This book offers basic theory on hypercomplex-analytic automorphic forms and functions for arithmetic subgroups of the Vahlen group in higher dimensional spaces. Vector- and Clifford algebra-valued Eisenstein and Poincaré series are constructed within this framework and a detailed description of their analytic and number theoretical properties is provided. It establishes explicit relationships to generalized variants of the Riemann zeta function and Dirichlet L-series, and introduces hypercomplex multiplication of lattices.
Author(s): Rolf S. Krausshar
Series: Frontiers in Mathematics
Edition: 1
Publisher: Birkhäuser Basel
Year: 2004
Language: English
Pages: 182
Contents......Page 8
Introduction......Page 10
1.1 Hypercomplex numbers and Cli.ord Algebras......Page 17
1.2 Vahlen groups and arithmetic subgroups......Page 21
1.3 Di.erentiability, conformality and analyticity in hypercomplex spaces......Page 25
1.4 Basic theorems of Cli.ord analysis......Page 33
1.5 Orders of isolated a-points, an argument principle and Rouch´e’s theorem......Page 44
1.6 The generalized negative power functions......Page 51
2.1 Multiperiodic Mittag-Le.er series......Page 65
2.2 Some results on the zeroes of the generalized cotangent and tangent......Page 74
2.3 Liouville type theorems for generalized elliptic functions......Page 76
2.4 Series expansions, divisor sums and Dirichlet series......Page 81
2.5 The integer multiplication of the Cli.ord-analytic Eisenstein series......Page 88
2.6 Characterization theorems......Page 95
2.7 Lattices with hypercomplex multiplication......Page 99
2.8 Bergman kernels of rectangular domains......Page 113
2.9 Szego kernels of strip domains......Page 124
2.10 Boundary value problems on conformally .at cylinders and tori......Page 126
2.11 Order theory and argument principles on cylinders and tori......Page 129
3 Clifford-analytic Modular Forms......Page 133
3.1 Rotation and translation invariant Eisenstein series......Page 134
3.2 Clifford-analytic modular forms in one hypercomplex variable......Page 140
3.3 Clifford-analytic modular forms in two and several hypercomplex variables......Page 147
3.4 Some remarks on Cli.ord-analytic modular forms in real and complex Minkowski spaces......Page 160
3.5 Some Perspectives......Page 163
Bibliography......Page 167
List of Symbols......Page 179
Index......Page 181
