Limiting distributions of theta series on Siegel half-spaces
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- by F. Götze and M. Gordin
- St. Petersburg Math. J. 15 (2004), 81-102
- DOI: https://doi.org/10.1090/S1061-0022-03-00803-3
- Published electronically: December 31, 2003
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Abstract:
Let $m \ge 1$ be an integer. For any $Z$ in the Siegel upper half-space we consider the multivariate theta series \begin{equation*}\Theta (Z)= \sum _{{\overline {n}} \in \mathbb {Z}^{m}} \exp (\pi i {}^{t} {\overline {n}} Z {\overline {n}}).\end{equation*} The function $\Theta$ is invariant with respect to every substitution $Z\longmapsto Z + P$, where $P$ is a real symmetric matrix with integral entries and even diagonal. Therefore, for any real matrix $Y > 0$ the function $\Theta _{Y} ( \cdot ) = (\det Y)^{1/4} \Theta (\cdot +iY)$ can be viewed as a complex-valued random variable on the torus $\mathbb {T}^{m(m+1)/2}$ with the Haar probability measure. It is proved that the weak limit of the distribution of $\Theta _{\tau Y}$ as $\tau \to 0$ exists and does not depend on the choice of $Y$. This theorem is an extension of known results for $m=1$ to higher dimension. Also, the rotational invariance of the limiting distribution is established. The proof of the main theorem makes use of Dani–Margulis’ and Ratner’s results on dynamics of unipotent flows.References
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Bibliographic Information
- F. Götze
- Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
- Email: goetze@mathematik.uni-bielefeld.de
- M. Gordin
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191011, Russia
- Email: gordin@pdmi.ras.ru
- Received by editor(s): September 2, 2002
- Published electronically: December 31, 2003
- Additional Notes: Supported in part by the DFG-Forschergruppe FOR 399/1-1.
M. Gordin was also partially supported by RFBR (grant no. 02.01-00265) and by Sc. Schools grant no. 2258.2003.1. He was a guest of SFB-343 and the Department of Mathematics at the University of Bielefeld while the major part of this paper was prepared. - © Copyright 2003 American Mathematical Society
- Journal: St. Petersburg Math. J. 15 (2004), 81-102
- MSC (2000): Primary 11Fxx, 37D30
- DOI: https://doi.org/10.1090/S1061-0022-03-00803-3
- MathSciNet review: 1979719