Wei, Dasheng; Xu, Fei
(2016):
Counting integral points in certain homogeneous spaces.
In: Journal of Algebra, Vol. 448: pp. 350398

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Abstract
The leading term of asymptotic formula of the number of integral points in noncompact symmetric homogeneous spaces of semisimple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the Brauer Manin obstruction. The similar result is also true for homogeneous spaces of reductive groups with some restriction. As application, we will give the explicit asymptotic formulae of the number of integral points of certain norm equations and prove the leading term of asymptotic formula of the number of integral matrices with a fixed irreducible characteristic polynomial over Z studied by Eskin Mozes Shah is equal to the product of the number of local integral solutions over all primes although the density function defined by Borovoi and Rudnick is not trivial in general. We also answer a question raised by Borovoi and Rudnick for comparing the number of integral symmetric matrices with the given determinant with the product of local densities. (C) 2015 ELSEVIER. All rights reserved.