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Results tagged with quadratic-forms
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user 48142
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.
6
votes
Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$
Since $\mathbb{Z}_{(p)}$ can be thought of as a subring of $\mathbb{Z}_{p}$,
if two quadratic forms $Q_{1}$ and $Q_{2}$ are equivalent over $\mathbb{Z}_{(p)}$, then they must be equivalence over $\mat …
- 20.4k
8
votes
Accepted
Calculating the explicit constant – Siegel zeros and class numbers
One place to find this worked out in detail is the paper "On the Siegel-Tatuzawa theorem" by Jeffrey Hoffstein (published in 1980 in Acta Arithmetica). Lemma 1 of that paper states that if $\chi$ is a …
- 20.4k
5
votes
Accepted
Can each natural number be represented by $2w^2+x^2+y^2+z^2+xyz$ with $x,y,z\in\mathbb N$?
The answer to question 1 is yes - the other questions seem to me to be more difficult.
If $n$ is odd, then there are non-negative integers $w$, $x$ and $y$ so that $n = 2w^{2} + x^{2} + y^{2}$. One wa …
- 20.4k
7
votes
Proof of Witt's result about quaternion extensions
The problem you raise is tied to the "embedding problem." There is chapter devoted to this in Malle and Matzat's "Inverse Galois Theory" text (here's a link to the publisher's page about this book).
…
- 20.4k
6
votes
Accepted
Ternary quadratic form theta series as Hecke eigenforms and class number one
There are many specific questions raised in this post. I will address the main one and show that if a $Q$ is a ternary quadratic form and the theta series $\theta_{Q}$ is a Hecke eigenform, then $Q$ h …
- 20.4k
7
votes
Accepted
Simple comparison of positive ternary quadratic form representation counts
No, this isn't true in general. You could, for example, take $f(x,y,z) = x^{2} + y^{2} + 13z^{2}$. Then $R(1) = 4$ and $R(25) = 12$. The form is isotropic at $5$ (since $5$ doesn't divide the discrimi …
- 20.4k
7
votes
Accepted
Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sq...
It seems to me that what Buell says about the narrow class group is not quite right (it's hard for me to say, as I don't have a copy of it). Magma tells me that in $\mathbb{Q}(\sqrt{210})$, the narrow …
- 20.4k
8
votes
The Chebotarev Density Theorem and the representation of infinitely many numbers by forms
The example of Heath-Brown's article is a good one. For a bit more elementary examples, you can fix a number field $K$ with $[K : \mathbb{Q}] = n$ and ring of integers $\mathcal{O}_{K}$ and pick a bas …
- 20.4k