All Questions
Tagged with brauer-groups nt.number-theory
15 questions
2
votes
1
answer
251
views
Ring structure on Brauer group
Class field theory defines an isomorphism between the Brauer group of a finite extension of p-adic fields and a cyclic group with a canonical generator. This in turn defines an isomorphism of the ...
- 447
3
votes
0
answers
230
views
A Brauer group of a double covering of a "well-understood" variety
Let $k$ be a field (it is possible to assume that $k = \mathbb{Q}$ or $= \overline{\mathbb{Q}}$) and $X, Y$ nice varieties over $k$.
Let $f \colon Y \to X$ be a finite flat surjective morphism of ...
- 1,364
3
votes
1
answer
315
views
Unitary involutions on a simple central algebra after a scalar extension
$\DeclareMathOperator{id}{id}$
Let $L/K$ be a quadratic separable extension of fields. Let $A$ be a central simple algebra over $L$ such that its norm $N_{L/K}(A)$ splits. Then we know that there ...
- 93
1
vote
0
answers
191
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Vanishing of the local étale cohomology sheaf (?)
Let $X$ be a locally noetherian regular scheme, and let $Z$ be a closed subscheme of $X$ whose codimension $d > 0$ at every point.
Let $U$ be the complement of $Z$ in $X$.
For a sheaf $\mathscr{F}$ ...
- 185
6
votes
1
answer
891
views
Brauer group of rational numbers
Reading about the calculation of the Brauer group of rational numbers, the calculations of the group are extremely lengthy and technical. First of all, it will be very helpful to me if someone can ...
- 119
9
votes
1
answer
702
views
Software for detecting Brauer-Manin obstructions?
In the context of another MO question, the following question arose: Does there exist any software for detecting Brauer–Manin obstructions to the existence of integer solutions to a single polynomial ...
- 82.6k
3
votes
1
answer
244
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A local-to global principle for splitting of Azumaya algebras
Let $S$ be a finitely generated domain with the field of fractions $F.$ Let X be a smooth,
geometrically connected affine variety over $S.$ Let $A$ be an Azumaya algebra over $X.$
Assume that for all ...
- 33
3
votes
0
answers
98
views
Multiplication law in a central simple algebra of dimension 9 over a global field
Let $k$ be a global field, for example $k=\Bbb Q$.
Let $D$ denote the central simple algebra of dimension 9 over $k$ with given local invariants $i_v$.
Here $v$ runs over the set $\Omega_f(k)$ of ...
- 14.1k
3
votes
0
answers
95
views
Multiplication law in a division algebra of dimension 9 over a non-archimedean local field
Let $k$ be a non-archimedean local field, for example, a $p$-adic field (a finite extension of the filed ${\Bbb Q}_p$ of $p$-adic numbers).
It is well known that there is a canonical isomorphism
$${\...
- 14.1k
4
votes
1
answer
314
views
Can base-change be non-surjective on Brauer groups?
Is there a finite-degree separable field extension $\mathbb{K} \subset \mathbb{L}$ such that the induced map on Brauer groups $\operatorname{Br}(\mathbb{K}) \to \operatorname{Br}(\mathbb{L})$ is not a ...
- 54.6k
6
votes
1
answer
360
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Brauer groups and field extensions
Let $k$ be a field and $\mathrm{Br}(k)$ the Brauer group of $k$. Let $k \subset L$ be a field extension. Let $b \in \mathrm{Br}(k)$ and denote by $b \otimes L \in \mathrm{Br}(L)$ the base-change of $b$...
- 21.1k
2
votes
1
answer
466
views
Do $PGL_n$-torsors induce elements of the Brauer group
Let $K$ be a field and let $n\geq 2$. If $n=2$, then the set of $K$-isomorphism classes of $PGL_n$-torsors is in bijection with the $n$-torsion of the Brauer group of $K$.
Is this only for $n=2$?
Is ...
- 23
6
votes
2
answers
516
views
Obstruction and rational points on curves
Is etale-Brauer the only obstruction to the existence of rational points on projective plane curves over number fields?
- 11.3k
4
votes
0
answers
148
views
Zeta functions with Brauer class
In algebraic geometry there are examples when a variety $X$ is somehow related (I call it double-mirror) to another variety $Y$, together with
a 2-torsion Brauer class. By "related" I mean statements ...
- 5,186
4
votes
2
answers
600
views
How do Brauer groups relate to zeta functions?
There are two approaches to class field theory that I was taught. The first, is the theory of $L$-functions, Dirichlet characters and so forth (which I described succintly in the question What are the ...
- 5,929