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

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(r\G 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