Generalized Analytic Automorphic Forms in Hypercomplex Spaces

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.

• Clifford-Analysis
• Eisenstein-Reihen
• Modulformen
• Potentialtheorie
• Zahlentheorie
• spezielle Funktionen
• arithmetic
• calculus
• Hilbert space
• index theory
• number theory
• Potential theory
• Riemann zeta function
• zeta function

From the reviews:

“Its remarkable feature is a masterful combination of deep and sophisticated results with very accessible form of presentation. I am sure the book will have numerous and far-reaching consequences and repercussions.”(ZENTRALBLATT MATH)

• Department of Mathematical Analysis, Ghent University, Ghent, Belgium

  Rolf Sören Krausshar

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