A Selberg trace formula for hypercomplex analytic cusp forms

A breakthrough in developing a theory of hypercomplex analytic modular forms over Clifford algebras has been the proof of the existence of non-trivial cusp forms for important discrete arithmetic subgroups of the Ahlfors–Vahlen group. Hypercomplex analytic modular forms in turn also include Maaß forms associated to particular eigenvalues as special cases. In this paper we establish a Selberg trace formula for this new class of automorphic forms. In particular, we show that the dimension of the space of hypercomplex-analytic cusp forms is finite. Finally, we describe the space of Eisenstein series and give a dimension formula for the complete space of k-holomorphic Cliffordian modular forms.