Generalized Analytic Automorphic Forms in Hypercomplex Space
S. Krausshar

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'The aim of this book is to provide a first comprehensive overview of the basic theory of hypercomplex-analytic automorphic forms and functions for arithmetic subgroups of the Vahlen group in higher dimensional spaces. It gives a summary on the research results obtained over the last five years and establishes a new field within the theory of functions of hypercomplex variables and within analytic number theory.' '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 Poincare series are constructed within this framework and a detailed description of their analytic and number theoretical properties is provided. In particular, explicit relationships to higher dimensional vector valued variants of the Riemann zeta function and Dirichlet 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 conformally flat manifolds are also included.' The book is directed to researchers as well as to graduate and postgraduate students with interest in the fields of the theory of generalized analytic functions in higher dimensional spaces, analytic number theory, function spaces and boundary value problems of partial differential equations of conformally flat manifolds, and some closely related fields in physics, such as instanton theory and quantum gravity.