The galois-cohomology tag has no wiki summary.

**26**

votes

**3**answers

2k views

### Why aren't there more classifying spaces in number theory?

Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested ...

**22**

votes

**1**answer

946 views

### Does the everywhere unramified extension of Q(mu_37) of degree 37 grow into a Z_37-extension?

Let $p=37$. Since $p$ divides the numerator of $B_{32}$, by Ribet's proof of the converse of Herbrand's theorem, we know that the class group of ${\bf Q}(\mu_p)$ has size divisible by $p$. More ...

**21**

votes

**1**answer

2k views

### Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...

**20**

votes

**1**answer

1k views

### How is etale cohomology of integer rings related to Galois cohomology?

In the paper of Bloch and Kato in the Grothendieck Festschrift, and some other papers relating to the Bloch-Kato conjecture and the ETNC, the cohomology groups
$H^i_{\mathrm{et}}(\operatorname{Spec} ...

**16**

votes

**1**answer

731 views

### Leopoldt's conjecture and cup-products

Among the many equivalent formulations of Leopoldt's conjecture, this one is probably the shortest: For any number field $K$, prime number $p$, finite set $S$ of primes of $K$ containing the primes ...

**15**

votes

**1**answer

333 views

### Computability of Brauer groups

A friend of mine and I were talking about computable algebra, and this question came up. The answer may already be known, but I couldn't find it with Google:
Suppose I have a countable field, $k$. ...

**14**

votes

**1**answer

1k views

### Are all Galois cohomology groups also étale cohomology groups?

Let $K$ be a field and $K^s$ a separable closure of $K$, and let $\mathcal{F}$ be a sheaf on $\mathrm{Spec}(K)$ (in the étale topology).
By Grothendieck's Galois Theory, we have the isomorphism
...

**12**

votes

**2**answers

819 views

### Compatibility of Bloch-Kato and Beilinson-Bloch

Suppose $V/K$ is a smooth projective variety. Let $\mathrm{Ch}^{j}(V)_{0}$
be the group of codimension-$j$ homologically trivial $K$-rational cycles on $V$, modulo rational equivalence. A conjecture ...

**12**

votes

**1**answer

628 views

### Explicit Bijection between Central Simple Algebras and twists of $\mathbb P^n$

The automorphism group of the algebra of $n$-dimensional matrices over a field $K$ is $PGL_n(K)$. The automorphism group of $n-1$-dimensional projective space over $K$ is also $PGL_n(K)$. Therefore, ...

**12**

votes

**1**answer

593 views

### Is there a canonical height on the Weil-Chatelet group?

Jon Hanke and I were just chatting and realized we didn't know the answer to the following question. If E is an elliptic curve over a number field, is there in any sense a "canonical height" on the ...

**11**

votes

**0**answers

620 views

### What is the Shafarevich-Tate group of GL(2)?

Let $K$ be a number field, $\bar{K}$ a fixed algebraic closure, $G_K$ the absolute galois group, $O_\bar{K}$ the ring of integers of the algebraic closure, $\mathfrak{p}$ runs over prime ideals and ...

**10**

votes

**2**answers

734 views

### Hilbert 90 for algebras

Let $L\diagup K$ be a Galois extension of fields satisfying $\left[L:K\right] < \infty$. Let $B$ be a finite-dimensional (as a $K$-vector space) $K$-algebra. Then, the Galois group $G$ of $L\diagup ...

**10**

votes

**0**answers

347 views

### Unramified Galois cohomology of number fields

I'm trying to understand unramified Galois cohomology of number fields a bit better.
Set-up: Let $k$ be a number field and $S$ a finite set of places of $k$ which contains all the archimedean ...

**8**

votes

**2**answers

372 views

### “Forms” of quadrics

The theory of Severi-Brauer varieties is well-known. Let $k$ be a (perfect) field. There may exist varieties not isomorphic to $\mathbf{P}^n$ over $k$, which are isomorphic to $\mathbf{P}^n$ over ...

**8**

votes

**1**answer

396 views

### Variant of Hilbert 90 for Galois extensions

Let $K/\mathbb F_q(x)$ be a finite Galois extension with Galois group $G$. Let $Aut(K)$ be the group of $\mathbb F_q$-automorphisms of $K$.
Obviously, $G\subseteq Aut(K)$. It is well known that
...

**8**

votes

**1**answer

247 views

### When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For ...

**8**

votes

**1**answer

626 views

### Galois descent for K-groups (or for étale cohomology groups)

Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$.
When is
$$
...

**7**

votes

**1**answer

354 views

### Is H^2(W_p,C^times) well-known?

Let $W_p$ be the Weil group of $\mathbf{Q}_p$. What is the Galois cohomology group $H^2(W_p,\mathbf{C}^{\times} )$ (with trivial action)? Is it zero, or something huge and complicated?
(This group ...

**7**

votes

**2**answers

392 views

### Forms of algebraic varieties

Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, ...

**6**

votes

**1**answer

1k views

### p-adic Hodge theory for varieties defined over \C _p ?

I have a question on p-adic Hodge theory:
When e.g. $X$ is a smooth proper scheme over a finite extension $K$ of $\mathbf{Q}_{p}$ then e.g. one variant of $p$-adic Hodge theory says that there is a ...

**6**

votes

**1**answer

279 views

### Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...

**6**

votes

**2**answers

642 views

### Peu or très ramifiée extension

Let $p$ be a prime number. Let $\mathbb{F}$ be a finite extension of $\mathbb{F}_p$. Let $\omega$ be the mod $p$ cyclotomic character and let $V$ be a representation of $G_{p} = Gal(\bar{\mathbb{Q}}_p ...

**6**

votes

**0**answers

106 views

### References for Gauss Composition using Galois Cohomology

Note: I have already posted this on stackexchange, but have not yet gotten a response.
What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois ...

**6**

votes

**0**answers

290 views

### Real approximation for homogeneous spaces of linear algebraic groups

Let $X$ be a smooth geometrically integral variety over $\mathbf{Q}$, having a $\mathbf{Q}$-point.
We say that $X$ has the real approximation property if $X(\mathbf{Q})$ is dense in $X(\mathbf{R})$.
...

**5**

votes

**1**answer

272 views

### rationality question while dealing with an isogeny

I don't think that the following is known, but before going to other things, I would like to know what can be said about it. Thanks in advance for any relevant comment !
So here is the situation. Let ...

**5**

votes

**1**answer

138 views

### $H^1$ and fractional ideals group

Let $L/K$ be a Galois extension with Galois group and $\mathfrak p$ be a prime of the ring of integers $\mathcal O_K$.
I would like to prove that $H^1(G, I_{\mathfrak p})=1$ where $I_{\mathfrak p}$ is ...

**5**

votes

**1**answer

183 views

### Rank four quadratic Form with non trivial discriminant in I(k)

Im sure this is a beginners question.
Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k).
The Arason-Pfister-Hauptsatz states:
"If $\varphi$ is any anisotropic class in ...

**5**

votes

**1**answer

856 views

### What kind of structures allow Galois descent?

EDIT: Question solved.
Let me explain what I mean.
The classical formulation of Galois descent, e. g. in Crawley-Boevey's "Cohomology and central simple algebras", uses the following notion:
...

**5**

votes

**1**answer

276 views

### Open subgroups of the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$

Let $G$ be the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$. Then $G$ is isomorphic to a semidirect product of $\widehat {\mathbb Z}(1)$ by $ Gal_\mathbb Q$.
Is it true that ...

**5**

votes

**1**answer

788 views

### Galois groups via cohomology

I would like to know about references for the following result (point 3):
Let $K/k$ be a normal extension (I am interested in number fields, but everything should work in fields of characteristic ...

**5**

votes

**1**answer

308 views

### Equality of Galois modules

Let $k$ be a number field. Let $M$ be a (continuous) $\text{Gal}(\overline{k}/k)$-module.
One can define two subgroups of the Galois cohomology group $H^i(k,M)$:
the group of elements of $H^i(k,M)$ ...

**5**

votes

**1**answer

748 views

### Relations between the cohomology of discrete groups and of profinite groups

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...

**4**

votes

**2**answers

606 views

### Galois cohomology H^1(Q_p, Z_p(2)) = 0?

For Tate twists Z_p(2), which is defined by the projective limit of
\mu_{p^m}(2) over all m>0, I would like to calculate H^1(Q_p, Z_p(2)).
I guess this is zero, but cannot prove it.
Is it possible ...

**4**

votes

**2**answers

236 views

### Generalization of Kummer isomorphism?

This is a question I asked on math.stackexchange without success.
Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action ...

**4**

votes

**1**answer

189 views

### What is the interpretation of this galois cohomology set?

Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$
The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are ...

**4**

votes

**2**answers

277 views

### Projectives in the category of discrete G-modules

If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5).
Since the cited ...

**4**

votes

**1**answer

233 views

### Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...

**4**

votes

**1**answer

166 views

### Twists of projective automorphisms

Let $X$ be a projective variety over a perfect field $k$. Recall that a twist of $X$ is a variety $Y$ over $k$ such that $$X_{\bar k} \cong Y_{\bar k}.$$
The twists of $X$ are classified by the Galois ...

**4**

votes

**0**answers

175 views

### geometric interpretation of the transgression map

Let $X$ be an algebraic variety over an algebraically closed field $k$ and let a finite group $G$ act on it so that it acts freely on the generic fibre of the projection $X \to X/G$, so ...

**3**

votes

**1**answer

196 views

### What's the minimum number of generators for the wild inertia?

Suppose $K$ is a finite extension of $\mathbb{Q}_p$ and $K^{nr}$ the maximal unramified extension of $K$ in some fixed algebraic closure. Let $G_K$ be the absolute Galois group of $K$ and let $I_w$ be ...

**3**

votes

**2**answers

290 views

### Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?

**3**

votes

**1**answer

241 views

### The cardinality of first non-abelian Galois cohomology

Let $G$ be a linear algebraic group over a non-archimedean local field $F$. Let $H^1(F,G)$ be the first non-abelian Galois cohomology. It is known that when $F$ is of characteristic 0, i.e. finite ...

**3**

votes

**1**answer

759 views

### Galois cohomology groups given by étale cohomology

What are cases when Galois cohomology groups are given by étale cohomology?
Example: $S = Spec(K)$ the spectrum of a field, $F \in Sh(K)$, then $H^p(K, F) = H^p(G_K, F_{\bar{K}})$.
What if $G = ...

**3**

votes

**2**answers

633 views

### Restriction of scalars of simple algebraic groups

I'm trying to understand the following basic property of the restriction of scalars:
Given an absolutely simple algebraic groups $G$ defined over a number field $k$, are there at most finitely (up-to ...

**3**

votes

**1**answer

116 views

### Classification of 3-forms in dimension 7

I'm looking for a classification of $3$-forms over a real vector space of dimension $7$ as for the $3$-forms in dimension $6$. References on the latter case are R. Bryant On the geometry of almost ...

**3**

votes

**1**answer

214 views

### A question on the cohomology of elliptic curves over local fields

Let $K$ be a number field,$\nu$ a nonarchimedian prime of $K$, $K_{\nu} $ the completion of $K $ at $\nu $ with maximal unramified extension $K_{\nu}^{unr} $. Let $E $ be an elliptic curve defined ...

**3**

votes

**1**answer

117 views

### On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants

In these Lecture Notes http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned:
Is the map
$g_3 ...

**3**

votes

**1**answer

284 views

### Langlands Paper on representations of abelian algebraic groups

I have been working through Langlands paper which you can see here http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf and I can understand why one of his maps is obvious and ...

**3**

votes

**1**answer

82 views

### Set of isomorphisms of Pfister forms corresponding to first cohomology of algebraic group

Assume $k_0$ is a field with char($k_0$) not $2$. Let us define functors from $\rm Field_{/k_0}\to \rm Sets$ as $\rm Pfister_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$;
...

**2**

votes

**2**answers

333 views

### Why does $H^1(G_p/I_p,\mathbb{F}(\delta\epsilon^{-1}))$ vanish?

For a finite field $\mathbb{F}$ of char $p$ $\geq 2$, let $f$ be a normalised eigenform of weight $k\geq 2$ that is ordinary at $p$ and $\overline{\rho}_f$ be the mod $p$ Gal representation attached ...