# Questions tagged [galois-cohomology]

The galois-cohomology tag has no usage guidance.

292
questions

5
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### Triviality of $\unicode{1064}(T_pE ⊗ T_pE)$ for elliptic curves and Bogomolov's lemma

Consider the case of an elliptic curve $E$ over Q, and let $S$ be a finite set of primes including all places of bad reduction and a place $p$ of good reduction.
Bogomolov's Lemma says that when $p$ ...

1
vote

0
answers

154
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### 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 ...

16
votes

1
answer

331
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### Galois cohomology for non-Galois extensions

If $L/K$ is a Galois extension with group $G$ then we can consider $H^*(G;L^\times)$. This is useful in algebraic number theory, and there are many results about it.
Now let $L/K$ be a finite ...

6
votes

0
answers

166
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### 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 $\mu_8$...

7
votes

2
answers

834
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### Hilbert's Satz 90 for real simply-connected groups?

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K/k$ be a Galois extension. Then one generalisation of Hilbert's Satz 90 states that $H^1(\Gal(K/k),\GL_n(K))=...

3
votes

0
answers

190
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### Cup product structure on Galois cohomology

Let $G_S$ denote the Galois group of the maximal extension of $\mathbb{Q}$ unramified outside a finite, non-empty set of primes, $S$. Let $p\in S$ and let $V, W$ be a pair of finite dimensional $p$-...

11
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0
answers

328
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### Interpretation of $H^3(\mathrm{Gal}(L/K),L^\times)$

During my work I came across the group $H^3(\mathrm{Gal}(L/K),L^\times)=H^3(L/K,L^\times)$ for certain (infinite) Galois extensions $L/K$ (for an arbitrary field $K$) and I wondered if there is an ...

0
votes

0
answers

76
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### Bounding the dimension of $H^1(G, V\otimes V^{\vee})$

Let $G_S$ denote the Galois group of the maximal extension of $\mathbb{Q}$ unramified away from a finite set of primes, $S$. Let $V$ be a finite dimensional, $G_S$-representation over $\mathbb{F}_p$ (...

2
votes

0
answers

146
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### Absolute Bloch-Kato Cohomology

The étale cohomology $R\Gamma_{\mathrm{ét}}(X;\mathbb{Z}_p(n))$ of a scheme $X/K$ can be computed by a Hochschild-Serre spectral sequence with terms of the form $H^i(K;H^j(X_{\overline{K}};\mathbb{Z}...

2
votes

0
answers

67
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### Finite dimensionality of Galois cohomology

Let $K_S$ denote the maximal extension of $\mathbb{Q}$, unramified outside a finite set of primes $S$, and let $G_S$ denote the Galois group of $K_S/\mathbb{Q}$.
It is known that for any finitely ...

7
votes

0
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238
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### What justifies the following isomorphism in Cassels' proof of the Cassels–Tate pairing?

In Cassels' paper Arithmetic on curves of genus 1. IV introducing the Cassels–Tate pairing the following lemma is stated.
Lemma 5.1: Let $q$ be a rational prime and $\Gamma$ the Galois group of the ...

5
votes

1
answer

159
views

### When is $G(R)\rightarrow G^{\textrm{ad}}(R)$ surjective?

Consider a reductive group $G$ over a field $k$. The adjoint group $G^{\textrm{ad}}$ is defined by the exact sequence $$1\rightarrow Z(G)\rightarrow G\rightarrow G^{\textrm{ad}}\rightarrow 1$$The ...

1
vote

0
answers

70
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### A possible generalization of Brauer's theorem about the prime factors of the period and index of a central simple algebra

Let $K$ be an arbitrary field, and let $K^s$ be a fixed separable closure of $K$.
Let $F/K$ a be a finite Galois extension in $K^s$.
Let $n>0$ be a natural number.
Let $A$ be a central simple ...

2
votes

0
answers

163
views

### Bounding dimensions of Galois cohomology

Let $G$ be the absolute Galois group of the rationals, and $V$ a finite dimensional $p$-adic representation.
Is there a way to provide a general bound on $H^i(G, V)$ in terms of $\dim_{\mathbb{Q}_p} V$...

5
votes

1
answer

252
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### Converse of "generalized Hilbert 90" / Galois descent

The following generalization of Hilbert 90 can be found in Serre's Corps Locaux (Chap. X, §1, ex.2, p.160 of the French edition), see also this question:
Theorem: If $L|K$ is a finite Galois extension ...

5
votes

1
answer

107
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### An isomorphic classification of non-associative division octonion algebras

A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and ...

1
vote

0
answers

41
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### When is $B^G\backslash(B/A)^G$ finite?

Let $G$ be a locally compact group, let $A,B$ be (not necessarily abelian) connected reductive complex groups equipped with continuous actions of $G$ via algebraic automorphisms. Let $\phi:A\to B$ be ...

4
votes

0
answers

103
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### Anisotropic semisimple groups with no real compact factor

Let $F$ be a number field, and let $G$ be a semi-simple connected, anisotropic algebraic group over $F$ which is $F$-simple (or almost simple, the question is agnostic to isogenies). Suppose further ...

2
votes

1
answer

251
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### Connecting homomorphism in non-abelian cohomology

Let $G$ be a simply connected, semisimple algebraic group over $\mathbb{R}$ and let $X$ be a homogeneous space for $G$ with finite commutative stabilizer $\mu$. There is a connecting homomorphism from ...

1
vote

1
answer

163
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### cokernel of $H^1(F_\Sigma/F,E[p^\infty])\to \prod_v H^1(F_v,E[p^\infty])/\operatorname{im}(\kappa_v)$

Let $F$ be a number field and $E/F$ an elliptic curve. Fix an odd prime $p$.Let $\kappa:E(F)\otimes \mathbb Q_p/\mathbb Z_p\to H^1(F,E[p^\infty])$ the Kummer map and $\kappa_v$ its reduction.
Let $\...

1
vote

0
answers

100
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### Bounding dimension of $H^1(G_{\mathbb{Q}}, (V_pE)^{\otimes n})$

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime of good reduction, $T_pE$ is its $p$-adic Tate module, $V_pE = T_pE\otimes \mathbb{Q}_p$, and $(V_pE)^{\otimes n}$ its $n$'th tensor ...

1
vote

0
answers

353
views

### Amitsur's theorem for generalized Severi–Brauer varieties

Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...

4
votes

0
answers

162
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### A computation of nearby cycles

I'm currently reading P.Scholze's paper "THE LANGLANDS-KOTTWITZ APPROACH FOR THE MODULAR
CURVE". In Lemma 7.7, he showed a (maybe simple) nearby cycle computation, which I can't follow.
Now ...

1
vote

0
answers

85
views

### Inflation-restrction sequence for maximal $S$-ramified extension

Let $K$ be a number field. Let $G_K$ be an absolute Galois group of $K$. Let $M$ be a $G_K$-module and $L/K$ be a finite extension.
There is a inflation-restriction exact sequence,
$0\to H^1(Gak(L/K), ...

2
votes

0
answers

126
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### Absolute Galois cohomology of function fields (of high-dimensional) varieties

What is known about the absolute Galois cohomology of function fields of varieties of dimension 2 or larger? Specifically, I am interested in multiplicative coefficients $\mathbb G_m$.
I have seen ...

6
votes

2
answers

256
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### Group homology for a metacyclic group

Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
We work with the first homology group
$$ H_1(G,M).$$
For any ...

8
votes

1
answer

607
views

### $\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{R})$

Let $\mathbf{G}$ be the image of the natural embedding of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_4(\mathbb{C})\subset \operatorname{SL}_8(\mathbb{R})$. Then $\mathbf{G}$ is an ...

9
votes

1
answer

369
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### For which subgroups the transfer map kills a given element of a group?

$\newcommand{\ab}{{\rm ab}}
\newcommand{\ord}{{\rm ord}}
$Let $G$ be a finite or profinite group. Consider the abelianized group
$$G^\ab=G/G'$$
where $G'$ is the commutator subgroup of $G$.
Let $H\...

6
votes

1
answer

243
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### Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$

This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763.
It got upvotes, but no answers or comments, and so I ask it here.
Let $G$ ...

3
votes

1
answer

317
views

### Pontryagin dual of cokernel, $(\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $

Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve.
Consider the natural map
$$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{...

1
vote

0
answers

123
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### Representability of twists of projective schemes

Let $K$ be a perfect field, and let $S$ be a projective $K$-scheme. Denote by $\text{Twist}(S/K)$ the set of twists of $S$ up to $K$-isomorphism. These are (apriori) sheaves $\mathcal{F}$ on the ...

4
votes

0
answers

61
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### Possible questions about the Tate-Shafarevich subgroup of a Galois hypercohomology group?

$\newcommand{\wt}{\widetilde}$
Let $n=1,2$. There are infinite torsion abelian groups $H^1$, $H^2$ killed by some natural number $m$.
There are finite subgroups
$$ {\rm Sha}^1 \subset H^1,\quad ...

1
vote

0
answers

132
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### Kernel of restriction map in Galois cohomology

Let $S$ be the algebraic group $SL_2/\mathbb{Q}_p$ with a $G=G_{\mathbb{Q}}$ action, (acts by conjugation with a representation $\rho: G\longrightarrow GL_2$.)
Let $G_p$ be the decomposition group at ...

1
vote

1
answer

188
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### Crystalline fibre of a morphism of Galois cohomology stacks

Let $K = \mathbb{Q}_p$, $G = G_K$ its absolute Galois group. Let
$$1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$$
be a split exact sequence of (not necessarily abelian) group ...

2
votes

1
answer

306
views

### Equivalence between twists of a curve and torsors of its automorphism group

Let $X$ be a curve defined over a number field $K$, and let $G_K$ be the absolute Galois group of $K$. Let $\text{Aut}(X)$ be the group of $\overline{K}$-defined automorphisms of $X$, and consider the ...

1
vote

0
answers

81
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### Algebraizable image of a morphism of Galois cohomology stacks

Assume I have a surjective morphism of algebraic group schemes over $\mathbb{Q}_p$, $\mathcal{G}\longrightarrow S$, equipped with a section, and assume both of these group schemes are equipped with an ...

6
votes

2
answers

362
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### Twisted forms with real points of a real Grassmannian

Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$.
We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{\...

2
votes

0
answers

103
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### Extensions of groups with a $G$-action

Let $1\longrightarrow A\longrightarrow \mathcal{G}\longrightarrow R\longrightarrow 1$ be an exact sequence of algebraic group schemes, with $\mathcal{G}$ being an extension of $R$, an affine reductive ...

3
votes

1
answer

254
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### The second Tate-Shafarevich group of a permutation module is trivial

Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $...

10
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0
answers

223
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### If $H$ is a quotient of $G$, does there exist an $H$-extension of $\mathbb{Q}$ not contained in a $G$-extension?

Let $\phi\colon G\rightarrow H$ be a surjective homomorphism between finite groups. Assume that $\phi$ is not split, in other words there exists no homomorphism $\sigma\colon H\rightarrow G$ such that ...

6
votes

1
answer

395
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### Ker of corestriction of Galois cohomology

Let $G$ be a Galois group and $H$ be its normal subgroup. Let $M$ be a $G$-module.
Consider the restriction map $res: H^1(G,M) \to H^1(H,M)$. Its kernel is given by $H^1(G/H,M^H)$.
On the other hand, ...

3
votes

0
answers

132
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### Cohomology of local fields in positive characteristic

It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...

13
votes

2
answers

645
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### Example of continuous cohomology vs cohomology

I am looking for an example of a locally compact group $G$ and a continuous $G$ module $M$, which also is locally compact, such that the continuous cochain cohomology differs from group cohomology (...

1
vote

0
answers

147
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### About the exact sequence $0\to $$Ш(E/\Bbb{Q})[\phi]\to Ш(E/\Bbb{Q})[2]\to Ш(E'/\Bbb{Q})[\hat{\phi}]$ in Silverman's book of elliptic curves

This question is about Silverman's book,$7$-th line from the bottom of $350$ page of 'The arithmetic of elliptic curves' (http://www.pdmi.ras.ru/~lowdimma/BSD/Silverman-Arithmetic_of_EC.pdf) .
Let $E$ ...

2
votes

1
answer

381
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### Galois cohomology of Tate modules

Let $E,E'$ be a pair of elliptic curves defined over $\mathbb{Z}$. Let $T_p[E], T_p[E']$ be their associated ($p$-adic) Tate modules. These are Galois representations for the absolute Galois group of $...

1
vote

0
answers

175
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### Crystalline exact sequence in Galois cohomology

Let $G$ be the absolute Galois group of $\mathbb{Q}_p$, and let $1\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 1$ be a short exact sequence of (non-abelian) algebraic group ...

3
votes

1
answer

220
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### Deformations of Galois cohomology

Let $M = (\mathbb{Z}_p)^2$ be a Galois representation, with Galois action given by $\rho: G\longrightarrow SL_2(\mathbb{Z}_p)$. I am trying to understand how sensitive the Galois cohomology group $H^1(...

3
votes

1
answer

355
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### Local Tate duality for F-vector space

A version of local Tate duality stated: Let $K$ be a finite extension of $\mathbb Q_p$, $A$ be a finite $G_K=Gal(\overline K/K)$ module. Then for $0\le i\le 2$, the cup product induces a perfect ...

4
votes

2
answers

279
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### 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 ...

1
vote

1
answer

170
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### 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}{\...