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Lifting, restricting and sifting integral points on affine homogeneous varieties

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

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: a.gorodnik@bristol.ac.uk)
Amos Nevo
Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel (email: anevo@tx.technion.ac.il)
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Abstract

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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

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