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 Poincar 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.
General
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