reciprocity result for central arrangements defined over fields of characteristic zero. We also prove combinatorial formulae for a specialization of the flag Hilbert-Poincaré series for irreducible Coxeter arrangements of types $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{D}$ i…

A problem of Petersson about weight 0 meromorphic modular forms

In this paper, we provide an explicit construction of weight $0$ meromorphic modular forms. Following work of Petersson, we build these via Poincaré series. There are two main aspects of our investigation which differ from his approach. Firstly, the naive definition of the Poincaré series diverges and one must analytically continue via Hecke's trick. Hecke's trick is further complicated in our situation by the fact that the Fourier expansion does not converge everywhere due to singularities in the upper half-plane so it cannot solely be used to analytically continue the functions. To explain t…

Irreducible vector-valued modular forms of dimension less than six

An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T-unitarizable representations of the full modular group, of dimension less than six. For representations of dimension less than four, it is shown that the associated space of vector-valued modular forms is a cyclic module over a certain skew polynomial ring of differential operators. For dimensions four and five, a complete list of possible Hilbert-Poincare series is given, using the fact that the space of vector-valued modular f…

On vector-valued Poincaré series of weight 2

Given a pair $(Γ,ρ)$ of a Fuchsian group of the first kind, and a unitary representation $ρ$ of $Γ$ of arbitrary rank, the problem of construction of vector-valued Poincaré series of weight 2 is considered. Implications in the theory of parabolic bundles are discussed. When the genus of the group is zero, it is shown how an explicit basis for the space of these functions can be constructed.

Integral of two-loop modular graph functions

The integral of an arbitrary two-loop modular graph function over the fundamental domain for $SL(2,Z)$ in the upper half plane is evaluated using recent results on the Poincaré series for these functions.

Poincaré square series for the Weil representation

We calculate the Jacobi Eisenstein series of weight $k \ge 3$ for a certain representation of the Jacobi group, and evaluate these at $z = 0$ to give coefficient formulas for a family of modular forms $Q_{k,m,β}$ of weight $k \ge 5/2$ for the (dual) Weil representation on an even lattice. The forms we construct always contain all cusp forms within their span. We explain how to compute the representation numbers in the coefficient formulas for $Q_{k,m,β}$ and the Eisenstein series of Bruinier and Kuss $p$-adically to get an efficient algorithm. The main application is in constructing automorphi…