Counting integral points in certain homogeneous spaces

JOURNAL OF ALGEBRA

卷:448

Article

Wei, Dasheng^1,2  Xu, Fei³ 

[1] Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China

[2] Univ Munich, Math Inst, Theresienstr 39, D-80333 Munich, Germany

[3] Capital Normal Univ, Sch Math Sci, Beijing 100048, Peoples R China

关键词: Integral point;    Homogeneous space;    Tamagawa measure;   

DOI  :  10.1016/j.jalgebra.2015.09.043

来源: Elsevier

【 摘 要 】

The leading term of asymptotic formula of the number of integral points in non-compact symmetric homogeneous spaces of semi-simple 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 Inc. All rights reserved.

【 授权许可】

Free