5
votes
1answer
160 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
1answer
119 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 ...
8
votes
1answer
353 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 ...
7
votes
2answers
345 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 $K$, ...
5
votes
1answer
247 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 ...
10
votes
0answers
290 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 ...
4
votes
1answer
171 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 ...
0
votes
0answers
154 views

on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field. $F=k((t))$ and $\mathcal{O}=k[[t]]$. Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$. Let $E=k((t^{1/n}))$ with $n$ prime to the ...
6
votes
2answers
584 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 ...
4
votes
2answers
219 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 ...
2
votes
0answers
244 views

Where does the name Euler System come from?

I've recently been reading about Euler systems, and was curious where the name comes from. In particular, while the notion of an Euler system is still not rigorously defined, does the idea resemble ...
1
vote
1answer
223 views

Biprincipal spaces and their composition, following Serre's “Galois Cohomology”

When reviewing my notes and Serre's book "Galois Cohomology" Chapter 5 dealing with non-abelian group cohomology, I realized that I don't fully understand the concept of biprincipal spaces such as ...
1
vote
1answer
282 views

Isomorphism between the set of classes of Principal Homogeneous spaces and non-Abelian H^1(G,A) cohomology

Let A be a G-group, i.e. a set on which G acts on, has a group structure and satisfies $^s(xy)=^s x ^s y$ for all $x,y \in A \ , s \in G$. A homogeneous principal space P is a non-empty G-set on which ...
16
votes
1answer
716 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 ...
6
votes
1answer
948 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 ...
0
votes
1answer
382 views

Poitou-Tate dualities for Galois representations into power series rings?

Suppose $K$ is a finite extension of $\mathbf{Q}_p$, $A=K[[T_1,\dots,T_n]]$, $V$ a finite-rank free $A$-module, and $\rho:G_{\mathbf{Q}} \to \mathrm{GL}(V)$ a continuous Galois representation. Are ...
2
votes
2answers
328 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 ...
6
votes
1answer
568 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 $$ ...
19
votes
2answers
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} ...
3
votes
1answer
720 views

Galois cohomology groups given by étale cohomology

What are cases when Galois cohomology groups are given by ├ętale 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 = ...
11
votes
1answer
556 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 ...
7
votes
1answer
351 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 ...
6
votes
0answers
260 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 $X(\mathbf{R})$. ...
22
votes
1answer
926 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 ...
1
vote
1answer
542 views

Is the direct limit of Weil restriction of an elliptic curve a scheme?

In a discussion today on the Shafarevich-Tate group of an elliptic curve, the following structure and question came up. I will abuse many notations and be very vague about some things, but am very ...
11
votes
0answers
602 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 ...