Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-27T12:59:20.130Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrepancy of rational points in simple algebraic groups

Part of: Ergodic theory Diophantine approximation, transcendental number theory Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  13 March 2024

Alexander Gorodnik  and
Amos Nevo
Alexander Gorodnik
Affiliation:
Institute für Mathematik, Universität Zürich, CH-8057 Zürich, Switzerland alexander.gorodnik@math.uzh.ch
Amos Nevo
Affiliation:
Faculty of Mathematics Technion, Israel Institute of Technology, 32000 Haifa, Israel amosnevo6@gmail.com and Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

Keywords

semisimple groupsarithmetic lattice subgroupsDiophantine inequalitiesautomorphic representations

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 37A17: Homogeneous flows 22E46: Semisimple Lie groups and their representations 11J83: Metric theory
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 4 , April 2024 , pp. 836 - 877
Check for updates
Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benoist, Y. and Oh, H., Effective equidistribution of S-integral points on symmetric varieties, Ann. Inst. Fourier (Grenoble) 62 (2012), 1889–1942.10.5802/aif.2738CrossRefGoogle Scholar
Burger, M., Li, J.-S. and Sarnak, P., Ramanujan duals and automorphic spectrum, Bull. Amer. Math. Soc. (N.S.) 26 (1992), 253–257.10.1090/S0273-0979-1992-00267-7CrossRefGoogle Scholar
Burger, M. and Sarnak, P., Ramanujan duals. II, Invent. Math. 106 (1991), 1–11.10.1007/BF01243900CrossRefGoogle Scholar
Clozel, L., Automorphic forms and the distribution of points on odd-dimensional spheres, Israel J. Math. 132 (2002), 175–187.10.1007/BF02784510CrossRefGoogle Scholar
Clozel, L., Démonstration de la conjecture $\tau$, Invent. Math. 151 (2003), 297–328.10.1007/s00222-002-0253-8CrossRefGoogle Scholar
Clozel, L., Spectral theory of automorphic forms, in Automorphic forms and applications, IAS/Park City Mathematics Series, vol. 12 (American Mathematical Society, Providence, RI, 2007), 43–93.Google Scholar
Clozel, L., Oh, H. and Ullmo, E., Hecke operators and equidistribution of Hecke points, Invent. Math. 144 (2001), 327–351.10.1007/s002220100126CrossRefGoogle Scholar
Clozel, L. and Ullmo, E., Équidistribution des points de Hecke, in Contributions to automorphic forms, geometry, and number theory (Johns Hopkins University Press, Baltimore, MD, 2004), 193–254.Google Scholar
Conrad, B., Modular forms and the Ramanujan conjecture (Cambridge University Press, 2011).Google Scholar
Cowling, M., Haagerup, U. and Howe, R., Almost $L^2$ matrix coefficients, J. Reine Angew. Math. 387 (1988), 97–110.Google Scholar
Duke, W., Rational points on the sphere. Rankin memorial issues, Ramanujan J. 7 (2003), 235–239.10.1023/A:1026203430418CrossRefGoogle Scholar
Erdös, P., Some results on Diophantine approximation, Acta Arith. 5 (1959), 359–369.10.4064/aa-5-4-359-369CrossRefGoogle Scholar
Flath, D., Decomposition of representations into tensor products, in Automorphic forms, representations and L-functions, Part 1, Proceedings of Symposia in Pure Mathematics, vol. XXXIII (American Mathematical Society, Providence, RI, 1979), 179–183.10.1090/pspum/033.1/546596CrossRefGoogle Scholar
Ghosh, A., Gorodnik, A. and Nevo, A., Diophantine approximation and automorphic spectrum, Int. Math. Res. Not. IMRN 21 (2013), 5002–5058.10.1093/imrn/rns198CrossRefGoogle Scholar
Ghosh, A., Gorodnik, A. and Nevo, A., Metric Diophantine approximation on homogeneous varieties, Comput. Math. 150 (2014), 1435–1456.Google Scholar
Ghosh, A., Gorodnik, A. and Nevo, A., Counting solutions to intrinsic Diophantine inequalities in simple algebraic groups, Israel J. Math. 251 (2022), 443–466.10.1007/s11856-022-2428-xCrossRefGoogle Scholar
Gorodnik, A. and Kadyrov, S., Diophantine approximation with restricted numerators and denominators on semisimple groups, J. Théor. Nombres Bordeaux 29 (2017), 1–28.10.5802/jtnb.967CrossRefGoogle Scholar
Gorodnik, A., Maucourant, F. and Oh, H., Manin's and Peyre's conjectures on rational points and adelic mixing, Ann. Sci. Éc. Norm. Supér. 41 (2008), 383–435.Google Scholar
Gorodnik, A. and Nevo, A., Counting lattice points, J. Reine Angew. Math. 663 (2012), 127–176.Google Scholar
Lang, S., Report on Diophantine approximation, Bull. Soc. Math. France 93 (1965), 177–192.10.24033/bsmf.1621CrossRefGoogle Scholar
LeVeque, W. J., On the frequency of small fractional parts in certain real sequences. III, J. Reine Angew. Math. 202 (1959), 215–220.10.1515/crll.1959.202.215CrossRefGoogle Scholar
Nevo, A., Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups, Math. Res. Lett. 5 (1998), 305–325.10.4310/MRL.1998.v5.n3.a5CrossRefGoogle Scholar
Platonov, V. and Rapinchuk, A., Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139 (Academic Press, Boston, MA, 1994).Google Scholar
Sakai, T., Riemannian geometry, Translations of Mathematical Monographs, vol. 149 (American Mathematical Society, Providence, RI, 1996).10.1090/mmono/149CrossRefGoogle Scholar
Sarnak, P., Notes on the generalized Ramanujan conjectures, in Harmonic analysis, the trace formula, and Shimura varieties, Clay Mathematics Proceedings, vol. 4 (American Mathematical Society, Providence, RI, 2005), 659–685.Google Scholar
Schmidt, W., A metrical theorem in Diophantine approximation, Canad. J. Math. 12 (1960), 619–631.10.4153/CJM-1960-056-0CrossRefGoogle Scholar