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

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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Arithmetic purity of strong approximation for semi-simple simply connected groups

Compositio Mathematica ◽

10.1112/s0010437x20007617 ◽

2020 ◽

Vol 156 (12) ◽

pp. 2628-2649

Author(s):

Yang Cao ◽

Zhizhong Huang

Keyword(s):

Quadratic Form ◽

Strong Approximation ◽

Homogeneous Spaces ◽

Algebraic Groups ◽

Integral Points ◽

Spin Group ◽

Linear Algebraic Groups ◽

Simply Connected ◽

Quadratic Hypersurfaces ◽

Almost Prime

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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FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

International Journal of Number Theory ◽

10.1142/s1793042107001103 ◽

2007 ◽

Vol 03 (04) ◽

pp. 541-556 ◽

Cited By ~ 1

Author(s):

WAI KIU CHAN ◽

A. G. EARNEST ◽

MARIA INES ICAZA ◽

JI YOUNG KIM

Keyword(s):

Quadratic Form ◽

Number Field ◽

Quadratic Forms ◽

Positive Definite ◽

Integral Quadratic Form ◽

Ring Of Integers ◽

Ternary Quadratic Forms ◽

Totally Positive ◽

Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

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Integral quadratic forms avoiding arithmetic progressions

International Journal of Number Theory ◽

10.1142/s1793042120501109 ◽

2020 ◽

Vol 16 (10) ◽

pp. 2141-2148

Author(s):

A. G. Earnest ◽

Ji Young Kim

Keyword(s):

Positive Integer ◽

Quadratic Form ◽

Quadratic Forms ◽

Positive Definite ◽

Arithmetic Progressions ◽

Integral Quadratic Form ◽

Positive Integers

For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

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The number of representations of integers by generalized Bell ternary quadratic forms

International Journal of Number Theory ◽

10.1142/s1793042121500135 ◽

2020 ◽

pp. 1-15

Author(s):

Kyoungmin Kim ◽

Yeong-Wook Kwon

Keyword(s):

Quadratic Form ◽

Class Number ◽

Quadratic Forms ◽

Positive Definite ◽

Cusp Forms ◽

Ternary Quadratic Form ◽

Closed Formula ◽

Integral Quadratic Form ◽

Ternary Quadratic Forms ◽

Representations Of Integers

For a positive definite ternary integral quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. A ternary quadratic form [Formula: see text] is said to be a generalized Bell ternary quadratic form if [Formula: see text] is isometric to [Formula: see text] for some nonnegative integers [Formula: see text]. In this paper, we give a closed formula for [Formula: see text] for a generalized Bell ternary quadratic form [Formula: see text] with [Formula: see text] and class number greater than [Formula: see text] by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight [Formula: see text] and level [Formula: see text] with [Formula: see text] consisting of eta-quotients.

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Quadratic Forms, Linear Algebraic Groups, and Cohomology

10.1007/978-1-4419-6211-9 ◽

2010 ◽

Cited By ~ 1

Keyword(s):

Quadratic Forms ◽

Algebraic Groups ◽

Linear Algebraic Groups

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REPRESENTATIONS OF ARITHMETIC PROGRESSIONS BY POSITIVE DEFINITE QUADRATIC FORMS

International Journal of Number Theory ◽

10.1142/s1793042111004915 ◽

2011 ◽

Vol 07 (06) ◽

pp. 1603-1614 ◽

Cited By ~ 2

Author(s):

BYEONG-KWEON OH

Keyword(s):

Positive Integer ◽

Quadratic Form ◽

Quadratic Forms ◽

Positive Definite ◽

Arithmetic Progressions ◽

Negative Integer ◽

Definite Integral ◽

Integral Quadratic Form ◽

Ternary Quadratic Forms

For a positive integer d and a non-negative integer a, let Sd,a be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be Sd,a-universal if it represents all integers in the set Sd, a, and is said to be Sd,a-regular if it represents all integers in the non-empty set Sd,a ∩ Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) ∈ ℚ[x,y] (R(x,y) ∈ ℚ[x,y]) such that the discriminant df for any Sd,a-universal (Sd,a-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).

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