Quadratic Forms, Linear Algebraic Groups, and Cohomology

2010 ◽  
Keyword(s):  
Quadratic Forms ◽  
Algebraic Groups ◽  
Linear Algebraic Groups

Quadratic Forms and Linear Algebraic Groups

2009 ◽  
pp. 1375-1426
Author(s):  
Detlev Hoffmann ◽  
Alexander Merkurjev ◽  
Jean-Pierre Tignol
Keyword(s):  
Quadratic Forms ◽  
Algebraic Groups ◽  
Linear Algebraic Groups

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

2009 ◽  
Vol 145 (2) ◽  
pp. 309-363 ◽  
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.

Quadratic Forms and Linear Algebraic Groups

2006 ◽  
pp. 1743-1794
Author(s):  
Detlev Hoffmann ◽  
Alexander Merkurjev ◽  
Jean-Pierre Tignol
Keyword(s):  
Quadratic Forms ◽  
Algebraic Groups ◽  
Linear Algebraic Groups

Quadratic Forms and Linear Algebraic Groups

2013 ◽  
Vol 10 (2) ◽  
pp. 1819-1859
Author(s):  
Detlev Hoffmann ◽  
Alexander Merkurjev ◽  
Jean-Pierre Tignol
Keyword(s):  
Quadratic Forms ◽  
Algebraic Groups ◽  
Linear Algebraic Groups

scholarly journals Small Degree Representations of Finite Chevalley Groups in Defining Characteristic

2001 ◽  
Vol 4 ◽  
pp. 135-169 ◽  
Author(s):  
Frank Lübeck
Keyword(s):  
Algebraic Groups ◽  
Small Degree ◽  
Irreducible Representations ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Linear Algebraic Groups ◽  
Large Rank ◽  
Projective Representations ◽  
Computer Calculations ◽  
Simply Connected

AbstractThe author has determined, for all simple simply connected reductive linear algebraic groups defined over a finite field, all the irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For large rank l, this bound is proportional to l3, and for rank less than or equal to 11 much higher. The small rank cases are based on extensive computer calculations.

scholarly journals Closedness of some subgroups in linear algebraic groups

1962 ◽  
Vol 14 (3) ◽  
pp. 272-275
Author(s):  
T. MIYATA ◽  
T. ODA ◽  
K. OTSUKA
Keyword(s):  
Algebraic Groups ◽  
Linear Algebraic Groups

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

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.

scholarly journals Analytic Dirac approximation for real linear algebraic groups

2010 ◽  
Vol 351 (2) ◽  
pp. 403-410 ◽  
Author(s):  
Christoph Lienau
Keyword(s):  
Algebraic Groups ◽  
Linear Algebraic Groups ◽  
Real Linear
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