Advanced search
1 file | 365.22 KB

A new class of hypercomplex analytic cusp forms

Author
Organization
Abstract
In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator D Delta(k/2) for some even k is an element of Z. They will be called k-holomorphic Cliffordian automorphic forms. k-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions also have the property that they are solutions to the homogeneous and inhomogeneous Weinstein equations. This function class includes the set of k-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing k-hypermonogenic cusp forms for k not equal 0, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms, we turn to the construction of Poincare series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of k-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of k-hypermonogenic Eisenstein series and of k-holomorphic Cliffordian cusp forms.
Keywords
CONGRUENCE SUBGROUPS, HYPERBOLIC SPACE, EISENSTEIN-SERIES, AUTOMORPHIC-FORMS, CLIFFORD ANALYSIS, VAHLEN GROUP, MANIFOLDS, OPERATORS, DOMAINS, Hypercomplex cusp forms, Poincare series, hyperbolic harmonic functions, Maass wave forms, Dirac type operators, Clifford algebras

Downloads

  • (...).pdf
    • full text
    • |
    • UGent only
    • |
    • PDF
    • |
    • 365.22 KB

Citation

Please use this url to cite or link to this publication:

Chicago
Constales, Denis, D Grob, and RS Krausshar. 2013. “A New Class of Hypercomplex Analytic Cusp Forms.” Transactions of the American Mathematical Society 365 (2): 811–835.
APA
Constales, D., Grob, D., & Krausshar, R. (2013). A new class of hypercomplex analytic cusp forms. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 365(2), 811–835.
Vancouver
1.
Constales D, Grob D, Krausshar R. A new class of hypercomplex analytic cusp forms. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. 2013;365(2):811–35.
MLA
Constales, Denis, D Grob, and RS Krausshar. “A New Class of Hypercomplex Analytic Cusp Forms.” TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 365.2 (2013): 811–835. Print.
@article{4129350,
  abstract     = {In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator D Delta(k/2) for some even k is an element of Z. They will be called k-holomorphic Cliffordian automorphic forms. k-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions also have the property that they are solutions to the homogeneous and inhomogeneous Weinstein equations. This function class includes the set of k-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing k-hypermonogenic cusp forms for k not equal 0, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms, we turn to the construction of Poincare series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of k-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of k-hypermonogenic Eisenstein series and of k-holomorphic Cliffordian cusp forms.},
  author       = {Constales, Denis and Grob, D and Krausshar, RS},
  issn         = {0002-9947},
  journal      = {TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY},
  keywords     = {CONGRUENCE SUBGROUPS,HYPERBOLIC SPACE,EISENSTEIN-SERIES,AUTOMORPHIC-FORMS,CLIFFORD ANALYSIS,VAHLEN GROUP,MANIFOLDS,OPERATORS,DOMAINS,Hypercomplex cusp forms,Poincare series,hyperbolic harmonic functions,Maass wave forms,Dirac type operators,Clifford algebras},
  language     = {eng},
  number       = {2},
  pages        = {811--835},
  title        = {A new class of hypercomplex analytic cusp forms},
  url          = {http://dx.doi.org/10.1090/S0002-9947-2012-05613-3},
  volume       = {365},
  year         = {2013},
}

Altmetric
View in Altmetric
Web of Science
Times cited: