Stephen D. Miller, Wilfried Schmid
We give an overview of classical summation formulations, such as Poisson's and Voronoi's, and then turn to modern versions involving modular form coefficients. A new formula involving the coefficients of cusp forms on GL(3) is described, and its proof sketched, followed by applications to L-functions. The main method used is the boundary value distribution of automorphic forms.
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In an earlier paper we derived an analogue of the classical Voronoi summation formula for automorphic forms on GL(3), by using the theory of automorphic distributions. The purpose of the present paper is to apply this theory to derive the analogous formulas for GL(n).
In this note, we establish a Voronoï--Oppenheim summation formula for divisor functions over an arbitrary number field.
A general Voronoï summation formula for the (metaplectic) double cover of $\text{GL}_2$ is derived via the representation theoretic framework à la Ichino--Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms.
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.
This paper is third in a series of three, following "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and Analytic Continuation of Dirichlet Series" (math.FA/0403030). The first is primarily an expository paper explaining the present one, whereas the second contains some distributional machinery used here as well. These papers concern the boundary distributions of automorphic forms, and how they can be applied to study questions about cusp forms on semisimple Lie groups. The main result of this paper is a Voronoi-style summation formula for the …
A Voronoi summation formula for the shifted triple divisor function
本文证明了由加性特征扭曲的移位3重除数函数的Voronoi求和公式，主要工具是提供了移位$GL(3)$ Estermann函数的函数方程。
S.D. Miller and F. Zhou have proved a balanced Voronoi summation formula for GL$_N$ over $\mathbb Q$, which allows one to control the dimensions of the Kloosterman sums appearing on either side of the Voronoi formula. In this note, we prove a balanced Voronoi formula over an arbitrary number field, starting with the Voronoi summation formula of A. Ichino and N. Templier over number fields, allowing one to extend recent results on spectral reciprocity laws to number fields, in special cases.
We consider the Fourier expansion of a Hecke (resp.\ Hecke--Maaß) cusp form of general level $N$ at the various cusps of $Γ_{0}(N)\bs\Hb$. We explain how to compute these coefficients via the local theory of $p$-adic Whittaker functions and establish a classical Voronoï summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin--Lehner relations.
In this paper we establish a very flexible and explicit Voronoi summation formula. This is then used to prove an almost Weyl strength subconvexity result for automorphic $L$-functions of degree two in the depth aspect. That is, looking at twists by characters of prime power conductor. This is the natural $p$-adic analogue to the well studied $t$-aspect.
We investigate the Voronoi summation problem for ${\rm GL}_n$ in the level aspect for $n\geq 2$. Of particular interest are those primes at which the level and modulus are jointly ramified - a common occurrence in analytic number theory when using techniques such as the Petersson trace formula. Building on previous legacies, our formula stands as the most general of its kind; in particular we extend the results of Ichino-Templier. We also give (classical) refinements of our formula and study the $p$-adic generalisations of the Hankel transform.
In this article we study the Poisson summation for Hankel transform in the sense of Braverman-Kazhdan-Ngo in the special case of $L$-embedding $ρ: GL_1\rightarrow GL_2$. We view such a summation formula as the generalization of the classical Voronoi summation formula.
Character analogues of Cohen type identities and related Voronoi summation formulas
本文针对 Cohen 型恒等式给出特征类比，推导相关 Voronoi 求和公式，并证明实原特征下 L(1,χ) 的正性。
A note on Poisson summation for GL(2)
本文从adelic视角重新审视GL(2)上通过修改泊松求和公式抵消平凡表示贡献的方法。
Braverman and Kazhdan introduced influential conjectures generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands $L$-functions have meromorphic continuations and functional equations as predicted by Langlands' functoriality conjecture. As evidence for their conjectures, Braverman and Kazhdan considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. In this paper we consider a special case o…
Plancherel and Poisson summation formulae for a family of affine $Ψ$-bundles
本文证明一类仿射Ψ丛的Plancherel公式，构造对应傅里叶变换与渐近Schwartz空间，在适当假设下证明泊松求和公式，其研究动机源于三重乘积L-函数的应用。
In this paper we show how the GL(N) Voronoi summation formula of [MiSc2] can be rewritten to incorporate hyper-Kloosterman sums of various dimensions on both sides. This generalizes a formula for GL(4) with ordinary Kloosterman sums on both sides that was considered by Xiaoqing Li and the first-named author, and later by the second-named author in [Zho].
Schubert Eisenstein series and Poisson summation for Schubert varieties
本文重新审视了关于 Schubert Eisenstein 级数亚纯延拓的猜想，通过证明与 Schubert 簇相关方案的 Poisson 求和公式，在许多情况下确立并改进了该猜想。
With analytic applications in mind, in particular Beyond Endoscopy ([13]), we initiate the study of the elliptic part of the trace formula. Incorporating the approximate functional equation to the elliptic part we control the analytic behavior of the volumes of tori that appear in the elliptic part. Furthermore by carefully choosing the truncation parameter in the approximate functional equation we smooth-out the singularities of orbital integrals. Finally by an application of Poisson summation we rewrite the elliptic part so that it is ready to be used in analytic applications, and in particu…
It is proved that a maximal abelian subalgebra of the noncommutative torus commutes with the Laplace operator on a complex torus. As a corollary, one gets an analog of the Poisson summation formula for noncommutative tori.
Beyond Endoscopy via Poisson Summation for GL(2,K)
本研究将朗兰兹超越内窥策略中针对GL(2,Q)的第一步结果推广到一类全实数域，通过正则椭圆项的加法泊松求和消去平凡与特殊表示对迹公式的贡献，推进函子性原理相关研究。
Euler--Maclaurin and Poisson analogues of the summations $\sum_{a < n \leq b} χ(n) f(n)$, $\sum_{a < n \leq b} d(n) f(n)$, $\sum_{a < n \leq b} d(n) χ(n) f(n)$ have been obtained in a unified manner, where $(χ(n))$ is a periodic complex sequence; $d(n)$ is the divisor function and $f(x)$ is a sufficiently smooth function on $[a,b]$. We also state a generalised Abel's summation formula, generalised Euler's summation formula and Euler's summation formula in several variables.
This paper is second in a series of three papers; the first of which is "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187), and the third of which is "Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)". The first paper is primarily an expository paper, while the third proves a Voronoi-style summation formula for the coefficients of a cusp form on GL(3,Z)\GL(3,R). This present paper contains the distributional machinery used in the third paper for rigorously deriving the summation formula, and also for the proof of the GL(3)xGL(1) converse theo…
The Voronoi Summation Formula for $\mathrm{GL}_n$ and the Godement-Jacquet Kernels
本文基于 Godement-Jacquet 理论进一步发展局部理论，提出 $\mathrm{GL}_n$ 上 Voronoi 求和公式的 Poisson 证明，并引入 Godement-Jacquet 核及其对偶核概念，揭示其与 $L$-函数零点的关联。
Let $G$ be a reductive algebraic group defined over $\bQ$, with anisotropic centre. Given a rational action of $G$ on a finite-dimensional vector space $V$, we analyze the truncated integral of the theta series corresponding to a Schwartz-Bruhat function on $V(\bA)$. The Poisson summation formula then yields an identity of distributions on $V(\bA)$. The truncation used is due to Arthur.
Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 1: Poisson summation
本文通过改进Alduğ的方法，解决了在$\mathbb{Q}$上隔离特定$\mathsf{GL}_2$表示的问题，利用Poisson求和公式和残差计算实现。
Beyond Endoscopy: Poisson Summation Formula and Kuznetsov Trace Formula on $\mathrm{GL}_2$
本文证明 $\mathrm{GL}_2$ Whittaker 空间上的泊松求和公式，并推导 Kuznetsov 型迹公式，揭示其自然导出标准 $L$-函数的函数方程。
Poisson summation formula and Index Transforms
本文研究了一类与指标变换相关的求和公式，主要结果涉及 Whittaker 函数第二个指标的求和与积分公式。
We obtain new Poisson type summation formulas with nodes $\pm \sqrt{n}$ and with weights involving the function $r_k(n)$ that gives the number of representations of a positive integer $n$ as the sum of $k$ squares. Our results extend summation formulas due to Guinand and Meyer that involve the sum-of-three-squares function $r_3(n)$.
We investigate the relations for $L$-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and half-integral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of $SL(2)$. Our goal is converse theorems for automorphic distributions and Maass forms of level $N$ characterizing them by analytic properties of the associated $L$-functions. As an application of our converse theorems, we construc…
We study the notion of formal duality introduced by Cohn, Kumar, and Schürmann in their computational study of energy-minimizing particle configurations in Euclidean space. In particular, using the Poisson summation formula we reformulate formal duality as a combinatorial phenomenon in finite abelian groups. We give new examples related to Gauss sums and make some progress towards classifying formally dual configurations.
Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities
本研究推导了刘维尔函数、莫比乌斯函数及除数函数平方的Voronoi求和公式，得到多类恒等式特例，并在相关数论猜想假设下证明了对应Riesz和及其误差项的振荡性质。
Voronoi summation formula for the generalized divisor function $σ_{z}^{(k)}(n)$
本文研究广义除数函数 $σ_{z}^{(k)}(n)$，在其 Dirichlet 级数不落入经典多 gamma 因子 Hecke 函数方程理论的情况下，建立显式 Voronoï 求和公式，并分析新型 Bessel 型核及其关键关系。
We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, ou…
We show that a combination of well-known operators, namely $\Iτ\circ{H}\circ\Ps$ is self-adjoint and {\em ad-hoc} related to the $ζ$ function. Here $τ$ is an involution appearing in Weil's positivity criteria needed for hermitrization, $H$ a regularization operator introduced by Connes \cite{Co2} and $\Ps$ essentially Poisson summation. We elaborate on the Hilbert-Pólya conjecture, discuss why the Hermite-Biehler theorem, uncertainty relations and cohomologies are interesting in our scenario.
By giving the definition of the sum of a series indexed by a set on which a group acts, we prove that the sum of the series that defines the Riemann zeta function, the Epstein zeta function, and a few other series indexed by $\Z^k$ has an intrinsic meaning as a complex number, independent of the requirements of analytic continuation. The definition of the sum requires nothing more than algebra and the concept of absolute convergence. The analytical significance of the algebraically defined sum is then explained by an argument that relies on the Poisson formula for tempered distributions.
Let $K=\mathbb{Q}(\sqrt{-D})$ be an imaginary number field, $(p)=\mathfrak{p}\mathfrak{p}'$ be a split odd prime and $ψ$ be a Hecke character of conductor $\mathfrak{p}$. Let $L(s,ψ)$ be the associated $L$-function. We prove the Burgess bound in $t$-aspect and a hybrid bound in conductor aspect, \begin{equation*} L(1/2+it,ψ)\ll_{D,\varepsilon} (1+|t|)^{3/8+\varepsilon}p^{1/8} \end{equation*} for $p\ll t$. In Appendix A, we present the ideas for an elementary proof of Voronoi summation formula for holomorphic cusp forms with CM and squarefree level. This is done by exploiting the lattice struct…
Recent work of Altuğ continues the preliminary analysis of Langlands' Beyond Endoscopy proposal for $GL(2)$ by removing the contribution of the trivial representation to the trace formula using a Poisson summation formula. We show that Altuğ's method of smoothing real elliptic orbital integrals by an approximate functional equation extends to $GL(n)$. We also discuss the case of an arbitrary reductive group, and remaining obstructions for applying Poisson summation.
Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms $\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let \begin{align*} Y \subset \prod_{i=1}V_i \end{align*} be the closed subscheme consisting of $(v_1,v_2,v_3)$ on which $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. Motivated by conjectures of Braverman and Kazhdan and related work of Lafforgue, Ngô, and Sakellaridis we prove an analogue of the Poisson summation formula for certain functions on this space.
Beyond Endoscopy for $\mathrm{GL}(3, \mathbb{Q})$: Isolation of the residual spectrum via Poisson Summation
本文在GL(3,Q)上推广Aldug为GL(2,Q)发展的Poisson求和方法，通过假设猜想A分离平凡表示的贡献，并通过Poisson求和得到具体展开。
On Berndt's summation formula
本文通过穆恩茨公式和泊松求和公式证明了等价于伯恩德特公式的求和公式，并利用梅林反演性质给出了不连续函数的例子，提出多个新应用。
Measures, modular forms, and summation formulas of Poisson type
本文建立 $k$-球面 Fourier 本征测度与 metaplectic 群下模型 Fourier 级数的对应，并据此构造测度、导出 Poisson 型求和公式及高维推广。
Geometrization of summation formulae for quadrics
通过明确Braverman-Kazhdan空间与theta提升定义的两种二次曲面Schwartz空间之间的关系，将数域上偶数个变量的分裂二次型零点集上的Poisson求和公式几何化。
We obtain resonances for short exponential sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(n,\mathbb Z)$. This involves deriving asymptotics for the integrals appearing in the $\mathrm{GL}(n)$ Voronoi summation formula. As an application, we also prove an $Ω$-result for short sums of Fourier coefficients.
In this note, we revisit an identity that Miller and Schmid showed in their article on a general Voronoi summation formula for $GL(n,\mathbb{Z})$ in 2009. For the proof, we mainly follow Cogdell and Piatetski-Shapiro's ideas in their work on converse theorems for $GL(n)$.
This article describes a general method for computing automorphic forms using Voronoi-type summation formulas. It gives a numerical example where the technique is successful in quickly finding a cusp form on GL(3,Z)\GL(3,R), albeit one whose existence was already known as a Langlands lift.
A classification of Fourier summation formulas and crystalline measures
本文对傅里叶求和公式和晶体测度进行了分类，并引入了eta-quotients方法推广了Guinand的例子。
We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, classical summation formulas as Euler-McLaurin or Abel-Plana formulas, explicit formulas in number theory and Selberg trace formulas in Riemannian geometry appear as special cases of our general Poisson-Newton formula. We also associate to finite order meromorphic functions general Poisson…
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