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Hermitian modular forms congruent to 1 modulo p

Authors
  • Hentschel, Michael1
  • Nebe, Gabriele2
  • 1 RWTH Aachen University, Lehrstuhl A für Mathematik, Aachen, 52056, Germany , Aachen (Germany)
  • 2 RWTH Aachen University, Lehrstuhl D für Mathematik, Aachen, 52056, Germany , Aachen (Germany)
Type
Published Article
Journal
Archiv der Mathematik
Publisher
Birkhäuser-Verlag
Publication Date
Mar 27, 2009
Volume
92
Issue
3
Pages
251–256
Identifiers
DOI: 10.1007/s00013-008-3072-3
Source
Springer Nature
Keywords
License
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Abstract

For any natural number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document} and any prime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \equiv 1$$\end{document} (mod 4) not dividing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document} there is a Hermitian modular form of arbitrary genus n over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L := {\mathbb{Q}}[\sqrt{-\ell}]$$\end{document} that is congruent to 1 modulo p which is a Hermitian theta series of an OL-lattice of rank p − 1 admitting a fixed point free automorphism of order p. It is shown that also for non-free lattices such theta series are modular forms.

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