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Results tagged with quadratic-forms
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user 39552
Algebraic and geometric theory of quadratic forms and symmetric bilinear forms, e.g., values attained by quadratic forms, isotropic subspaces, the Witt ring, invariants of quadratic forms, the discriminant and Clifford algebra of a quadratic form, Pfister forms, automorphisms of quadratic forms.
2
votes
Mass of spinor genus, positive integral quadratic forms
I'm late on that question ... I found it while looking at the posts related to this other question on spinor genera.
It seems to me that the answer should be yes, at least in dimension $d\geq 4$. Her …
- 1,980
4
votes
Accepted
A quadratic Diophantine equation
Call $q$ your quadratic form, and $M$ the $\mathbf Z$-quadratic space $(\mathbf Z^2,q)$. It's discriminant is $-20$. Now let $p$ be a prime. Then $V_p:=M\otimes \mathbf F_p$ is a $2$-dimensional quadr …
- 1,980
7
votes
Accepted
Integral orthogonal group for indefinite ternary quadratic form
Edit : this is a new answer, after more computations.
Let $H$ be the subgroup of your orthogonal group that preserve globally each connected component of the (two-sheeted) space $q(x,y,z)=-1$.
Up t …
- 1,980
3
votes
2-dimensional sublattices with all vectors having very big square (in absolute value)
This doesn't solve the question : it shows that a non-degenerate lattice always contains a primitive rank 2 sublattice with the required property regarding norms ... but the sublattice found is defini …
- 1,980
2
votes
genus 2 Siegel theta series of 3-dimensional lattices
Edit : @WKC furnished the valuable reference Yoshiyuki Kitaoka, Arithmetic of quadratic forms, where it is proved that two $n$-dim. positive definite lattices that have the same determinant and repre …
- 1,980
5
votes
3
answers
470
views
genus 2 Siegel theta series of 3-dimensional lattices
Let $(V,f)$ be a $3$-dimensional positive definite quadratic space over $\mathbf Q$.
Let $G(V)$ be a set of representatives of the isometry classes of maximal integral lattices on $V$.
To an eleme …
- 1,980
4
votes
Accepted
S genus of quadratic forms
Let $L=(\mathbf Z^n,b)$ be a bilinear module such that $L\otimes \mathbf Q$ is non-degenerate.
For a set of ultrametric places $S$, let us write $\mathbf Z[S^{-1}]$ for the set of rationals that are …
- 1,980
4
votes
Accepted
Given a positive-definite integral unimodular Gram matrix, how to find a basis of the associ...
Here is a method.
Let $\mathrm{V}$ be the bilinear space $(\mathbf Q^n, b)$ with Gram Matrix $G$ in the canonical basis.
Let $M_0$ be the lattice $\mathbf Z^n$ on $\mathbf Q^n$. Find a primitive ve …
- 1,980
1
vote
Accepted
Number of vectors of fixed norm
It is known at least since Hermite that, given integers $N$ and $D$ there are only finitely many equivalence classes of positive definite quadratic modules rank $n<N$ and discriminant $d<D$.
(His pro …
- 1,980
0
votes
Accepted
Orbits of the maximal compact subgroup on the light cone for $p$-adic groups
Let $(V,(x,y)\mapsto x.y)$ be a bilinear space. Let $L$ be a lattice on $V$, and let $G$ be its stabilizer in $O(V)$.
To an isotropic line $\ell$ in $V$ one can associate the ideal $I(\ell):=(\ell\c …
- 1,980
7
votes
0
answers
253
views
Question on some coverings of the euclidean space
Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has d …
- 1,980
7
votes
Accepted
"Pythagoras number" for integral matrices
Here is an answer (see the last point). It differs from what I had been claiming in my first post. There I was saying that any positive bilinear module $\Lambda$ over $\mathbf Z[\frac 12]$ was represe …
- 1,980