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2
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0answers
34 views

Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
3
votes
1answer
70 views

Definition of Selmer-Group for Elliptic Curves

Im facing a problem in Silvermans Book "Arithmetic of elliptic Curves" at the beginning of chapter X.4 concerning the exact sequences. Let $K$ be a number field with a valutaion $v$. I'm ...
0
votes
0answers
18 views

First cohomology of direct product (in the coefficients)

Let $k$ be a field and let $G = A \times B$ be the product of two algebraic groups over $k$ ($G$ is not necessarily finite nor abelian). Is there a nice way to express $H^1(Gal(k^s/k), G(k^s))$ in ...
1
vote
1answer
26 views

a simple example of crossed homomorphism for the proof of Hilbert Theorem 90

For a field extension $K/F$ and a subgroup $G$ of $Aut(K)$, A crossed homomorphism is defined to be a function $f:G \rightarrow K^*$ satisfying $$f(\sigma\tau)= f(\sigma)\cdot \sigma(f(\tau)) $$ I ...
2
votes
1answer
57 views

(non Galois) correspondence

If $L/K$ is a field extension we note $\textrm{Aut}_K (L)$ the group of field automorphisms of $L$ that are fixing each element of $K$. Fix an extension $L/K$ and note $G:=\textrm{Aut}_K (L)$. If $H$ ...
3
votes
1answer
53 views

image of Kummer isomorphism

Let $K$ be a finite extension of the field of $p$-adic numbers which contains the $p$-roots of unity. There is an isomorphism $K^{\times} / (K^{\times})^p \to \mathrm{Hom}(\mathrm{Gal} (\bar{K} / K), ...
2
votes
1answer
65 views

Proof for a theorem on Cohomology by Tate

I am searching for a reference for the proof of the following theorem. Let $G$ be a finite group, let $C$ be a $G$-module, and let $u$ be an element of $\hat{H}^2(G,C)$. Assume that $\hat{H}^1(H,C) = ...
2
votes
0answers
39 views

Splitting varieties of two Galois cohomology symbols

One important property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$: For some $\alpha \in H^n(k,\mu_p)$ there is a ...
1
vote
0answers
36 views

Albert- Algebras and Traceforms

Im new to the topic so this could be basic nonsense to you. Any Albert-Algebra $A$ has a trace map $T:A \rightarrow k$ and thus one can assign a quadratic form $q_A$ of rank $27$ by setting $q_A(x) = ...
6
votes
2answers
157 views

Galois theory, had it solved any major problems beside its original applications to classical problems?

Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible ...
2
votes
0answers
40 views

Classification of quadrtic forms over Q_p

I need some one to recap the topic with me and correct me when i am wrong. There are basically just a few questions at the end,but its important to also show what i know and not just what i dont. ...
1
vote
1answer
83 views

Selmer and Shafarevich-Tate Groups

I'm currently trying to under the Selmer and Shafarevich-Tate Groups from Silverman's Arithmetic of Elliptic Curves (2nd edition), pg. 331 onwards. I have a couple of questions I think is derived from ...
2
votes
0answers
87 views

Studying Galois Cohomology from Category Theory.

I am a masters student with background in Category Theory - I also know some Topos Theory. I would like to ask if you think it would be possible to start studying Galois Cohomology without any ...
3
votes
0answers
45 views

$\operatorname{Br}(\Bbb Q_{47})$

I'm looking for division algebras over $\Bbb Q_{47}$. I guess my best bet is to calculate the Brauer group $\operatorname{Br}(\Bbb Q_{47})$. What's the best way of performing this calculation? Should ...
2
votes
0answers
72 views

References for Composition Law on Binary Quadratic Forms

What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois Cohomology? It is my understanding that there is a cohomological approach, and I am studying ...
1
vote
2answers
178 views

Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
1
vote
1answer
57 views

Explicit Kummer isomorphism

Let $K$ be a characteristic $0$ field containing $\mu_n$ (the $n$-th roots of unity). Then it known that the map $K^{\times} / (K^{\times})^n \to \mathrm{Hom}(G, \mu_n)$ which sends $x$ to $\sigma ...
0
votes
0answers
59 views

First cohomology of a Galois group with finite base field

Let $l/k$ be a (may be infinite) galois extension with galois group $G$ and $k$ a finite field with size $q$. Also $k$ and $l$ are given the discrete topology. $G$ is given the Krull topology. Then ...
3
votes
0answers
99 views

Dirichlet characters as coboundaries of Gauss sums

Let $p$ be a prime number, and consider a Dirichlet character $\chi : (\mathbf Z/p\mathbf Z)^\times \to \mathbf C^\times$. Its image lands in the group $\mu_{p-1}$ of $(p-1)$-st roots of unity. The ...
2
votes
0answers
39 views

Regarding splitting in the Brauer group over fields of prime characteristic.

Let $k$ be a field of characteristic $p$, and let $K=k^{1/p}$. We want to show : $[A]\in\rm Br(k)$ splits over $K$, i.e. $[A\otimes_k K]=1\in\rm Br(K)$ iff $p. [A]=1\in\rm Br(k)$. Here $[A]$ denotes ...
6
votes
0answers
77 views

Generalization of Kummer isomorphism?

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 of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
4
votes
0answers
87 views

Surjective map in Galois cohomology?

Let $p$ be a prime number, $\mathbb{F}_p$ the field with $p$ element and $\omega$ the mod $p$ cyclotomic character. Let $K$ be a finite extension of $\mathbb{Q}_p$ (the field of $p$-adic numbers) and ...
3
votes
0answers
48 views

computation of an $H^2$

Let $p$ be a prime number and $\mathbb{Q}_p$ the field of $p$-adic numbers. Let $G_p$ be the absolute Galois group of $\mathbb{Q}_p$ and fix an absolutely irreducible representation $$ \rho : G_p \to ...
0
votes
0answers
64 views

Group Cohomology Vs Profinite group Cohomology

What is the difference of the group cohomology and the profinite group cohomology? I think one reason is that in the profinite group situation, the G-module must be continuous. Does any other ...
6
votes
2answers
184 views

Motivation for the relations defining $H^1(G,A)$ for non-commutative cohomology

First let me review the definition of first non-commutative cohomology. Let $G$ be a group and $A$ a left $G$-group, i.e. for any $\sigma, \tau\in G$ and $a, b\in A$, one has ...
2
votes
1answer
186 views

Galois groups of maximal unramified extensions

I have been workin throught Cassels-Frohlich Algebraic number theory. Im looking at local fields and their cohomology, in particular the Brauer group. On page 133 of casses-frohlich they make the ...
3
votes
0answers
53 views

Why do the two topologies on a Galois group coincide?

In the following one is referred to the book. At this page, the author defines the Krull topology on a Galois group (not necessarily finite); on the 22nd page of the same book, the author defines the ...
2
votes
0answers
115 views

Question on $\operatorname{res}^G_U \circ \operatorname{cor}^U_G = N_{G/U}$

This is probably a very basic question, but I can't wrap my head around it. Given a normal open subgroup $U$ of a profinite group $G$ and a $G$-Module $A$ we have the following equation in group ...
4
votes
1answer
87 views

Local invariants of the discrete Galois module associated to a $p$-ordinary newform

Let $f=\sum_{n=1}^\infty a_nq^n$ be a $p$-ordinary newform of weight $k\geq 2$, level $N$, and character $\chi$, and let $\rho_f:G_\mathbf{Q}\rightarrow\mathrm{GL}_2(K_f)$ be the associated $p$-adic ...
10
votes
1answer
311 views

On Grunwald-Wang theorem

Consider (roughtly speaking) the following statements (the Grunwald-Wang theorem) Theorem 1 (see here for details Wiki): Let $K$ be a number field and $x \in K$. Then under some conditions : $x$ is ...
1
vote
1answer
121 views

Lifting additive characters

Let $K$ a finite extension of $\mathbb{Q}_p$ ($p$ prime different from 2) and let $G_K$ the absolute Galois group of $K$. Let $\bar{u} : G_K \longrightarrow \mathbb{F}_p$ a continuous additive ...
2
votes
2answers
298 views

Are absolute Galois groups compact topological groups

Let $T$ be a finite set of primes and let $K$ be the maximal extension of $\mathbf{Q}$ unramified outside $T$. We have three Galois groups: $G_{\mathbf{Q}} = ...
21
votes
0answers
296 views

Projective profinite groups

I'm reading the first chapter of Serre's Galois Cohomology. On p. 58, He gives two examples of projective profinite groups: the profinite completion of free (discrete) groups; the cartesian product ...
4
votes
0answers
129 views

Why has the category of all discrete $G$-modules not enough projectives when $G$ is profinite?

I found this article about Galois Cohomology. In it, it says that, when $G$ is a profinite group and $\mathbf{C}_G$ is the category of all discrete $G$-modules, $\mathbf{C}_G$ doesn't have enough ...
6
votes
0answers
181 views

Galois cohomology of Unitary groups

From Hilbert 90 (or, more precisely, a generalisation thereof), we know that the first (Galois) cohomology group of $GL_n$ is trivial, no matter the field of definition. However, for unitary groups, ...
10
votes
1answer
134 views

Which number fields allow higher genus curves with everywhere good reduction

The field of rational numbers is not such a number field. That is, there does not exist a smooth projective morphism $X\to\text{Spec } \mathbf{Z}$ such that the generic fibre is a curve of genus ...