Geodesic
Limiting distributions of theta series on Siegel half-spaces
Algebra i analiz, Tome 15 (2003) no. 1, pp. 118-147.

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Let $m>1$ be an integer. For any $Z$ from the Siegel upper half-space we consider the multivariate theta series $$ \Theta(Z)=\sum_{\bar n\in\mathbb Z^m}\exp(\pi i^t\bar n Z\bar n). $$ 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 $\Theta_Y(\cdot)=(\det Y)^{1/4}\Theta(\cdot+iY)$ may be viewed as a complex-valued random variable on the torus $\mathbb T^{m(m+1)/2}$ with the probability Haar measure. We prove that there exists a weak limit of the distribution of $\Theta_{\tau Y}$ as $\tau\to0$, and this limit does not depend on the choice of $Y$. This theorem is an extension of known results for $m=1$ to higher dimension. We also establish the rotational invariance of the limiting distribution. The proof of the main theorem makes use of Dani–Margulis' and Ratner's results on dynamics of unipotent flows.
Keywords: theta series, Siegel's half-space, convergence in distribution, closed horospheres, unipotent flows. 
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F. Götze; M. Gordin. Limiting distributions of theta series on Siegel half-spaces. Algebra i analiz, Tome 15 (2003) no. 1, pp. 118-147. http://geodesic.mathdoc.fr/item/AA_2003_15_1_a3/

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