Jason Levy
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.
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In this paper we introduce the so-called truncated very-well-poised $_6ψ_6$ series and set up an explicit recurrence relation for it by means of the classical Abel lemma on summation by parts. This new recurrence relation implies an elementary proof of Bailey's well-known $_6ψ_6$ summation formula.
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.
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。
A poly-log time method to compute the truncated theta function, its derivatives, and integrals is presented. The method is elementary, rigorous, explicit, and suited for computer implementation. We repeatedly apply the Poisson summation formula to the truncated theta function while suitably normalizing the linear and quadratic arguments after each repetition. The method relies on the periodicity of the complex exponential, which enables the suitable normalization of the arguments, and on the self-similarity of the Gaussian, which ensures that we still obtain a truncated theta function after ea…
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.
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。
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。
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.
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…
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.
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.
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 $r_3(n)$.
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。
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。
In this paper, we introduce the degenerate zero-truncated Poisson random variables whose probability mass functions are a natural extension of the zero-truncated Poisson distributions, and investigate various properties of those random variables.
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。
The zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers, which are also known as the conditional Poisson distributions or the positive Poisson distributions. In this paper, we introduce the r-truncated degenerate Poisson random variable with parameter a and investigate various properties of this random variable
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…
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.
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。
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.
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。
Carlitz [2] initiated a study on degenerate versions of Bernoulli and Euler numbers which has been extended recently to the researches on various degenerate versions of quite a few special numbers and polynomials. They have been explored by using several different tools including generating functions, combinatorial methods, $p$-adic analysis, umbral calculus, special functions, differential equations and probability theory as well. \par The degenerate Poisson random variables are degenerate versions of the Poisson random variables. In [6], studied are the degenerate binomial and degenerate Poi…
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。
Let $n\ge 1$ be an integer and $e_n(x)$ denote the truncated exponential Taylor polynomial, i.e. $e_{n}(x)=\sum_{i=0}^n\frac{x^i}{i!}$. A well-known theorem of Schur states that the Galois group of $e_n(x)$ over $\Q$ is the alternating group $A_n$ if $n$ is divisible by 4 or the symmetric group $S_n$ otherwise. In this paper, we study algebraic properties of the summation of two truncated exponential Taylor polynomials $\E_n(x):=e_n(x)+e_{n-1}(x)$. We show that $\frac{x^n}{n!}+\sum_{i=0}^{n-1}c_i\frac{x^i}{i!}$ with all $c_i \ (0\le i\le n-1)$ being integers is irreducible over $\Q$ if either …
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。
Let $F$ be a number field and let $\mathbb{A}_F$ be its ring of adeles. Let $B$ be a quaternion algebra over $F$ and let $ν:B \to F$ be the reduced norm. Consider the reductive monoid $M$ over $F$ whose points in an $F$-algebra $R$ are given by \begin{align*} M(R):=\{(γ_1,γ_2) \in (B \otimes_F R)^{2}:ν(γ_1)=ν(γ_2)\}. \end{align*} Motivated by an influential conjecture of Braverman and Kazhdan we prove a summation formula analogous to the Poisson summation formula for certain spaces of functions on the monoid. As an application, we define new zeta integrals for the Rankin-Selberg $L$-function a…
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.
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。
We continue the analysis of the elliptic part of the trace formula initiated in \cite{Altug:2015aa}. In that reference Poisson summation was applied to the elliptic part and the dominant term was analyzed. The main aim of this paper is to study the remaining terms after Poisson summation. We analyze the the Fourier transforms of (smoothed) orbital integrals and obtain exact asymptotic expansions. As an application we recover, using the Arthur-Selberg trace formula, Kuznetsov's result (cf. \cite{Kuznetsov:1980aa}) that the trace of the $p$th Hecke operator on cuspidal automorphic representation…
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…
Certain Fourier Operators and their Associated Poisson Summation Formulae on $\mathrm{GL}_1$
本文推广Tate的工作，利用$\GL_1$上的调和分析研究Langlands自守$L$-函数，引入了$(\sig,ρ)$-Schwartz函数与Fourier算子，证明了$\GL_n$标准表示下的Poisson求和公式猜想，并给出了标准$L$-函数临界零点的谱解释。
Summation formulae for quadrics
本文证明偶数变量二次曲面零点集上无支集限制的 Poisson 求和公式，边界项由常数或相关较小二次曲面求和表示，并借助 Arthur 截断 theta 提升建立其与正交群表示及二次型解数估计的联系。
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 …
Computing the truncated theta function via Mordell integral
本文提出简化Hiary算法，用莫尔德尔积分替代泊松求和公式，高效计算截断θ函数，提升实现便捷性。
On triple product $L$-functions and the fiber bundle method
本文引入多变量 zeta 积分展开为欧拉乘积以表示三重积 $L$-函数，提出推广的泊松求和猜想并证明其蕴含解析性质，最后用纤维丛方法将猜想简化。
Extending ideas of A.N. Parshin and D.V. Osipov, Harmonic Analysis is developed for filtered infinite dimensional modules over a ring. We establish Pontryagin duality, the Fourier inversion formula, Plancherel formula and Poisson summation formula for all dimensions.
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 Fourier transform for triples of quadratic spaces
本文针对数域上三个带非退化二次型的偶维向量空间对应的闭子概形，通过引入边界项、采用全局到局部论证，证明其傅里叶变换在施瓦茨空间上良定义，并推广了泊松求和公式，为非Braverman-Kazhdan空间的球面簇建立了首个带边界项的求和公式。
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