Prime and almost prime integral points on principal homogeneous spaces

In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field $k$. For instance, for any such group $G$ and for any open subset $U$ of $G$ with ${\mathrm {codim}}(G\setminus U, G)\geqslant 2$, we prove that (i) if $G$ is $k$-simple and $k$-isotropic, then $U$ satisfies strong approximation off any finite number of places; and (ii) if $G$ is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then $U$ satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of $G\setminus U$, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.

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Lifting, restricting and sifting integral points on affine homogeneous varieties

Compositio Mathematica ◽

10.1112/s0010437x12000516 ◽

2012 ◽

Vol 148 (6) ◽

pp. 1695-1716 ◽

Cited By ~ 1

Author(s):

Alexander Gorodnik ◽

Amos Nevo

Keyword(s):

Homogeneous Spaces ◽

Acta Math ◽

Upper Bounds ◽

Lattice Points ◽

List Type ◽

Integral Points ◽

Counting Problem ◽

Symmetric Varieties ◽

Almost Prime ◽

Homogeneous Varieties

AbstractIn [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 semisimpleS-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action ofGonG/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)effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;(2)effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;(3)effective lower bounds on the number of almost prime points on symmetric varieties;(4)effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

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Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

Compositio Mathematica ◽

10.1112/s0010437x0800376x ◽

2009 ◽

Vol 145 (2) ◽

pp. 309-363 ◽

Cited By ~ 25

Author(s):

Jean-Louis Colliot-Thélène ◽

Fei Xu

Keyword(s):

Quadratic Form ◽

Quadratic Forms ◽

Homogeneous Spaces ◽

Algebraic Groups ◽

Integral Points ◽

Integral Quadratic Form ◽

Linear Algebraic Groups ◽

Global Principle

AbstractAn integer may be represented by a quadratic form over each ring ofp-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

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ON S-HARDY–LITTLEWOOD HOMOGENEOUS SPACES

International Journal of Mathematics ◽

10.1142/s0129167x98000312 ◽

1998 ◽

Vol 09 (06) ◽

pp. 723-757 ◽

Cited By ~ 1

Author(s):

MASANORI MORISHITA ◽

TAKAO WATANABE

Keyword(s):

Number Field ◽

Homogeneous Spaces ◽

Algebraic Number Field ◽

Number Fields ◽

Positive Definite ◽

Integral Points ◽

Distribution Property ◽

Totally Positive ◽

Finite Set ◽

Totally Real

We study the asymptotic distribution of S-integral points on affine homogeneous spaces in the light of the Hardy–Littlewood property introduced by Borovoi and Rudnick. We introduce the S-Hardy–Littlewood property for affine homogeneous spaces defined over an algebraic number field and a finite set S of places of the base field. We work with the adelic harmonic analysis on affine algebraic groups over a number field to determine the asymptotic density of S-integral points under congruence conditions. We give some new examples of strongly or relatively S-Hardy–Littlewood homogeneous spaces over number fields. As an application, we prove certain asymptotically uniform distribution property of integral points on an ellipsoid defined by a totally positive definite tenary quadratic form over a totally real number field.

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