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26
votes
3answers
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 in...)....
23
votes
1answer
2k 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} ...
23
votes
1answer
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 ...
22
votes
1answer
966 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
4answers
1k views

Hasse principle for rational times square

Does a Hasse principle hold for the property of being a rational times a square ? Let $a \in \mathbb{K}$ be an element of a number field. Assume that at every place $\mathbb{K}_v$ of $\mathbb{K}$, $a$...
16
votes
1answer
746 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
1answer
371 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
1answer
2k 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 $$H_{...
13
votes
3answers
454 views

Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module

Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite? This would complete the answer of Daniel Loughran. There is a ...
12
votes
2answers
863 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
1answer
726 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
1answer
614 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
2answers
797 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 ...
11
votes
1answer
201 views

A Galois extension over $\mathbb{Q}$ with Galois group $A_4$ and with cyclic decomposition groups

Does there exist a Galois extension $L/\mathbb{Q}$ with Galois group $A_4$ (the alternating group on four letters) such that all the decomposition groups are cyclic? This question is motivated by the ...
11
votes
1answer
423 views

Elements of arbitrary large order in the first Galois cohomology of an elliptic curve

Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$. In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...
11
votes
0answers
645 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
0answers
414 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 places....
9
votes
1answer
250 views

Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$. Is the set of $S$-isomorphism classes of $G$-torsors ...
9
votes
1answer
688 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 $$ K_{2n-1}(\...
8
votes
2answers
392 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
1answer
425 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 $H^1(G,...
8
votes
1answer
304 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 $...
7
votes
2answers
457 views

Galois cohomologies of an elliptic curve

I asked this question at math stackexchange but did not get any answer and I was suggested to post the question here. I am studying basic theory of elliptic curves. I encountered Galois cohomology. ...
7
votes
1answer
355 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
2answers
407 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 $K$, ...
7
votes
1answer
258 views

Weight filtration on certain Galois representations

Let $G$ be the absolute Galois group of a number field $K$. Let $\ell$ be a prime number. There are representations $\mathbb{Z}_\ell(n)$ of $G$ on the group of $\ell$-adic integers given by the ...
7
votes
1answer
313 views

A vanishing condition for cup products in Galois cohomology

Let $k$ be a field of characteristic $\neq 2$. For a non-zero element $a \in k^*$, let us write $[a] \in H^1(k,\mathbb{Z}/2)$ for the Galois cohomology class corresponding to the quadratic extension $...
7
votes
1answer
1k 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 ...
6
votes
2answers
411 views

First Galois cohomology of Weil restriction of $\mathbb{G}_m$

Let $L/K$ be a finite Galois extension, write $G:= Gal(L/K)$. Denote by $R = Res(\mathbb{G}_m)$ the Weil restriction of $\mathbb{G}_m$, from $L$ to $K$. I want to show that its first Galois cohomology ...
6
votes
1answer
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
1answer
313 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
1answer
454 views

Non-vanishing of the Tate-Shafarevich kernel in group cohomology

Let $G$ be a finite group. Let $M$ be a finite $G$-module (a finite abelian group with an action of $G$). We consider a special kind of $G$-modules; in particular, our $M$ is a finite dimensional ...
6
votes
1answer
64 views

Bilinearity of the Cassels-Tate pairing

Let $K$ be a number field and let $A$ be an abelian variety over $K$ (I'm mostly interested in the case that $A$ is an elliptic curve). We use $v$ to denote places of $K$ and we write $H^i(k, A)$ for ...
6
votes
2answers
686 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
0answers
117 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
0answers
329 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
1answer
294 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
1answer
144 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
1answer
208 views

Constructing groups of Type E7 with certain Tits Index

In a new survey on $E_8$, namely Skip Garibaldi - E8 the most exceptional group , the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-...
5
votes
1answer
179 views

Applications of the Galois embedding problem

Given a finite Galois extension of number fields $L/K$ with Galois group $G$ and a surjection $E\twoheadrightarrow G$ of finite groups, the Galois embedding problem is the question of whether there ...
5
votes
1answer
187 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 $I^n(...
5
votes
1answer
993 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
1answer
292 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
1answer
807 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
1answer
269 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 ...
5
votes
1answer
309 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 $H^i(k,M)$ ...
4
votes
2answers
624 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
2answers
250 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
1answer
242 views

Lifting torsors in characteristic $p$ to characteristic zero

Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with ...
4
votes
1answer
207 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 ...