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Results tagged with quadratic-forms
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user 3635
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.
119
votes
8
answers
34k
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Zagier's one-sentence proof of a theorem of Fermat
Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an involution of the set $S= \lbrace (x,y,z) …
- 6,224
6
votes
1
answer
455
views
The number of integral solutions to $x^2+y^2-az^2=0$
I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation
$$ x^2+y^ …
- 6,224
4
votes
2
answers
298
views
Orbits of the maximal compact subgroup on the light cone for $p$-adic groups
It is known that if $Q$ is an indefinite non-degenerate quadratic form on $ \mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the …
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4
votes
1
answer
500
views
$p$-adic analogues of $\mathrm{SO}(3)$
I read in the paper " From Laplace to Langlands via representations of orthogonal groups" by Benedict Gross and Mark Reeder that there are, up to isomorphism, two orthogonal groups of the (non-degener …
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2
votes
0
answers
109
views
Bounds on the number of zeros of a quadratic form
Let $Q(x_1, \dots, x_n)$ be a non-degenerate indefinite quadratic form with integer coefficients. Let $N(Q,T)$ be the set of vectors $x=(x_1, \dots, x_n) \in {\mathbb Z}^n$ such that $|x|<T$ and $Q(x) …
- 6,224
0
votes
1
answer
423
views
$p$-adic orthogonal groups in four variables
Let $p>2$ be prime. By the classification of quadratic forms, there are $8$ pairwise non-equivalent isotropic orthogonal groups in $4$ variables. Is there a concrete classification of orthogonal group …
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