Automorphic Forms and Zeta Functions : Proceedings of the Conference in Memory of Tsuneo Arakawa Rikkyo University, Japan 4-7 September 2004.
Title:
Automorphic Forms and Zeta Functions : Proceedings of the Conference in Memory of Tsuneo Arakawa Rikkyo University, Japan 4-7 September 2004.
Author:
Böcherer, Siegfried.
ISBN:
9789812774415
Personal Author:
Physical Description:
1 online resource (400 pages)
Contents:
CONTENTS -- Preface -- Tsuneo Arakawa and His Works -- 1 Tsuneo Arakawa (1949 - 2003) -- 2 Arakawa's works on Siegel and Jacobi modular forms -- 3 Arakawa's works on Selberg zeta functions -- 4 Arakawa's works on special values of zeta and L- functions -- List of Publications of Tsuneo Arakawa -- Estimate of the Dimensions of Hilbert Modular Forms by Means of Differential Operators -- 1 Definitions and notations -- 2 Estimate of the dimension -- 3 A differential operator of Rankin-Cohen-Ibukiyama type -- 4 Remark -- References -- Marsden-Weinstein Reduction Orbits and Representations of the Jacobi Group -- 1 Some General Remarks on the Orbit Method and Marsden-Weinstein Reduction -- 2 Discrete Series Representations of SL(2 R) and the Jacobi Group -- 3 Coadjoint Orbits of SL(2 R) and GJ -- 4 Marsden-Weinstein Reduction and Symplectic Volumes -- 5 Appendix: Explicit Expressions of Symplectic Forms for Elliptic Coadjoint Obits of GJ -- References -- On Eisenstein Series of Degree Two for Squarefree Levels and the Genus Version of the Basis Problem I -- 1 Introduction -- 2 Preliminaries -- 3 Cusps and Eisenstein series for rno(N) -- 4 Coset decompositions for r2o(N) -- 5 Unfolding I -- 6 Double cosets for r0(N) -- 7 Unfolding II -- 8 The basis problem for squarefree level -- References -- Double Zeta Values and Modular Forms -- 1 Introduction and main results -- 2 The formal double zeta space -- 3 Using the action of PGL2(Z) -- 4 Representing even double zeta values in terms of odd ones -- 5 Double zeta values and period polynomials -- 6 Double zeta values and modular forms -- 7 Double Eisenstein series -- Acknowledgements -- References -- Type Numbers and Linear Relations of Theta Series for Some General Orders of Quaternion Algebras -- 1 Introduction -- 2 Type numbers of split and non-split orders.
3 Construction of orders of level (q N) -- 4 Theta series -- 5 Examples for split orders of high power levels -- 6 Linear relations for split orders with T 0: split orders -- 9 Table of T(q N) with qN 0: non-split orders -- Acknowledgments -- References -- Skew-Holomorphic Jacobi Forms of Higher Degree -- 1 Introduction -- 2 Holomorphic Jacobi forms and skew-holomorphic Jacobi forms of higher degree -- 3 Siegel modular form of half-integral weight and generalized plus space -- 4 Siegel's formula -- 5 Klingen type Eisenstein series -- Acknowledgments -- References -- A Hermitian Analog of the Schottky Form -- 1 Introduction -- 2 Hermitian modular forms -- 3 Even unimodular Gaussian lattices -- Acknowledgement -- References -- The Siegel Series and Spherical Functions on 0(2n)/(0(n) x 0(n)) -- 1 Introduction -- 2 An integral representation of the Siegel series -- 3 Spherical functions on 0(Hn)/(0(T) X 0(T)) and the relation to the Siegel series -- 4 Functional equation of the Siegel series -- 5 Degenerate principal series representation for 0(Hn) -- 6 Proof of the functional equation of spherical functions -- References -- Koecher-Maafi Series for Real Analytic Siegel Eisenstein Series -- 1 Introduction -- 2 Siegel's formula -- 3 Proof of main theorems -- 4 Functional equations and special values of Koecher-Maafi series -- References -- A Short History on Investigation of the Special Values of Zeta and L-Functions of Totally Real Number Fields -- Introduction -- 1 Before 1950: Hecke Siegel and others -- 2 From 1951 untill 1969 -- 3 From 70's to 80's -- 4 After 1990 cocycles on GL(n Q) -- 5 Further problems -- References.
Genus Theta Series Hecke Operators and the Basis Problem for Eisenstein Series -- 1 Introduction -- 2 Preliminaries -- 3 Eisenstein series and theta series -- 4 Action of T(p) and local densities -- 5 Spaces of genus theta series for odd prime level -- 6 Connection with Kudla's matching principle -- References -- The Quadratic Mean of Automorphic L-Functions -- 1 Introduction -- 2 Notations and preliminaries -- 3 Some Lemmas -- 4 Proof of the Theorems -- 5 Examples -- Acknowledgment -- References -- Inner Product Formula for Kudla Lift -- 0 Introduction -- 1 Main results -- 2 Metaplectic representations -- 3 Kudla lift -- 4 Inner product formula -- 5 The basic identity -- 6 Local spherical function -- 7 Local zeta integral -- 8 Local calculation (I) -- 9 Local calculation (II) -- 10 Local calculation (III) -- References -- On Certain Automorphic Forms of Sp(1 q) (Arakawa's Results and Recent Progress) -- 0 Introduction -- 1 Structure of Sp(1 q) -- 2 Reviews on automorphic forms of Sp(1 q) introduced by Arakawa -- 3 Dimension formula of Ao(r\G wK) -- 4 Theta lifting from elliptic cusp forms to automorphic forms on Sp(1 q) -- Acknowledgments -- References -- On Modular Forms for the Paramodular Groups -- 1 Introduction -- 2 Definitions -- 3 Linear independence at different levels -- 4 The level raising operators -- 5 Oldforms and newforms -- 6 Saito-Kurokawa liftings -- 7 Two theorems -- Appendix. Paramodular vectors in Iwahori-spherical representations -- References -- SL(2 Z)-Invariant Spaces Spanned by Modular Units -- 1 Introduction -- 2 Statement of results -- 3 The group of units generated by the [r]l -- 4 Properties of modular sets -- 5 Computing modular sets -- 6 The [r]l in terms of l-division values of the Weierstrass o-function.
7 Appendix: The Weierstrass o-function as Jacobi form -- Acknowledgments -- References.
Abstract:
This volume contains a valuable collection of articles presented at a conference on Automorphic Forms and Zeta Functions in memory of Tsuneo Arakawa, an eminent researcher in modular forms in several variables and zeta functions. The book begins with a review of his works, followed by 16 articles by experts in the fields including H Aoki, R Berndt, K Hashimoto, S Hayashida, Y Hironaka, H Katsurada, W Kohnen, A Krieg, A Murase, H Narita, T Oda, B Roberts, R Schmidt, R Schulze-Pillot, N Skoruppa, T Sugano, and D Zagier. A variety of topics in the theory of modular forms and zeta functions are covered: Theta series and the basis problems, Jacobi forms, automorphic forms on Sp(1, q) , double zeta functions, special values of zeta and L -functions, many of which are closely related to Arakawa's works. This collection of papers illustrates Arakawa's contributions and the current trends in modular forms in several variables and related zeta functions. Contents: Tsuneo Arakawa and His Works; Estimate of the Dimensions of Hilbert Modular Forms by Means of Differential Operators (H Aoki); Marsden-Weinstein Reduction, Orbits and Representations of the Jacobi Group (R Berndt); On Eisenstein Series of Degree Two for Squarefree Levels and the Genus Version of the Basis Problem I (S Böcherer); Double Zeta Values and Modular Forms (H Gangl et al.); Type Numbers and Linear Relations of Theta Series for Some General Orders of Quaternion Algebras (K-I Hashimoto); Skew-Holomorphic Jacobi Forms of Higher Degree (S Hayashida); A Hermitian Analog of the Schottky Form (M Hentschel & A Krieg); The Siegel Series and Spherical Functions on O (2 n) / (O (n) x O (n) ) (Y Hironaka & F Sato); Koecher-Maaß Series for Real Analytic Siegel Eisenstein Series (T Ibukiyama & H Katsurada); A Short History on Investigation of the Special Values of Zeta and L -Functions of Totally Real
Number Fields (T Ishii & T Oda); Genus Theta Series, Hecke Operators and the Basis Problem for Eisenstein Series (H Katsurada & R Schulze-Pillot); The Quadratic Mean of Automorphic L -Functions (W Kohnen et al.); Inner Product Formula for Kudla Lift (A Murase & T Sugano); On Certain Automorphic Forms of Sp (1, q ) (Arakawa's Results and Recent Progress) (H-A Narita); On Modular Forms for the Paramodular Groups (B Roberts & R Schmidt); SL(2,Z)-Invariant Spaces Spanned by Modular Units (N-P Skoruppa & W Eholzer). Readership: Researchers and graduate students in number theory or representation theory as well as in mathematical physics or combinatorics.
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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