# Questions tagged [galois-cohomology]

The galois-cohomology tag has no usage guidance.

262
questions

3
votes

1
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### 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

107
views

### 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

51
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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

106
views

### 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

156
views

### 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

189
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

70
views

### 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

339
views

### 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

97
views

### 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

213
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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
votes

0
answers

204
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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

274
views

### 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

111
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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

405
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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

140
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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

294
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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

163
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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

202
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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

258
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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

254
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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

131
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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}{\...

2
votes

0
answers

248
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### Galois cohomology of $\breve{\mathbb Q}_p \otimes_{\mathbb Q_p} \breve{\mathbb Q}_p$

Let $\breve{\mathbb Q}_p$ denote the completion of the maximal unramified extension of $\mathbb Q_p$. I‘d like to compute Galois cohomology groups and sets related to $\breve{\mathbb Q}_p \otimes_{\...

5
votes

1
answer

218
views

### Torus gerbes over curves

Setup: Let $k$ be an algebraically closed field. Let $C$ be a smooth connected curve over $k$. Let $K(C)$ be the function field of $C$.
Tsen's Theorem implies that every $\mathbb{G}_m$-gerbe over $K(C)...

4
votes

1
answer

93
views

### Real forms of the general linear Lie superalgebra

I'm interested in a classification of the real forms of the general linear Lie superalgebra $\mathfrak{gl}_{m|m}(\mathbb{C})$.
The real forms of the simple complex Lie superalgebras were classified by ...

4
votes

0
answers

142
views

### Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$

$\newcommand{\GL}{\operatorname{GL}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\C}{{\Bbb C}}
$For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle
of $G=\GL_{n,\R}\,$, that is,
an invertible ...

5
votes

0
answers

247
views

### Do algebraic tori have no $H^1$?

If $G$ is an algebraic group over a field $K$, we can consider the Galois (or flat) cohomology $H^1(K, G)$. If $G = \mathbb{G}_a$ or $\mathbb{G}_m$, it is well known that $H^1(K, G) = 0$ (the latter ...

3
votes

1
answer

152
views

### Why an isogeny induces a surjection between points over maximal unramified extension?

Let $E$ and $E'$ be elliptic curves over $\mathbb Q$, and let $\phi\colon E\to E'$ be an isogeny defined over $\mathbb Q$. Let $p$ be a prime relatively prime to the degree of $\phi$. Let $\mathbb Q_p^...

3
votes

0
answers

64
views

### Finiteness for Galois cohomology for $\mathbb{Z}_p$-module coefficients

I am looking for a general survey on the finite generation properties of
$$H^i(F,\mathbb{Z}_p(j))$$
for fields $F$. Here I refer to Galois cohomology (continuous group cohomology) and the group is ...

2
votes

0
answers

82
views

### Galois cohomology with coefficients in the integers of the Lubin-Tate extension

Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...

2
votes

1
answer

127
views

### Vanishing of the degree 2 cohomology of a p-adic field with coefficients Q/Z and action of the Frobenius and the Pontryagin dual of the inertia

Let $K$ be a $p$-adic field with Galois group $G$ and inertia subgroup $I\subset G$. Denote $(-)^\ast=\mathrm{Hom}_{cont}(-,\mathbb{Q}/\mathbb{Z})$. Using Tate local duality, we can compute $$H^2(G,\...

5
votes

1
answer

163
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### Restriction vs. multiplication by $n$ in Tate cohomology

$\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Cor}{Cor}$
This question was asked in MSE.
It got no answers or comments, and so I post it here.
Let $H$ be a subgroup of a finite group $G$, and ...

4
votes

1
answer

291
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### Why descend a representation (of a finite group) over $K$ to a representation over $k$ with $k$ a subfield of $K$ is useful?

I heard that Schur was trying to answer the following question
Given a representation of a finite group $G \overset{\rho}{\rightarrow} \operatorname{GL}_{n}(K)$, how to find the smallest subfield $k$ ...

7
votes

2
answers

803
views

### Is this exact sequence known?

$\newcommand{\Tors}{{\rm Tors}}
\newcommand{\tf}{{\rm\, t.f.}}
\newcommand{\Gt}{{\Gamma\!,\,\Tors}}
\newcommand{\Gtf}{{\Gamma\!,\tf}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\...

4
votes

1
answer

515
views

### Generalizations of global Euler characteristic formula

Let $ K $ be a number field, $ S $ a finite set of primes of $K $ including the archimedean primes and $ G_{K,S} $ be the Galois group of the maximal extension of $K$ unramified outside $ S $. Assume ...

2
votes

1
answer

149
views

### Image of Kummer map for CM Elliptic curves

Let $K$ be an imaginary quadratic field and let $F$ be a finite extension of $K$. Let $E$ be an elliptic curve over $F$ with CM by $K$. Suppose that $p$ is a prime that splits as $p=\pi\pi^*$ in $K$. ...

5
votes

0
answers

179
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### Essence of relations between central simple algebras and Galois cohomology in canonical morphism of class field theory

I was somewhat puzzled after I finished learning class field theory for several times. My question is about the relations between "classical simple algebras, Brauer groups" and "modern ...

11
votes

1
answer

268
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### Galois cohomology class of a reductive group not coming from a torus

Let $G$ be a (connected) reductive group over a perfect field $k$, and let $\xi\in H^1(k,G)$ be a cohomology class.
By a theorem of Steinberg (Serre, Galois cohomology, Appendix 1 to Chapter III, ...

3
votes

0
answers

206
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### Relation between the Tate-Shafarevich group of a number field and the Tate-Shafarevich group of an elliptic curve

Let $E$ be an elliptic curve over a number field $K$, and let $sha^1(K,E)$ be the Tate-Shafarevich group, defined by:
Let $v$ be a valuation on $K$, and denote by $K_v$ the completion of $K$ by $v$, ...

3
votes

1
answer

227
views

### The torsion subgroup of the coinvariants for a $G$-module

Let $G$ be a finite group and $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
Consider the functor
$$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm ...

2
votes

0
answers

111
views

### Hasse principle for $H^2$ of a maximal torus of a connected quasisplit group?

Let $k$ be a number field and let $G$ be a quasisplit reductive algebraic group over $k$. Does there exist a maximal torus in $G$ such that the Hasse principle in dimension $2$ holds, i.e., such that ...

4
votes

0
answers

175
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### Alternative formulation of the Ferrero-Washington Theorem

The Ferrero-Washington theorem says that if $K/\mathbf{Q}$ is an abelian extension, then the cyclotomic $\mathbf{Z}_p$ extension $K^{\text{cyc}}/K$ has $\mu=0$.
In the paper "Iwasawa invariants ...

9
votes

0
answers

329
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### What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?

Consider the following statement:
If $K\subseteq L$ is a Galois extension of fields with Galois group $G$ and $x \in \mathbb{P}^n(L)$ is such that $\sigma(x)=x$ for all $\sigma\in G$, then $x \in \...

3
votes

1
answer

358
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### Galois cohomology of abelian varieties

Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action.
For the first Galois cohomology of $M$, ...

2
votes

0
answers

464
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### Confusion regarding Proposition 1.1 in Wiles's Fermat paper

This is from p. 459 of Wiles's Fermat paper.
Theorem: If $D_{p}$ is a decomposition group at $p$, $A$ is an Artinian local ring with maximal ideal $\mathfrak{m}$ and finite residue field $k$ of ...

3
votes

1
answer

223
views

### Proof of $V\cong \overline{K} \otimes_{K} V_K$ using $H^1(G_{\overline{K}/K},\operatorname{GL}_n(K))＝0$

This is from Silverman's book "The arithmetic of elliptic curves" (AEC), p.36, lemma 5.8.1.
Lemma 5.8.1 states
Let $V$ be a $\overline{K}$-vector space, and assume that
$G_{\overline{K}/K}$ ...

2
votes

1
answer

229
views

### Calculating the Galois cohomology group $H^1(K_v, \, E[p^{\infty}])$

Suppose $K$ is a number field and $E$ is an elliptic curve defined over $K$. My question is: how do you compute the local cohomology group $H^1(K_v, \, E[p^{\infty}])$?
As to why I'm asking this, it ...

5
votes

0
answers

237
views

### Exactness of a term after taking Pontryagin dual: a step in the proof of Poitou-Tate duality

I'm reading the proof of Poitou-Tate duality in the book Galois Cohomology and Class Field Theory by David Harari.
After some arguments, we get a exact sequence
$$
\mathbf{P}^1_S(k,M^{'})^* \...

5
votes

0
answers

365
views

### The Tate-Nakayama theorem and inflation

Let $K$ be a nonarchimedean local field,
and $L/K$ be a finite Galois extension with Galois group $G={\rm Gal}(L/K)$ of order $n=[L:K]$.
By local class field theory, there is a canonical isomorphism
$$...

5
votes

2
answers

661
views

### Embedding torsors of elliptic curves into projective space

Suppose I have a genus 1 curve $C$ over a field $k$. If $C$ has a point, then we can embed it into the projective plane by a Weierstrass equation. Now let us suppose that $C$ does not have a point (so ...

1
vote

0
answers

143
views

### Inflation in degrees $0$, $-1$, and $-2$ for Tate cohomology of finite groups

Let $\pi\colon G'\to G$ be a surjective homomorphism of finite groups, and let $A$ be a $G$-module.
I need explicit formulas for the inflation maps
$${\rm Inf}^{r}\colon H^{r}(G,A)\to H^{r}(G',A)$$ ...