Jitendra Bajpai, Subham Bhakta, Renan Finder
In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated to even integer weight vector-valued automorphic forms of Fuchsian groups of the first kind. At the end, their $L$-functions and exponential sums have been discussed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Lifting of vector-valued automorphic forms
本文研究对数向量值自守形式的提升问题，在已有可容许向量值自守形式提升结果的基础上，显式计算了提升后向量值自守形式的傅里叶系数。
We present bases for certain spaces of meromorphic vector-valued rational-weight mock modular forms constructed using Rademacher sums.
We prove Siegel-Walfisz type theorems (over long and short intervals) for the Fourier coefficients of certain automorphic $L$-functions and Rankin-Selberg $L$-functions over number fields.
Construction of vector-valued adelic automorphic forms
本文构造了Scheithauer提升映射的adele版本，引入向量值adele自守形式空间并定义adele提升映射，证明其与经典映射在adelization下相容，将标量与向量值模形式的关系推广至adele框架。
Constructing vector-valued automorphic forms on unitary groups
本文引入一种从标量值自守形式构造酉群上向量值自守形式的方法，并给出显式例子，该方法使用微分算子并克服了Siegel情形中未出现的挑战。
In this paper we study the images of certain families $\{ρ_{π,\ell} \}_\ell$ of $G_2$-valued Galois representations of $\mbox{Gal}(\overline{F}/F)$ associated to $L$-algebraic regular, self-dual, cuspidal automorphic representations $π$ of $\mbox{GL}_7(\mathbb{A}_F)$, where $F$ is a totally real field. In particular, we prove that, under certain automorphic conditions, the images of the residual representations $\overlineρ_{π,\ell}$ are as large as possible for infinitely many primes $\ell$. Moreover, we apply our result to some examples constructed by Chenevier, Renard and Taïbi.
On vector valued automorphic forms for the Weil representation
本文发展了与 Weil 表示相关的向量值自守形式理论，并确定了每个素数 p 下相应的球面 Hecke 代数结构。
Quasiautomorphic forms are isomorphic to vector-valued automorphic forms
本文研究了拟自守形式与向量值自守形式之间的同构关系，通过定义Hecke向量形式并证明其功能方程和乘数系统，建立了两者的双射关系。
Computation of Fourier coefficients of automorphic forms of type $G_2$
本文介绍了$G_2$型自守形式Fourier系数的计算公式及其在SAGE中的实现，并推导了相关的理论推论。
We classify the simple even lattices of square free level and signature (2,n) for n > 3. A lattice is called simple if the space of cusp forms of weight 1+n/2 for the dual Weil representation of the lattice is trivial. For a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form. Using this, we determine all holomorphic Borcherds products of singular weight (arising from vector valued modular forms with non-negative principal part) for the simple lattices. We construct the corresponding vector valued modular forms by eta prod…
We study Whittaker--Fourier coefficients of automorphic forms on a quasi-split unitary group. We reduce the analogue of the Ichino--Ikeda conjectures to a conjectural local statement using the descent method of Ginzburg--Rallis--Soudry.
In [J14], a conjecture was proposed on a relation between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. In this paper, we discuss the recent progress on this conjecture and certain problems which lead to better understanding of Fourier coefficients of automorphic forms. At the end, we extend a useful technical lemma to a few versions, which are more convenient for future applications.
We study the structures of Fourier coefficients of automorphic forms on symplectic groups based on their local and global structures related to Arthur parameters. This is a first step towards the general conjecture on the relation between the structure of Fourier coefficients and Arthur parameters given by the first named author in [J14].
We study certain cases of convoluted Fourier coefficients of $GL_n$-automorphic functions. We establish identities that express them in terms of Fourier coefficients related to unipotent orbits. The most general case that is studied is $(n)\circ(k,2^{n-1})$. The conclusions for this case is only up to a conjecture that I state. However there are certain special cases and other examples that are not based on any conjecture.
Fourier-Jacobi expansion of automorphic forms generating quaternionic discrete series
针对某类单伴随群上的自守形式建立Fourier-Jacobi展开理论，证明生成四元数离散级数的自守形式除两类群外自动满足适度增长，还证明对应尖点形式Fourier展开项无Jacobi群离散谱贡献。
A vanishing theorem for vector-valued Siegel automorphic forms in characteristic $p$
本文证明了特征 p 下向量值 Siegel 自守形式在权处于显式轨迹之外时为零，从而验证了与 Goldring 提出的 Hodge 型 Shimura 簇一般猜想的一个特例。
We extend the Fourier expansion of automorphic functions of Piateski-Shapiro and Shalika in a way that the Euclidean algorithm appears nontrivially. Then, we discuss a potential application in the Rankin-Selberg method.
While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group $Γ$ and an arbitrary representation $R$ of $Γ$ in GL$(n,{\mathbb C})$, their existence has been established in the literature only when restrictions are imposed on both $Γ$ and $R$. In this paper, we prove the existence of $n$ linearly independent vector-valued automorphic forms for any Fuchsian group $Γ$ and any $n$-dimensional complex representation $R$ of $Γ$. To this end, we realize these automorphic forms as global sections of a special rank $n$ vector bundle built using solutions to the Riemann-Hilbert p…
We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of ${\mathrm{SL}}_n(\mathbb{A})$ are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro-Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory.
In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of $\mathrm{GL}_n(\mathbb{A})$ of form $$ Δ(τ_1, b_1) \boxplus Δ(τ_2, b_2) \boxplus \cdots \boxplus Δ(τ_r, b_r)\,, $$ where $Δ(τ_i,b_i)$'s are Speh representations in the discrete spectrum of $\mathrm{GL}_{a_ib_i}(\mathbb{A})$ with $τ_i$'s being unitary cuspidal representations of $\mathrm{GL}_{a_i}(\mathbb{A})$, and $n = \sum_{i=1}^r a_ib_i$. Endoscopic lifting i…
Kurokawa-Mizumoto congruence and differential operators on automorphic forms
该研究给出与Klingen-Eisenstein级数相关的向量值黑川-水本同余成立的充分条件，并借助表示论对自守形式上的微分算子作出新的诠释。
We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blo…
Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix coefficients of integrable discrete series. We use these results to construct explicit automorphic cuspidal realizations, which have appropriate Fourier coefficients $\neq 0$, of integrable discrete series in families of congruence subgroups. In the case of $G=Sp_{2n}(\mathbb R)$, we relate our work to that of Li [15]. For $\mathcal G$ quasi--split over $\mathbb Q$, …
Automorphic products that are singular modulo primes
本文利用O(n, 2)上的Rankin--Cohen括号，证明了反射Borcherds乘积的傅里叶系数通常满足模某些素数的同余关系。
We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all N larger than 2, satisfy a central limit theorem in a suitable range, generalizing the case N=2 treated by E. Fouvry, S. Ganguly, E. Kowalski and P. Michel. Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.
We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a `hidden' invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier-Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class o…
The Arithmetic of the Fourier Coefficients of Automorphic Forms
本论文从经典与阿代尔视角研究模形式，推导其傅里叶系数的算术性质：计算带非平凡中心特征的局部惠特克新形式的$p$进赋值下界，刻画希尔伯特新形式在任意尖点处傅里叶系数的生成域，并讨论两部分关联与未来方向。
The cone conjecture for vector-valued Siegel automorphic forms
本文证明了特征$p$下秩为$n$的Siegel模簇上自守向量丛的有效锥由$G$-zip叠编码，证明了零阶上同调群在zip锥外消失，并在$n=3$时给出显式消失条件，同时定义了平凡型自守形式。
Let $m$ be an integer such that $m \geq 7$ and $m \equiv 0,1,7 \mod 8$. We construct strictly compatible systems of representations of $Γ_{\mathbb Q} \to \mathrm{Spin}_m(\overline{\mathbb Q}_l) \xrightarrow{\mathrm{spin}} \mathrm{GL}_N(\overline{\mathbb Q}_l)$ that is potentially automorphic and motivic. As an application, we prove instances of the inverse Galois problem for the $\mathbb F_p$--points of the spin groups. For odd $m$, we compare our examples with the work of A. Kret and S. W. Shin, which studies automorphic Galois representations valued in $\mathrm{Spin}_m$.
Fourier coefficients of automorphic representations $π$ of $\Sp_{2n}(\BA)$ are attached to unipotent adjoint orbits in $\Sp_{2n}(F)$, where $F$ is a number field and $\BA$ is the ring of adeles of $F$. We prove that for a given $π$, all maximal unipotent orbits, which gives nonzero Fourier coefficients of $π$ are special, and prove, under a well acceptable assumption, that if $π$ is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is $F$-anisotropic as algebraic group over $F$. These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and …
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, and applies in particular to trace functions of small conductor modulo primes.
We study Fourier coefficients of $GL_n(\A)$-automorphic functions $ϕ$, for $\A$ being the adele group of a number field $\kkk$. Let FC be an abbreviation for such a Fourier coefficient (and FCs for plural). Roughly speaking, in the present paper we process FCs by iteratively using the operations: Fourier expansions, certain exchanges of Fourier expansions, and conjugations. In Theorem 3.1 we express any FC in terms of---degenerate in many cases---Whittaker FCs. For FCs obtained from the trivial FC by choosing a certain "generic" term in each Fourier expansion involved, we establish a shortcut …
Square-root cancellation for sums of coefficients of $\operatorname{GL}(m)$-automorphic forms over values of random polynomials
本文证明了对任意 $m\geq 2$ 和 $d\geq 1$，$\mathrm{GL}(m)$-自守形式的傅里叶系数在随机 $d$ 次多项式值上的平均和具有平方根抵消。
We consider the Fourier expansions of automorphic forms on general Lie groups, with a particular emphasis on exceptional groups. After describing some principles underlying known results on GL(n), Sp(4), and G_2, we perform an analysis of the expansions on split real forms of E_6 and E_7 where simplifications take place for automorphic realizations of real representations which have small Gelfand-Kirillov dimension. Though the character varieties are more complicated for exceptional groups, we explain how the nonvanishing Fourier coefficients for small representations behave analogously to Fou…
We consider a general class of Fourier coefficients for an automorphic form on a finite cover of a reductive adelic group ${\bf G}(\mathbb{A}_{\mathbb{K}})$, associated to the data of a `Whittaker pair'. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are `Levi-distinguished' Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fou…
The goal of this paper is to study certain p-adic differential operators on automorphic forms on U(n,n). These operators are a generalization to the higher-dimensional, vector-valued situation of the p-adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the p-adic case of the C^{\infty}-differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain p-adic L-functions attached to p-adic families of automorphic forms on the unitary grou…
In this paper we analyze Fourier coefficients of automorphic forms on a finite cover $G$ of an adelic split simply-laced group. Let $π$ be a minimal or next-to-minimal automorphic representation of $G$. We prove that any $η\in π$ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $GL_n$. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient in terms of these Whitt…
We give a geometrical construction of the canonical automorphic factor for the Jacobi group and construct new vector valued modular forms from Jacobi forms by differentiating them with respect to toroidal variables and then evaluating at zero.
The zeros and poles of standard automorphic $L$-functions attached to representations of classical groups are linked to the nonvanishing of lifts in the theory of the theta correspondence. The results of this paper show that when a cuspidal representation $σ$ of a symplectic group $G$ lifts to a cuspidal representation $π= θ_ψ(σ)$ of an orthogonal group $H$ attached to a quadratic space $V$ of dimension $m$, the Fourier coefficients of automorphic forms in $σ$ are linked to periods of automorphic forms in $π$. Consequently, when our results are combined with the Rallis inner product formula in…
In the theory of automorphic descents developed by Ginzburg, Rallis and Soudry in [GRS11], the structure of Fourier coefficients of the residual representations of certain special Eisenstein series plays important roles. Started from [JLZ13], the authors are looking for more general residual representations, which may yield more general theory of automorphic descents. In this paper, we investigate the structure of Fourier coefficients of certain residual representations of symplectic groups, corresponding to certain interesting families of global Arthur parameters. On one hand, the results par…
Automorphic Green functions on Hilbert modular surfaces
本文将 Hilbert 模曲面上的自守 Green 函数理论推广至任意理想，给出 Fourier 展开、正则化、尖点增长、局部 Borcherds 乘积及生成级数模性等结果。
This article is a research exposition based on the author's talk at the International Colloquium on Automorphic Representations and L-Functions, 2012, held at TIFR, Mumbai. We consider some special cases of the following question: when is a natural subset of the Fourier coefficients sufficient to uniquely determine a "modular form"? Two kinds of modular forms are considered in this article: a) classical modular forms of half-integral weight, and b) Siegel modular forms of genus 2 and integral weight. These two apparently different scenarios turn out to be closely related. Our results were moti…
This is a continuation of our previous paper published in Israel J. Math 187 (2012), 317-369. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic L-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central $L$-values, we explicitly determine the constant of proportionality relating the square norm …
We investigate Fourier coefficients of automorphic forms on split simply-laced Lie groups G. We show that for automorphic representations of small Gelfand-Kirillov dimension the Fourier coefficients are completely determined by certain degenerate Whittaker vectors on G. Although we expect our results to hold for arbitrary simply-laced groups, we give complete proofs only for G=SL(3) and G=SL(4). This is based on a method of Ginzburg that associates Fourier coefficients of automorphic forms with nilpotent orbits of G. Our results complement and extend recent results of Miller and Sahi. We also …
Pullback formula for vector-valued Hermitian modular forms on $\mathrm{U}_{n,n}$
本文研究了在CM域上向量值Hermitian模形式的拉回公式，并给出了保持自守性质的微分算子的等价条件。
In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. Instead of relying on the trace formula, which was the primary tool in preceding studies on the subject, we use Fourier integral operator methods. This allows us to treat not only spherical, but also non-spherical Hecke-Maass forms with corresponding remainder estimates. Our asymptotic formulas are available for arbitrary simple and connected algebraic groups over number fields with cocompact arithmetic subgroups…
Primitive Fourier Coefficients of Hermitian Klingen-Eisenstein Series
本文确定了CM扩张上Hermitian Klingen-Eisenstein级数的本原Fourier系数，通过拉回公式将其表示为Siegel Eisenstein级数系数与Petersson内积的乘积，并在假设下证明了$p$-整性。
The Bernstein-Gelfand tensor product functors are endofunctors of the category of Harish-Chandra modules provided by tensor products with finite dimensional modules. We provide an automorphic analogue of these tensor product functors, implemented by vector-valued automorphic representations that are trivial at all finite places. They naturally explain the role of vector-valued modular forms in recent work by Bringmann-Kudla on Harish-Chandra modules associated with harmonic weak Maaß forms. We give a detailed account of the image $\mathrm{sym}^1 \otimes \varpi(E_2)$ of the automorphic represen…
It is well-known that there are automorphic eigenfunctions on SL(2,Z)\SL(2,R)/SO(2,R) -- such as the classical $j$-function -- that have exponential growth and have exponentially growing Fourier coefficients (e.g., negative powers of $q=e^{2πi z}$, or an I-Bessel function). We show that this phenomenon does not occur on the quotient SL(3,Z)\SL(3,R)/SO(3,R) and eigenvalues in general position (a removable technical assumption). More precisely, if such an automorphic eigenfunction has at most exponential growth, it cannot have non-decaying Whittaker functions in its Fourier expansion. This confi…
We establish (Theorem 3.6) polynomial-growth estimates for the Fourier coefficients of holomorphic logarithmic vector-valued modular forms.
In this paper we deal with monogenic and $k$-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between $k$-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vanishing monogenic automorphic form a Clifford algebra valued Maaß wave form. In the second part of the paper we compute the Fourier expansion of the $k$-hypermonogenic Eisenstein series which provide u…
Let $λ_ϕ(n)$ be the Fourier coefficients of a Hecke holomorphic or Hecke--Maass cusp form on ${\rm SL}_2(\mathbb Z)$, and $f$ be any multiplicative function that satisfies two mild hypotheses. We establish a non-trivial upper bound for the correlation $\sum_{n \leq X}f(n)λ_ϕ(n+h)$ uniformly in $0<|h|\ll X$. As applications, we consider some special cases, including $λ_π(n), \,μ(n)λ_π(n)$ and any divisor-bounded multiplicative function. Here $λ_π(n)$ denotes the $n$-th Dirichlet coefficient of $\text{GL}_m$ automorphic $L$-function $L(s,π)$ for an automorphic irreducible cuspidal representation…
We discover new Voronoi formulae for automorphic forms on GL($n$) for $n\geq 4$. There are $[n/2]$ different Voronoi formulae on GL($n$), which are Poisson summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums.
Lattice points on determinant surfaces and the spectrum of the automorphic Laplacian
利用经典傅里叶分析和自守形式谱论，推导了扩张盒子[-X,X]^4中行列式方程ad-bc=r的整数解个数的渐近公式，并应用于加权和的研究。
This paper studies the Fourier expansion of Hecke-Maass eigenforms for $GL(2, \mathbb Q)$ of arbitrary weight, level, and character at various cusps. Translating well known results in the theory of adelic automorphic representations into classical language, a multiplicative expression for the Fourier coefficients at any cusp is derived. In general, this expression involves Fourier coefficients at several different cusps. A sufficient condition for the existence of multiplicative relations among Fourier coefficients at a single cusp is given. It is shown that if the level is 4 times (or in some…
Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $λ_f(n)$ and $λ_g(n)$, respectively. In this paper, we prove nontrivial estimates for the sum $$ \sum_{n=1}^{\infty}λ_f(n) λ_g(n)e\left(t φ\left(\frac{n}{X}\right)\right)V\left(\frac{n}{X}\right), $$ where $e(x)=e^{2πix}$, $V(x)\in \mathcal{C}_c^{\infty}(1,2)$, $t\geq 1$ is a large parameter and $φ(x)$ is some nonlinear real valued smooth function. Applications of these estimates include a subconvex bound for the Rankin-Selberg $L$-function $L(s,f\otimes g)$ in the $t$-aspect, an …
In this paper we discuss the existence of certain classes of cuspidal automorphic representations having non-zero Fourier coefficients for general semisimple algebraic group $G$ defined over a number field $k$ such that its Archimedean group $G_\infty$ is not compact. When $G$ is quasi--split over $k$, we obtain a result on existence of generic cuspidal automorphic representations which generalize a result of Vign\' eras, Henniart, and Shahidi. We also discuss the existence of cuspidal automorphic forms with non--zero Fourier coefficients for congruence of subgroups of $G_\infty$.
We use results about Fourier coefficients appearing in [T] (and some more obtained here), to obtain information for certain among the integrals of the form $$I=\int_{GL_n(\kkk)Z_n(\A)\s GL_n(\A)}φ(g)ϕ(g)\F(E)(\tj(g))dg$$ where: $\A$ is the adele ring of a number field $\kkk$; $φ$ is a $GL_n(\A)$-cuspidal automorphic form; $ϕ$ is a $GL_n(\A)$-automorphic function (even the trivial for some results); $E$ is a $GL_{N}(\A)$-automorphic form for a multiple $N$ of $n$; $\F(E)$ is a Fourier coefficient of $E$ for certain choices of additive functions $\F$ in a set $\BBnk[N]$ which we defined in $[T];…
We look at genera of even unimodular lattices of rank $12$ over the ring of integers of $\mathbb{Q}(\sqrt{5})$ and of rank $8$ over the ring of integers of $\mathbb{Q}(\sqrt{3})$, using Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find instances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank $12$ over the Eisenstein integers, even and unimodular over $\mathbb…
H. Aoki showed that any symmetric formal Fourier-Jacobi series for the symplectic group Sp_2(Z) is the Fourier-Jacobi expansion of a holomorphic Siegel modular form. We prove an analogous result for vector valued symmetric formal Fourier-Jacobi series, by combining Aoki's theorem with facts about vector valued modular forms. Recently, this result was also proved independently by M. Raum using a different approach. As an application, by means of work of W. Zhang, modularity results for special cycles of codimension 2 on Shimura varieties associated to orthogonal groups can be derived.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。