Automorphoric forms and zeta functions : proceedings of the conference in memory of Tsuneo Arakawa, Rikkyo University, Japan, Tokyo, 4-7 September 2004

AUTOMORPHIC FORMS AND ZETA FUNCTIONS Proceedings of the Conference in Memory of Tsuneo Arakawa Rikkyo University, Japan 4 - 7 September 2004 edited by Siegfried Bocherer (Universitat Mannheim, Germany), Tomoyoshi Ibukiyama (Osaka University, Japan), Masanobu Kaneko (Kyushu University, Japan) & Fumihiro Sato (Rikkyo University, Japan) 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 Preface -- Tsuneo Arakawa and his works -- Estimate of the dimensions of Hilbert modular forms by means of differential operators / Hiroki Aoki -- Marsden-Weinstein reduction, orbits and representations of the Jacobi group / Rolf Berndt -- On Eisenstein series of degree two for squarefree levels and the genus version of the basis problem I / Siegfried Böcherer -- Double zeta values and modular forms / Herbert Gangl, Masanobu Kaneko and Don Zagier -- Type numbers and linear relations of theta series for some general orders of quaternion algebras / Ki-Ichiro Hashimoto -- Skew-holomorphic Jacobi forms of higher degree / Shuichi Hayashida -- A hermitian analog of the Schottky form / M. Hentschel and A. Krieg -- The Siegel series and spherical functions on O(2n)/(O(n) x O(n)) / Yumiko Hironaka and Fumihiro Sato -- Koecher-Maa[symbol] series for real analytic Siegel Eisenstein series / Tomoyoshi Ibukiyama and Hidenori Katsurada -- A short history on investigation of the special values of zeta and L-functions of totally real number fields / Taku Ishii and Takayuki Oda -- Genus theta series, Hecke operators and the basis problem for Eisenstein series / Hidenori Katsurada and Rainer Schulze-Pillot -- The quadratic mean of automorphic L-functions / W. Kohnen, A. Sankaranarayanan and J. Sengupta -- Inner product formula for Kudla lift / Atsushi Murase and Takashi Sugano -- On certain automorphic forms of Sp(1,q) (Arakawa's results and recent progress) / Hiro-Aki Narita -- On modular forms for the paramodular groups / Brooks Roberts and Ralf Schmidt -- SL(2, Z)- invariant spaces spanned by modular units / Nils-Peter Skoruppa and Wolfgang Eholzer 5 Klingen type Eisenstein seriesAcknowledgments; 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 1 Some General Remarks on the Orbit Method and Marsden-Weinstein Reduction2 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) Preface; CONTENTS; 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 7 Unfolding II8 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

this Volume Contains A 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. This Collection Of Papers Illustrates Arakawa's Contributions And The Current Trends In Modular Forms In Several Variables And Related Zeta Functions.