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Results tagged with algebraic-number-theory
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user 4149
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
1
vote
Non-commuting elements of finite orders in a reductive group over a p-adic field
Here I give details of the reduction in LSpice's comment.
I write it as an answer rather than a string of comments in order to have an editable text.
The reduction goes as follows. According to Will's …
5
votes
2
answers
175
views
Non-commuting elements of finite orders in a reductive group over a p-adic field
Let $k$ be a $p$-adic field and $G$ be a connected non-abelian reductive algebraic group over $k$. I am asking for a proof of the following lemma:
Lemma. Assuming that $p$ is "good" for $G$, there ex …
- 14.2k
1
vote
0
answers
156
views
Computer computation of the first Galois cohomology of a $p$-adic torus?
Let $T\subset {\rm GL}(N,{\mathbb Q})$ be an $n$-dimensional ${\mathbb Q}$-torus
given by its Lie algebra $\mathfrak{t}\subset \mathfrak{gl}(N,{\mathbb Q})$.
I want to compute, in some sense explicit …
- 14.2k
3
votes
0
answers
107
views
Describing the primes with each cyclic decomposition group in a given finite Galois extensio...
$\newcommand{\Q}{{\mathbb Q}}
$Let $f\in \Q[x]$ be a polynomial,
and let $L/\Q$ be the finite Galois extension
obtaining by adjoining to $\Q$ all roots of $f$.
Magma knows how to compute $\Gamma:={\r …
- 14.2k
6
votes
0
answers
167
views
Computer programs for decomposition groups?
There is quite a lot of work on computing Galois groups of splitting fields of polynomials over $\Bbb Q$. Magma is quite good at it.
In this answer to Decomposition groups for the Galois module $\mu_8 …
- 14.2k
6
votes
Hilbert's Satz 90 for real simply-connected groups?
In addition to the answer of Gro-Tsen: Let $G=\operatorname{SU}(3)$. Then $H^1({\mathbb R},G)$ classifies matrices of Hermitian forms in 3 variables with determinant 1. There are two equivalence class …
- 14.2k
2
votes
1
answer
86
views
Cyclic extensions of a number field of full local degree in a given set $S$
Let $K$ be a number field, and let $S=S_f\cup S_{\mathbb R}\cup S_{\mathbb C}$
be a finite set of places of $K$, where $S_f$ denotes the set of finite places in $S$, $S_{\mathbb R}$ denotes the set of …
- 14.2k
2
votes
0
answers
97
views
Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of...
1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$.
Note that any subgroup of $G$ contains $1_G$, and so the set of all s …
- 14.2k
2
votes
0
answers
62
views
Constructing a cyclic extension $L$ with given local behavior of a global field $K$ such tha...
Let $F$ be a global field without real places
(that is, a function field or a totally imaginary number field).
Let $K/F$ be a cyclic extension of degree $n$.
Let $S$ be a ${\rm Gal}(K/F)$-invariant fi …
- 14.2k
1
vote
0
answers
56
views
Normality in a tower of cyclic extensions of global fields, as in Artin-Tate
Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field,
and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of …
- 14.2k
4
votes
0
answers
135
views
A normal extension of a number field of given degree that does not split over a given set of...
Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.
Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\otime …
- 14.2k
1
vote
$\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{...
$\newcommand{\nN}{{\mathcal N}}
\newcommand{\SL}{{\rm SL}}
\newcommand{\G}{{\bf G}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\C}{{\Bbb C}}
$The answer is No.
Write $\nN$ for the …
- 14.2k
6
votes
Accepted
The second Tate-Shafarevich group of a permutation module is trivial
We write $G_w={\rm Gal}(L_w/K_v)$.
Definition. For $n\ge 1$, we denote
$$Ш_\omega^n(G,M)=\ker\Big(H^n(G,M)\to\prod_C H^n(C,M)\Big)$$
where $C$ runs over the cyclic subgroups of $G$.
Remark. $Ш^2(L …
- 14.2k
4
votes
2
answers
291
views
Biquadratic extension of global function fields with cyclic decomposition groups
Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$.
Question. What would be an example of a gl …
- 14.2k
1
vote
1
answer
180
views
Decomposition groups for the Galois module $\mu_8$
$\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Gal}{Gal}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Fbar}{{\overline F}}
\newcommand{\G}{\ …
- 14.2k