The Experts below are selected from a list of 5466 Experts worldwide ranked by ideXlab platform

Gautam Chinta - One of the best experts on this subject based on the ideXlab platform.

  • constructing weyl group multiple Dirichlet Series
    arXiv: Number Theory, 2008
    Co-Authors: Gautam Chinta, Paul E Gunnells
    Abstract:

    Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet Series for Phi is a Dirichlet Series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, that has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. A heuristic definition of such Series was given in [2], and they have been investigated in certain special cases in [2-6, 11-14]. In this paper we generalize results in [13] to construct Weyl group multiple Dirichlet Series by a uniform method, and show in all cases that they have the expected properties.

  • weyl group multiple Dirichlet Series constructed from quadratic characters
    Inventiones Mathematicae, 2007
    Co-Authors: Gautam Chinta, Paul E Gunnells
    Abstract:

    We construct multiple Dirichlet Series in several complex variables whose coefficients involve quadratic residue symbols. The Series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet Series in greater than two variables having the properties predicted in [2].

  • weyl group multiple Dirichlet Series i
    2006
    Co-Authors: Benjamin Brubaker, Daniel Bump, Gautam Chinta, Solomon Friedberg
    Abstract:

    Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet Series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet Series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet Series are fundamental objects. The goal of this paper is to define these Series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet Series are multiplicative, but the multiplicativity is twisted , so the Dirichlet Series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta Series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet Series resembling our An+1 multiple Dirichlet Series. There is no a priori reason that we are

Tian Hong-gen - One of the best experts on this subject based on the ideXlab platform.

Paul E Gunnells - One of the best experts on this subject based on the ideXlab platform.

  • constructing weyl group multiple Dirichlet Series
    arXiv: Number Theory, 2008
    Co-Authors: Gautam Chinta, Paul E Gunnells
    Abstract:

    Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet Series for Phi is a Dirichlet Series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, that has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. A heuristic definition of such Series was given in [2], and they have been investigated in certain special cases in [2-6, 11-14]. In this paper we generalize results in [13] to construct Weyl group multiple Dirichlet Series by a uniform method, and show in all cases that they have the expected properties.

  • weyl group multiple Dirichlet Series constructed from quadratic characters
    Inventiones Mathematicae, 2007
    Co-Authors: Gautam Chinta, Paul E Gunnells
    Abstract:

    We construct multiple Dirichlet Series in several complex variables whose coefficients involve quadratic residue symbols. The Series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet Series in greater than two variables having the properties predicted in [2].

Sun Dao-chun - One of the best experts on this subject based on the ideXlab platform.

Daniel Bump - One of the best experts on this subject based on the ideXlab platform.

  • introduction multiple Dirichlet Series
    2012
    Co-Authors: Daniel Bump
    Abstract:

    This introductory article aims to provide a roadmap to many of the interrelated papers in this volume and to a portion of the field of multiple Dirichlet Series, particularly emerging new ideas. It is both a survey of the recent literature, and an introduction to the combinatorial aspects of Weyl group multiple Dirichlet Series, a class of multiple Dirichlet Series that are not Euler products, but which may nevertheless be reconstructed from their p-parts. These p-parts are combinatorially interesting, and may often be identified with p-adic Whittaker functions.

  • multiple Dirichlet Series l functions and automorphic forms
    2012
    Co-Authors: Daniel Bump, Solomon Friedberg, Dorian Goldfeld
    Abstract:

    Preface.- Introduction: Multiple Dirichlet Series.- A Crystal Description for Symplectic Multiple Dirichlet Series.- Metaplectic Whittaker Functions and Crystals of Type B.- Metaplectic Ice.- Littelmann patterns and Weyl Group Multiple Dirichlet Series of Type D.- Toroidal Automorphic Forms, Waldspurger Periods and Double Dirichlet Series.- Natural Boundaries and Integral Moments of L-functions.- A Trace Formula of Special Values of Automorphic L-functions.- The Adjoint L-function of SU(2,1).- Symplectic Ice.- On Witten Multiple Zeta-Functions Associated with Semisimple Lie Algebras III.- A Pseudo Twin-Prime Theorem.- Principal Series Representations of Metaplectic Groups over Local Fields.- Two-Dimensional Adelic Analysis and Cuspidal Automorphic Representations of GL(2).

  • weyl group multiple Dirichlet Series type a combinatorial theory
    2011
    Co-Authors: Benjamin Brubaker, Daniel Bump, Solomon Friedberg
    Abstract:

    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 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.

  • weyl group multiple Dirichlet Series i
    2006
    Co-Authors: Benjamin Brubaker, Daniel Bump, Gautam Chinta, Solomon Friedberg
    Abstract:

    Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet Series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet Series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet Series are fundamental objects. The goal of this paper is to define these Series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet Series are multiplicative, but the multiplicativity is twisted , so the Dirichlet Series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta Series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet Series resembling our An+1 multiple Dirichlet Series. There is no a priori reason that we are

  • on kubota s Dirichlet Series
    Crelle's Journal, 2006
    Co-Authors: Benjamin Brubaker, Daniel Bump
    Abstract:

    Kubota [19] showed how the theory of Eisenstein Series on the higher metaplectic covers of SL2 (which he discovered) can be used to study the analytic properties of Dirichlet Series formed with n-th order Gauss sums. In this paper we will prove a functional equation for such Dirichlet Series in the precise form required by the companion paper [2]. Closely related results are in Eckhardt and Patterson [10]. The Kubota Dirichlet Series are the entry point to a fascinating universe. Their residues, for example, are mysterious if n > 3, though there is tantalizing evidence that these residues exhibit a rich structure that can only be partially glimpsed at this time. When n = 4 the residues are the Fourier coefficients of the biquadratic theta function that were studied by Suzuki [23]. Suzuki found that he could only determine some of the coefficients. This failure to determine all the coefficients was explained in terms of the failure of uniqueness of Whittaker models for the generalized theta Series by Deligne [9] and by Kazhdan and Patterson [15]. On the other hand, Patterson [22] conjectured that the mysterious coefficients are essentially square roots of Gauss sums. Evidence for Patterson’s conjecture is discussed in Bump and Hoffstein [6] and in Eckhardt and Patterson [10], where the conjecture is refined in light of numerical data. Partial proofs were given by Suzuki in [24] and [25]. Another set of conjectures relevant to the mysterious coefficients of n-th order theta functions were given by Bump and Hoffstein, who considered theta functions on the n-fold covers of GLr for arbitrary r. They are expressed as identities between Rankin-Selberg convolutions of generalized theta Series and Whittaker coefficients of Eisenstein Series on the metaplectic group, but they boil down to properties of the residues of Kubota Dirichlet Series, and their higher rank generalizations. See Bump and Hoffstein [6], Bump [4] and Hoffstein [12]. These conjectures are different from the Patterson conjecture, and there are other considerations which suggest that there may be further unproved relations beyond those described in the conjectures of Patterson and Bump and Hoffstein.