Automorphic Forms and Zeta Functions: Proceedings of the Conference in Memory of Tsuneo Arakawa Rikkyo University

Siegfried Bocherer, Tomoyoshi Ibukiyama, Masanobu Kaneko, Fumihiro Sato9789812566324, 981-256-632-5

“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.

---

Table of contents :
CONTENTS……Page 10
Preface……Page 8
1 Tsuneo Arakawa (1949 – 2003)……Page 12
2 Arakawa’s works on Siegel and Jacobi modular forms……Page 14
3 Arakawa’s works on Selberg zeta functions……Page 16
4 Arakawa’s works on special values of zeta and L- functions……Page 27
List of Publications of Tsuneo Arakawa……Page 29
1 Definitions and notations……Page 31
2 Estimate of the dimension……Page 34
3 A differential operator of Rankin-Cohen-Ibukiyama type……Page 37
4 Remark……Page 38
References……Page 39
Marsden-Weinstein Reduction Orbits and Representations of the Jacobi Group……Page 40
1 Some General Remarks on the Orbit Method and Marsden-Weinstein Reduction……Page 41
2 Discrete Series Representations of SL(2 R) and the Jacobi Group……Page 42
3 Coadjoint Orbits of SL(2 R) and GJ……Page 44
4 Marsden-Weinstein Reduction and Symplectic Volumes……Page 46
5 Appendix: Explicit Expressions of Symplectic Forms for Elliptic Coadjoint Obits of GJ……Page 49
References……Page 53
1 Introduction……Page 54
2 Preliminaries……Page 57
3 Cusps and Eisenstein series for rno(N)……Page 58
4 Coset decompositions for r2o(N)……Page 62
5 Unfolding I……Page 66
6 Double cosets for r0(N)……Page 68
7 Unfolding II……Page 70
8 The basis problem for squarefree level……Page 72
References……Page 80
1 Introduction and main results……Page 82
2 The formal double zeta space……Page 90
3 Using the action of PGL2(Z)……Page 95
4 Representing even double zeta values in terms of odd ones……Page 97
5 Double zeta values and period polynomials……Page 99
6 Double zeta values and modular forms……Page 105
7 Double Eisenstein series……Page 110
References……Page 116
1 Introduction……Page 118
2 Type numbers of split and non-split orders……Page 119
3 Construction of orders of level (q N)……Page 121
4 Theta series……Page 124
5 Examples for split orders of high power levels……Page 128
6 Linear relations for split orders with T < 12……Page 130
7 Linear relations for non-split orders with T < 20……Page 132
8 Table of T(q N) with qN < 1000 G > 0: split orders……Page 136
9 Table of T(q N) with qN < 5000 G > 0: non-split orders……Page 138
References……Page 139
1 Introduction……Page 141
2 Holomorphic Jacobi forms and skew-holomorphic Jacobi forms of higher degree……Page 142
3 Siegel modular form of half-integral weight and generalized plus space……Page 143
4 Siegel’s formula……Page 146
5 Klingen type Eisenstein series……Page 147
References……Page 149
1 Introduction……Page 151
2 Hermitian modular forms……Page 152
3 Even unimodular Gaussian lattices……Page 154
References……Page 159
1 Introduction……Page 161
2 An integral representation of the Siegel series……Page 163
3 Spherical functions on 0(Hn)/(0(T) X 0(T)) and the relation to the Siegel series……Page 167
4 Functional equation of the Siegel series……Page 168
5 Degenerate principal series representation for 0(Hn)……Page 170
6 Proof of the functional equation of spherical functions……Page 174
References……Page 180
1 Introduction……Page 181
2 Siegel’s formula……Page 186
3 Proof of main theorems……Page 190
4 Functional equations and special values of Koecher-Maafi series……Page 205
References……Page 207
Introduction……Page 209
1 Before 1950: Hecke Siegel and others……Page 211
2 From 1951 untill 1969……Page 220
3 From 70’s to 80’s……Page 229
4 After 1990 cocycles on GL(n Q)……Page 234
References……Page 240
1 Introduction……Page 245
2 Preliminaries……Page 246
3 Eisenstein series and theta series……Page 248
4 Action of T(p) and local densities……Page 249
5 Spaces of genus theta series for odd prime level……Page 253
6 Connection with Kudla’s matching principle……Page 268
References……Page 272
1 Introduction……Page 273
3 Some Lemmas……Page 276
4 Proof of the Theorems……Page 278
5 Examples……Page 283
References……Page 288
0 Introduction……Page 291
1 Main results……Page 292
2 Metaplectic representations……Page 299
3 Kudla lift……Page 301
4 Inner product formula……Page 303
5 The basic identity……Page 306
6 Local spherical function……Page 308
7 Local zeta integral……Page 310
8 Local calculation (I)……Page 312
9 Local calculation (II)……Page 318
10 Local calculation (III)……Page 320
References……Page 323
0 Introduction……Page 325
1 Structure of Sp(1 q)……Page 326
2 Reviews on automorphic forms of Sp(1 q) introduced by Arakawa……Page 328
3 Dimension formula of Ao(rG wK)……Page 331
4 Theta lifting from elliptic cusp forms to automorphic forms on Sp(1 q)……Page 335
References……Page 343
1 Introduction……Page 345
2 Definitions……Page 347
3 Linear independence at different levels……Page 351
4 The level raising operators……Page 353
5 Oldforms and newforms……Page 358
6 Saito-Kurokawa liftings……Page 362
7 Two theorems……Page 370
Appendix. Paramodular vectors in Iwahori-spherical representations……Page 372
References……Page 374
1 Introduction……Page 376
2 Statement of results……Page 378
3 The group of units generated by the [r]l……Page 383
4 Properties of modular sets……Page 387
5 Computing modular sets……Page 391
6 The [r]l in terms of l-division values of the Weierstrass o-function……Page 393
7 Appendix: The Weierstrass o-function as Jacobi form……Page 396
References……Page 398