A theory of modular forms in Clifford analysis, their applications and perspectives

This chapter is devoted to monogenic, harmonic and polymonogenic Clifford-valued automorphic forms that are related to discrete subgroups of Vahlen groups acting on half-spaces of real and complex Minkowski type spaces. In particular, Eisenstein and Poincare type series are constructed within this framework. We discuss their basic properties and provide a short overview about their range of applications to several areas from pure and applied mathematics, as for example, to number theory, to functional analysis and to order theory and pde's on manifolds.