A Selberg trace formula for hypercomplex analytic cusp forms. (English) Zbl 1380.11074
Summary: 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.
MSC:
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 11F03 | Modular and automorphic functions |
| 11F55 | Other groups and their modular and automorphic forms (several variables) |
| 30G35 | Functions of hypercomplex variables and generalized variables |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
Selberg trace formula; dimension formulas for modular forms; hypercomplex cusp forms; hyperbolic harmonic functions; Dirac type operators; Clifford algebrasReferences:
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