Limiting distributions of theta series on Siegel halfspaces
Authors:
F. Götze and M. Gordin
Translated by:
Original publication:
Algebra i Analiz, tom 15 (2003), nomer 1.
Journal:
St. Petersburg Math. J. 15 (2004), 81102
MSC (2000):
Primary 11Fxx, 37D30
Published electronically:
December 31, 2003
MathSciNet review:
1979719
Fulltext PDF Free Access
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Abstract: Let be an integer. For any in the Siegel upper halfspace we consider the multivariate theta series
The function is invariant with respect to every substitution , where is a real symmetric matrix with integral entries and even diagonal. Therefore, for any real matrix the function can be viewed as a complexvalued random variable on the torus with the Haar probability measure. It is proved that the weak limit of the distribution of as exists and does not depend on the choice of . This theorem is an extension of known results for to higher dimension. Also, the rotational invariance of the limiting distribution is established. The proof of the main theorem makes use of DaniMargulis' and Ratner's results on dynamics of unipotent flows.
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 G. H. Hardy and J. E. Littlewood, Some problems of Diophantine approximation. II, Acta Math. 37 (1914), 193239.
 [HL2]
 , A further note on the trigonometrical series associated with the elliptic thetafunctions, Proc. Cambridge Philos. Soc. 21 (1922), 15.
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 H. Klingen, Introductory lectures on Siegel modular forms, Cambridge Stud. Adv. Math., vol. 20, Cambridge Univ. Press, Cambridge, 1990. MR 91a:11021
 [LV]
 G. Lion and M. Vergne, The Weil representation, Maslov index and theta series, Progr. Math., vol. 6, Birkhäuser, Boston, MA, 1980. MR 81j:58075
 [Ma1]
 J. Marklof, Theta sums, Eisenstein series, and the semiclassical dynamics of a precessing spin, Emerging Applications of Number Theory (Minneapolis, MN, 1996), IMA Vol. Math. Appl., vol. 109, Springer, New York, 1999, pp. 405450. MR 2000k:81094
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 D. Mumford, Tata lectures on theta. I, Progr. Math., vol. 28, Birkhäuser, Boston, MA, 1983. MR 85h:14026
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Additional Information
F. Götze
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D33501 Bielefeld, Germany
Email:
goetze@mathematik.unibielefeld.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
DOI:
http://dx.doi.org/10.1090/S1061002203008033
PII:
S 10610022(03)008033
Keywords:
Theta series,
Siegel's halfspace,
convergence in distribution,
closed horospheres,
unipotent flows
Received by editor(s):
September 2, 2002
Published electronically:
December 31, 2003
Additional Notes:
Supported in part by the DFGForschergruppe FOR 399/11.
M. Gordin was also partially supported by RFBR (grant no. 02.0100265) and by Sc. Schools grant no. 2258.2003.1. He was a guest of SFB343 and the Department of Mathematics at the University of Bielefeld while the major part of this paper was prepared.
Article copyright:
© Copyright 2003
American Mathematical Society
