All Questions
Tagged with quadratic-forms algebraic-groups
15 questions
3
votes
1
answer
330
views
Strong Approximation for solutions to quadratic Diophantine equations
Can anyone either direct me to an relatively elementary proof in the literature--or show me why this (Conjecture 1 stated below) is true--or if I am mistaken and it is not true:
For any 4-tuple $\xi =...
- 4,219
1
vote
1
answer
343
views
Constructing groups of Type E^{66}_{7,1} having non trivial Tits algebra
This can be considered as a continuation of my last useful question:
Constructing groups of Type E7 with certain Tits Index
It is known that a quadratic form $q$ of dimension $12$, having splitting ...
- 2,178
1
vote
0
answers
90
views
The splitting pattern of the Killing form of an algebraic group and the Tits index
Let us assume that $G$ is an anisotropic semisimple, connected algebraic group over a field $k$ of characteristic zero.
Let $K_G$ denote the class of its Killing form in the Witt ring of $k$.
Let $X$ ...
- 12.9k
2
votes
0
answers
192
views
Maximal connected subgroup of orthogonal group
Let $(Q,V)$ be a quadratic space over an algebraically closed field $k$ with $\dim(V) \geq 3$
Define
$$ SO_Q:= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\...
- 63
1
vote
0
answers
53
views
Stabilizer group uniquely determines subspace
Let $(Q,V)$ be a quadratic space over an algebraically closed field $k$.
Let
$$ SO_Q(k):= \{ \sigma \in GL(V) : Q(\sigma v) = Q(v) \ \text{for all} \ v \in V \ \text{and} \det(\sigma) = 1 \}$$
Let $L \...
- 63
7
votes
1
answer
276
views
Is a 8-dimensional quadratic form recognized by its Lie algebra, modulo equivalence and scalar multiplication?
Question. Let $K$ be a field of characteristic zero (large characteristic should be fine too). Let $q,q'$ be two non-degenerate quadratic forms on $K^n$ with $n=8$. Suppose that the Lie algebras $\...
- 2,178
4
votes
1
answer
185
views
On the orthogonal group of a lattice on a quadratic space over dyadic local field
Let $F$ be a local field with valuation ring $R$. $V$ is a n dimensional non-singular quadratic space over $F$ with bilinear form $B$ and quadratic map $Q$.
As usual, $O(V)$ denotes the orthogonal ...
- 63.9k
4
votes
0
answers
161
views
Quadrics contained in the (complex) Cayley plane
In the paper
Ilev, Manivel - The Chow ring of the Cayley plane
we can learn, that $CH^8(X)$, with $X := E_6/P_1$, denoting the Cayley plane, has three generators with one of them being the class of ...
- 17.4k
5
votes
1
answer
450
views
Constructing groups of Type E7 with certain Tits Index
In a new survey on $E_8$, namely
Skip Garibaldi - E8 the most exceptional group
, the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-...
- 1,479
3
votes
1
answer
355
views
cubic forms and finiteness of $k^*/(k^*)^3$
In some recent computation I came across certain cubic forms and was wondering about analogue of following result for quadratic forms.
If $k^*/(k^*)^2$ is finite then there are only finitely many ...
- 47.4k
4
votes
0
answers
121
views
Norm variety for n=5, p=2 not isomorphic to a quadric
In the paper "Motivic construction of cohomological invariants", the author displays a list of known norm varieties for several $n,p$ on page $11$. For $p=2, n=5$ it says that a norm variety is given ...
- 17.6k
4
votes
1
answer
277
views
Automorphisms of SO_n(k,f)
Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let
$$O_n(k,f):=\{ g\in GL_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$
and
$$SO_n(k,...
- 5,654
2
votes
0
answers
123
views
What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?
We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$
$$
F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2)
$$
on the vector space $V^{\mathbb{R}}:={\...
- 14.2k
3
votes
1
answer
127
views
Set of isomorphisms of Pfister forms corresponding to first cohomology of algebraic group
Assume $k_0$ is a field with char($k_0$) not $2$. Let us define functors from $\rm Field_{/k_0}\to \rm Sets$ as $\rm Pfister_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$;
$\...
- 1,602
3
votes
1
answer
355
views
Indefinite orthogonal groups over p-adics
Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...
- 6,349