scholarly journals Weil type representations and automorphic forms

1980 ◽  
Vol 77 ◽  
pp. 145-166 ◽  
Author(s):  
Toshiaki Suzuki
Keyword(s):  
Dirichlet Series ◽  
Functional Equations ◽  
Theta Series ◽  
Type Representation ◽  
Generalized Poisson ◽  
Summation Formulae ◽  
Reciprocity Law ◽  
Weil Type ◽  
Generalized Theta Series ◽  
Poisson Summation

During 1934-1936, W. L. Ferrar investigated the relation between summation formulae and Dirichlet series with functional equations, inspired by Voronoi’s works (1904) on summation formula with respect to the numbers of divisors. In [11] A. Weil showed that the automorphic properties of theta series are expressed by generalized Poisson summation formulae with respect to the so-called Weil representation. On the other hand, T. Kubota, in his study of the reciprocity law in a number field, defined a generalized theta series and a generalized Weil type representation of SL(2, C) and obtained similar results to those of A. Weil (1970-1976, [5], [6], [7]). The methods, used by W. L. Ferrar and T. Kubota, to obtain (generalized Poisson) summation formulae depend similarly on functional equations of Dirichlet series (attached to the generalized theta series).

scholarly journals Poisson Summation Formulae Associated with the Special Affine Fourier Transform and Offset Hilbert Transform

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Zhi-Hai Zhuo
Keyword(s):  
Fourier Transform ◽  
Hilbert Transform ◽  
Fractional Fourier Transform ◽  
The Novel ◽  
Generalized Poisson ◽  
Summation Formulae ◽  
Points Of View ◽  
The Relationship ◽  
Transform Domains ◽  
Poisson Summation

This paper investigates the generalized pattern of Poisson summation formulae from the special affine Fourier transform (SAFT) and offset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. Then, the generalized Poisson sum formulae are obtained based on above relationships. The novel results can be regarded as the generalizations of the classical results in several transform domains such as FT, fractional Fourier transform, and the linear canonical transform.

scholarly journals Dirichlet series and automorphic functions associated to a quadratic form

2003 ◽  
Vol 171 ◽  
pp. 1-50 ◽  
Author(s):  
Manfred Peter
Keyword(s):  
Quadratic Form ◽  
Dirichlet Series ◽  
Functional Equations ◽  
The Other ◽  
Converse Theorem ◽  
Integral Quadratic Form ◽  
Special Values ◽  
Reciprocity Law ◽  
Automorphic Functions ◽  
Gaussian Sums

AbstractStarting from the reciprocity law for Gaussian sums attached to an integral quadratic form we prove functional equations for a new kind of Dirichlet series in two variables. For special values of one variable they are of Hecke type with respect to the other variable. With Weil’s converse theorem we derive automorphic functions which generalize Siegel’s genus invariant and the automorphic functions of Cohen and Zagier.

scholarly journals Subconvexity for a double Dirichlet series

2010 ◽  
Vol 147 (2) ◽  
pp. 355-374 ◽  
Author(s):  
Valentin Blomer
Keyword(s):  
Dirichlet Series ◽  
Functional Equations ◽  
Mean Square ◽  
Double Dirichlet Series ◽  
Image Position

AbstractFor two real characters ψ,ψ′ of conductor dividing 8 define where $\chi _d = (\frac {d}{.})$ and the subscript 2 denotes the fact that the Euler factor at 2 has been removed. These double Dirichlet series can be extended to $\Bbb {C}^2$ possessing a group of functional equations isomorphic to D12. The convexity bound for Z(s,w;ψ,ψ′) is |sw(s+w)|1/4+ε for ℜs=ℜw=1/2. It is proved that Moreover, the following mean square Lindelöf-type bound holds: for any Y1,Y2≥1.

On the space of generalized theta-series for certain quadratic forms in any number of variables

2019 ◽  
Vol 69 (1) ◽  
pp. 87-98
Author(s):  
Ketevan Shavgulidze
Keyword(s):  
Upper Bound ◽  
Quadratic Forms ◽  
Theta Series ◽  
Vector Spaces ◽  
Generalized Theta Series

Abstract An upper bound of the dimension of vector spaces of generalized theta-series corresponding to some nondiagonal quadratic forms in any number of variables is established. In a number of cases, an upper bound of the dimension of the space of theta-series with respect to the quadratic forms of five variables is improved and the basis of this space is constructed.

Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Riemann Zeta Function ◽  
Functional Equations ◽  
Lattice Points ◽  
Baxter Equation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Multiple Dirichlet Series

Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara's crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang–Baxter equation.

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