Generalized Analytic Automorphic Forms in Hypercomplex SpacesSpringer Science & Business Media, 2004 M02 23 - 168 páginas This book describes the basic theory of hypercomplex-analytic automorphic forms and functions for arithmetic subgroups of the Vahlen group in higher dimensional spaces. Hypercomplex analyticity generalizes the concept of complex analyticity in the sense of considering null-solutions to higher dimensional Cauchy-Riemann type systems. 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. In particular, explicit relationships to generalized variants of the Riemann zeta function and Dirichlet L-series are established and a concept of hypercomplex multiplication of lattices is introduced. Applications to the theory of Hilbert spaces with reproducing kernels, to partial differential equations and index theory on some conformal manifolds are also described. |
Contenido
Function theory in hyper complex spaces | xiii |
12 Vahlen groups and arithmetic subgroups | 1 |
13 Dijfferentiability conformality and analyticity in hypercomplex spaces | 5 |
14 Basic theorems of Clifford analysis | 11 |
15 Orders of isolated apoints an argument principle and Rouches theorem | 22 |
16 The generalized negative power functions | 29 |
Cliffordanalytic Eisenstein series associated to translation groups | 43 |
22 Some results on the zeroes of the generalized cotangent and tangent | 52 |
29 Szego kernels of strip domains | 102 |
210 Boundary value problems on conformally flat cylinders and tori | 104 |
211 Order theory and argument principles on cylinders and tori | 107 |
Cliffordanalytic Modular Forms | 111 |
31 Rotation and translation invariant Eisenstein series | 112 |
32 Cliffordanalytic modular forms in one hypercomplex variable | 118 |
33 Cliffordanalytic modular forms in two and several hypercomplex variables | 125 |
34 Some remarks on Cliffordanalytic modular forms in real and complex Minkowski spaces | 138 |
23 Liouville type theorems for generalized elliptic functions | 54 |
24 Series expansions divisor sums and Dirichlet series | 59 |
25 The integer multiplication of the Cliffordanalytic Eisenstein series | 66 |
26 Characterization theorems | 73 |
27 Lattices with hypercomplex multiplication | 77 |
28 Bergman kernels of rectangular domains | 91 |
35 Some Perspectives | 141 |
Bibliography | 145 |
| 157 | |
| 159 | |
Otras ediciones - Ver todas
Generalized Analytic Automorphic Forms in Hypercomplex Spaces Rolf S. Krausshar Sin vista previa disponible - 2014 |
Generalized Analytic Automorphic Forms in Hypercomplex Spaces Rolf S. Krausshar Sin vista previa disponible - 2005 |
Términos y frases comunes
analogues arbitrary argument principle associated automorphic forms Bergman kernel Cauchy Cauchy kernel classical complex Clifford analysis Clifford-analytic Clifford-valued Clok complex variables complexified conformally flat context convergence defined Definition denote domain Eisenstein series elliptic functions Euclidean spaces explicit finite Fueter function f function series H+(R higher dimensional Hilbert modular forms hypercomplex hypercomplex variable invariant isolated a-points k₁ kernel functions lattice in Ak+1 left monogenic left quaternionic Lemma manifolds matrix Minkowski type spaces Möbius transformations modular group monogenic functions multi-index obtains octonions p-dimensional lattice paravector particular positive integer primitive periods proof provides qm² quaternionic quaternionic differentiable real Euclidean spaces representation formula Riemann zeta s-monogenic s-monogenic Eisenstein series satisfy singularities space Ak+1 subgroups Szegö kernel Theorem translation group translative Eisenstein series upper half-space Vahlen group vector variables Weierstraß function Σ Σ
Pasajes populares
Página vii - Maafi proceeded to introduce in [117] another type of automorphic forms in a hypercomplex variable. The class of automorphic forms introduced by him consists of complex-valued non-analytic eigenfunctions to the higher dimensional hyperbolic Laplace-Beltrami operator.
Página vii - Fueter constructed in his paper [54] automorphic forms and functions related to Picard's group in a three-dimensional hypercomplex variable. However, the functions treated in [54] and also in his follow-up paper [55] , are not in kernels of higher dimensional Cauchy-Riemann type operators.
Página viii - In the previous five years, we started then to fill this gap step by step and to develop systematically a theory of monogenic and polymonogenic hypercomplexvalued...