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  • Daniel Loughran (a1), Ramin Takloo-Bighash (a2) and Sho Tanimoto (a3)


We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over $\mathbb{Q}$ whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.



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1.Arthur, J., A trace formula for reductive groups I. Terms associated to classes in G (ℚ), Duke Math. J. 45 (1978), 911952.
2.Arthur, J., Eisenstein series and the trace formula, in Automorphic Forms, Representations and L-Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, pp. 253274 (American Mathematical Society, Providence, RI, 1979).
3.Batyrev, V. V. and Manin, Y. I., Sur le nombre des points rationnels de hauteur bornée des variétés algébriques, Math. Ann. 286 (1990), 2743.
4.Bloch, S. and Ogus, A., Gersten’s conjecture and the homology of schemes, Ann. Sci. Éc. Norm. Supér. (4) 7 (1975), 181201.
5.Brion, M., The total coordinate ring of a wonderful variety, J. Algebra 313(1) (2007), 6199.
6.Chambert-Loir, A. and Tschinkel, Y., Igusa integrals and volume asymptotics in analytic and adelic geometry, Confluentes Math. 2(3) (2010), 351429.
7.Colliot-Thélène, J.-L., Birational invariants, purity and the Gersten conjecture, in K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Proceedings of Symposia in Pure Mathematics, Volume 58, Part I, pp. 164 (AMS Summer Research Institute, Santa Barbara, 1995).
8.De Concini, C. and Procesi, C., Complete symmetric varieties, in Invariant Theory (Montecatini, 1982), Lecture Notes in Mathematics, Volume 996, pp. 144 (Springer, Berlin, 1983).
9.Delange, H., Généralisation du théorème de Ikehara, Ann. Sci. Éc. Norm. Supér. 71(3) (1954), 213242.
10.Evens, S. and Jones, B. F., On the wonderful compactification, Preprint, 2008, arXiv:0801.0456; Lecture Notes based on Lectures given at HKUST and Notre Dame.
11.Flath, D., Decomposition of representations into tensor products, in Automorphic Forms, Representations and L-functions, Proceedings of Symposia in Pure Mathematics, Volume 33, Part 1, pp. 179183 (1979).
12.Hassett, B., Tanimoto, S. and Tschinkel, Y., Balanced line bundles and equivariant compactifications of homogeneous spaces, Int. Math. Res. Not. IMRN 15 (2015), 63756410.
13.Hooley, C., On ternary quadratic forms that represent zero, Glasg. Math. J. 35(1) (1993), 1323.
14.Hooley, C., On ternary quadratic forms that represent zero. II, J. reine angew. Math. 602 (2007), 179225.
15.Iversen, B., Brauer group of a linear algebraic group, J. Algebra 42 (1976), 295301.
16.Jacquet, H. and Langlands, R. P., Automorphic Forms on GL (2), Lecture Notes in Mathematics, Volume 114 (Springer, Berlin-New York, 1970).
17.Landau, E., Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. der Math. und Physik (3) 13 (1908), 305312.
18.Langlands, R. P., Eisenstein series, in Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., pp. 235252 (American Mathematical Society, Boulder, CO, 1965).
19.Langlands, R. P., On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, Volume 544 (Springer, Berlin–New York, 1976).
20.Loughran, D., The number of varieties in a family which contain a rational point, J. Eur. Math. Soc (JEMS) 20(10) (2018), 25392588.
21.Loughran, D. and Smeets, A., Fibrations with few rational points, Geom. Funct. Anal. 26(5) (2016), 14491482.
22.Gorodnik, A., Maucourant, F. and Oh, H., Manin’s conjecture on rational points of bounded height and adelic mixing, Ann. Sci. Éc. Norm. Supér. 41 (2008), 4797.
23.Kausz, I., A modular compactification of the general linear group, Doc. Math. 5 (2000), 553594.
24.Malle, G. and Testerman, D., Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge Studies in Advanced Mathematics, Volume 133 (Cambridge University Press, Cambridge, 2011).
25.Milne, J. S., Étale Cohomology, Princeton Mathematical Series, Volume 33 (Princeton University Press, Princeton, NJ, 1980).
26.Moeglin, C. and Waldspurger, J.-L., Spectral Decomposition and Eisenstein Series. Une Paraphrase de l’Écriture, Cambridge Tracts in Mathematics, Volume 113 (Cambridge University Press, Cambridge, 1995).
27.Müller, W., The trace class conjecture in the theory of automorphic Forms. II, Geom. Funct. Anal. (GAFA) 8(2) (1998), 315355.
28.Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of Number Fields, 2nd edn, Grundlehren der Mathematischen Wissenschafte, Volume 323 (Springer, Berlin, 2008).
29.Oh, H., Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J. 113 (2002), 133192.
30.Ono, T., On the relative theory of Tamagawa numbers, Ann. of Math. (2) 82 (1965), 88111.
31.Peyre, E., Hauteurs et mesures de Tamagawa sur les variétiés de Fano, Duke Math. J. 79(1) (1995), 101218.
32.Platonov, V. and Rapinchuk, A., Algebraic Groups and Number Theory, Pure and Applied Mathematics, Volume 139 (Academic Press, Boston, MA, 1994).
33.Poonen, B., Rational points on varieties, Grad. Stud. Math. 186 (2017), xv+377.
34.Sansuc, J.-J., Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 1280.
35.Serre, J.-P., Spécialisation des éléments de Br 2(ℚ(T 1, …, T n)), C. R. Acad. Sci. Paris Sér. I Math. 311(7) (1990), 397402.
36.Shalika, J., Takloo-Bighash, R. and Tschinkel, Y., Rational points on compactifications of semi-simple groups of rank 1, in Progr. Math., Arithmetic of Higher-Dimensional Algebraic Varieties (Palo Alto, CA, 2002), Volume 226, pp. 205233 (Birkhauser, Boston, 2004).
37.Shalika, J., Takloo-Bighash, R. and Tschinkel, Y., Rational points on compactifications of semi-simple groups, J. Amer. Math. Soc. 20(4) (2007), 11351186.
38.Sofos, E., Serre’s problem on the density of isotropic fibres in conic bundles, Proc. Lond. Math. Soc. (3) 113 (2016), 128.
39.Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory (Cambridge University Press, Cambridge, 1995).
40.Warner, G., Harmonic Analysis on Semi-Simple Lie Groups I (Springer, New York–Heidelberg, 1972).
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  • Daniel Loughran (a1), Ramin Takloo-Bighash (a2) and Sho Tanimoto (a3)


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