Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T17:30:21.220Z Has data issue: false hasContentIssue false
  • English
  • Français

Lifting, restricting and sifting integral points on affine homogeneous varieties

Part of: Special varieties Algebraic groups Multiplicative number theory Arithmetic algebraic geometry Diophantine equations

Published online by Cambridge University Press:  11 October 2012

Alexander Gorodnik  and
Amos Nevo
Alexander Gorodnik
Affiliation:
School of Mathematics and Statistics, University of Bristol, Bristol BS8 1TW, UK (email: [email protected])
Amos Nevo
Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [Gorodnik and Nevo, Counting lattice points, J. Reine Angew. Math. 663 (2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimple S-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action of G on G/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak, Prime and almost prime integral points on principal homogeneous spaces, Acta Math. 205 (2010), 361–402] and use them to establish several useful consequences of this property, including:

  1. (1) effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;

  2. (2) effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;

  3. (3) effective lower bounds on the number of almost prime points on symmetric varieties;

  4. (4) effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

Keywords

effective strong approximationhomogeneous varietiesintegral points on varietiesalmost primessieving in orbitsLinnik problem

MSC classification

Secondary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values 11D79: Congruences in many variables 11G35: Varieties over global fields 11N36: Applications of sieve methods 14L30: Group actions on varieties or schemes (quotients) 14M17: Homogeneous spaces and generalizations
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1695 - 1716
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[BO07]Benoist, Y. and Oh, H., Effective equidistribution of S-integral points on symmetric varieties, Preprint (2007), math.NT/0706.1621.Google Scholar
[BH62]Borel, A. and Harish-Chandra, , Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535.CrossRefGoogle Scholar
[BD09]Borovoi, M. and Demarche, C., Manin obstruction to strong approximation for homogeneous spaces, Comment. Math. Helv., to appear, available at math.NT/0912.0408v1.Google Scholar
[BGS10]Bourgain, J., Gamburd, A. and Sarnak, P., Affine linear sieve, expanders, and sum-product, Invent. Math. 179 (2010), 559–644.Google Scholar
[Bro09]Browning, T., Quantitative arithmetic of projective varieties, Progress in Mathematics, vol. 277 (Birkhäuser, Basel, 2009).CrossRefGoogle Scholar
[BHS06]Browning, T., Heath-Brown, D. R. and Salberger, P., Counting rational points on algebraic varieties, Duke Math. J. 132 (2006), 545–578.CrossRefGoogle Scholar
[BS91]Burger, M. and Sarnak, P., Ramanujan duals. II, Invent. Math. 106 (1991), 1–11.Google Scholar
[Clo03]Clozel, L., Démonstration de la conjecture τ, Invent. Math. 151 (2003), 297–328.CrossRefGoogle Scholar
[CX09]Colliot-Thélène, J.-L. and Xu, F., Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms, Compositio Math. 145 (2009), 309–363.Google Scholar
[CW75]Cooke, G. and Weinberger, P., On the construction of division chains in algebraic number rings, with applications to SL2, Comm. Algebra 3 (1975), 481–524.Google Scholar
[Die63]Dieudonné, J., La géométrie des groupes classiques (Springer, Berlin, 1963).CrossRefGoogle Scholar
[DRS93]Duke, W., Rudnick, Z. and Sarnak, P., Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993), 143–179.Google Scholar
[EM93]Eskin, A. and McMullen, C., Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), 181–209.CrossRefGoogle Scholar
[FI10]Friedlander, J. and Iwaniec, H., Opera de cribo, American Mathematical Society Colloquium Publications, vol. 57 (American Mathematical Society, Providence, RI, 2010).Google Scholar
[FI73]Fossum, R. and Iversen, B., On Picard groups of algebraic fibre spaces, J. Pure Appl. Algebra 3 (1973), 269–280.CrossRefGoogle Scholar
[GL02]Ghorpade, S. and Lachaud, G., Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J. 2 (2002), 589–631.CrossRefGoogle Scholar
[GN10]Gorodnik, A. and Nevo, A., The ergodic theory of lattice subgroups, Annals of Mathematics Studies, vol. 172 (Princeton University Press, Princeton, NJ, 2010).Google Scholar
[GN12]Gorodnik, A. and Nevo, A., Counting lattice points, J. Reine Angew. Math. 663 (2012), 127–176.Google Scholar
[GOS09]Gorodnik, A., Oh, H. and Shah, N., Integral points on symmetric varieties and Satake compactifications, Amer. J. Math. 131 (2009), 1–57.CrossRefGoogle Scholar
[GW07]Gorodnik, A. and Weiss, B., Distribution of lattice orbits on homogeneous varieties, Geom. Funct. Anal. 17 (2007), 58–115.CrossRefGoogle Scholar
[HR74]Halberstam, H. and Richert, H., Sieve methods (Academic Press, New York, NY, 1974).Google Scholar
[Har08]Harari, D., Le défaut d’approximation forte pour les groupes algébriques commutatifs, Algebra Number Theory 2 (2008), 595–611.CrossRefGoogle Scholar
[Har77]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, NY, 1977).Google Scholar
[Hea97]Heath-Brown, D. R., The density of rational points on cubic surfaces, Acta Arith. 79 (1997), 17–30.Google Scholar
[Hea02]Heath-Brown, D. R., The density of rational points on curves and surfaces, Ann. of Math. (2) 155 (2002), 553–595.CrossRefGoogle Scholar
[Hea06]Heath-Brown, D. R., Counting rational points on algebraic varieties, in Analytic number theory, Lecture Notes in Mathematics, vol. 1891 (Springer, Berlin, 2006), 51–95.CrossRefGoogle Scholar
[HS94]Heckman, G. and Schlichtkrull, H., Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics, vol. 16 (Academic Press, San Diego, CA, 1994).Google Scholar
[KKV89]Knop, F., Kraft, H. and Vust, T., The Picard group of a G-variety, in Algebraische Transformationsgruppen und Invariantentheorie, DMV Seminar, vol. 13 (Birkhäuser, Basel, 1989), 77–87.CrossRefGoogle Scholar
[LW54]Lang, S. and Weil, A., Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827.CrossRefGoogle Scholar
[Li95]Li, J.-S., The minimal decay of matrix coefficients for classical groups, in Harmonic analysis in China, Mathematics and its Applications, vol. 327 (Kluwer Academic, Dordrecht, 1995), 146–169.CrossRefGoogle Scholar
[Lin44a]Linnik, Y., On the least prime in an arithmetic progression. I. The basic theorem, Recueil Mathŭmatique [Mat. Sb.] N.Ser. 15 (1944), 139–178.Google Scholar
[Lin44b]Linnik, Y., On the least prime in an arithmetic progression. II. The Deuring–Heilbronn phenomenon, Recueil Mathŭmatique [Mat. Sb.] N.Ser. 15 (1944), 347–368.Google Scholar
[LS10]Liu, J. and Sarnak, P., Integral points on quadrics in three variables whose coordinates have few prime factors, Israel J. Math. 178 (2010), 393–426.CrossRefGoogle Scholar
[Mau07]Maucourant, F., Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices, Duke Math. J. 136 (2007), 357–399.CrossRefGoogle Scholar
[MV07]Montgomery, H. and Vaughan, R., Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97 (Cambridge University Press, Cambridge, 2007).Google Scholar
[Nar88]Narkiewicz, W., Units in residue classes, Arch. Math. (Basel) 51 (1988), 238–241.Google Scholar
[NS10]Nevo, A. and Sarnak, P., Prime and almost prime integral points on principal homogeneous spaces, Acta Math. 205 (2010), 361–402.Google Scholar
[Odo79]Odoni, R. W. K., A proof by classical methods of a result of Ax on polynomial congruences modulo a prime, Bull. Lond. Math. Soc. 11 (1979), 55–58.CrossRefGoogle Scholar
[Oh02]Oh, H., Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J. 113 (2002), 133–192.Google Scholar
[O'Me00]O’Meara, T., Introduction to quadratic forms, in Classics in mathematics (Springer, Berlin, 2000).Google Scholar
[PR94]Platonov, V. and Rapinchuk, A., Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139 (Academic Press, Boston, MA, 1994).Google Scholar
[Pop74]Popov, V. L., Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 294–322.Google Scholar
[Sar05]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
[Vos98]Voskresenski, V. E., Algebraic groups and their birational invariants, Translations of Mathematical Monographs, vol. 179 (American Mathematical Society, Providence, RI, 1998).Google Scholar