Questions tagged [absolute-galois-group]
The absolute-galois-group tag has no usage guidance.
23
questions with no upvoted or accepted answers
30
votes
0
answers
2k
views
What do dessins tell us about the absolute Galois group?
I have sometimes seen it asserted that one manifestiation of how complicated the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is is that one can not "pin down" any single ...
13
votes
0
answers
235
views
Dessins d'enfant of Dynkin diagrams?
Dessins d'enfant have a nice particular case of Shabat trees, where we take a tree, bicolor it, and get a polynomial map.
A very famous set of trees are the Dynkin diagrams. I wonder what are the ...
11
votes
0
answers
235
views
Grothendieck-Teichmüller conjecture and tropicalization of moduli of curves
Abramovich, Caporaso and Payne (2014) have constructed functorial tropicalization maps from the Berkovich analytification of the moduli spaces of stable curves, $\overline{M}_{g,n}$, to the moduli ...
10
votes
0
answers
220
views
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 ...
7
votes
0
answers
263
views
Connections between Borger's absolute geometry and Connes' and Consani's $\Gamma$-spaces
As the idea of an absolute geometry over the field with one element $\mathbb{F}_1$ becomes more clear, two approaches seem to have crystallized, being based on different assumptions and going into ...
7
votes
0
answers
197
views
Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$
$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...
7
votes
0
answers
249
views
Is this Related to Tannakian Formalism?
I am wondering how I might be able to express the following phenomenon, which is essentially equivalent to Artin's linear independence of characters, in Tannakian formalism. Any help would be much ...
6
votes
0
answers
771
views
field automorphism action on $Ext^1(\mathbb{C}^*,\mathbb{Z})$ or $Ext^1(\bar{\mathbb Q}^\star,\mathbb{Z})$
I am interested in the action of field automorphism group $Aut(\mathbb{C}/\mathbb{Q})$ on
$Ext^1(\mathbb{C}^{\star},\mathbb{Z})$ or $Ext^1(\mathbb{\bar Q}^*,\mathbb{Z})$, and, more generally, on
$Ext^...
5
votes
0
answers
222
views
Is the group $\mathrm{Gal}(\mathbb{C}/\overline{\mathbb{Q}})$ known?
The automorphism group of the complex numbers $\mathbb{C}$ and the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ are amongst the most mysterious and worst understood objects in Galois ...
4
votes
0
answers
790
views
Haar measure on Galois groups
Galois groups are nice compact Hausdorff groups, and therefore possess a bounded Haar measure, unique if we insist that the total volume be $1$. What is the Haar measure on the absolute Galois group ...
3
votes
0
answers
214
views
Dessins d'enfants and the absolute Galois group
If I am not mistaken, Alexander Grothendieck introduced dessins d'enfants as they are known today, in order to better understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}...
3
votes
0
answers
177
views
The Brauer group and the second Galois cohomology group
I know that for any finite Galois extension K of $\mathbb{Q}$ the group $H^2(Gal(K/\mathbb{Q}),K^*)$ is isomorphic to the Brauer group $Br(K/\mathbb{Q})$. The isomorphism goes as follows: to a 2-...
3
votes
0
answers
425
views
How are problems about number fields reduced to problems about their absolute Galois groups?
The article on Wikipedia about Neukirch–Uchida theorem claims right from the beginning the statement in my question. I have seen similar claims elsewhere before.
I am a little puzzled by this ...
3
votes
0
answers
1k
views
English version of "Quasi-Hopf Algebras"
I was wondering where I can find a pdf of Drinfeld's paper "Quasi-Hopf Algebras," which formulated the Grothendieck-Teichmuller group. The Russian version is in Algebra i Analiz, 1:6 (1989), 114–148, ...
3
votes
0
answers
797
views
Children's drawings and Seiberg-Witten curves
This physics (bear with me for a while) paper seems to say something about Gal \bar Q/Q:
Children's Drawings From Seiberg-Witten Curves, hep-th/061108.
Let's ...
2
votes
0
answers
106
views
Absolute Galois cohomology of function fields (of high-dimensional) varieties
What is known about the absolute Galois cohomology of function fields of varieties of dimension 2 or larger? Specifically, I am interested in multiplicative coefficients $\mathbb G_m$.
I have seen ...
2
votes
0
answers
112
views
Elementary abelian 2-subgroups of $\mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$ (with and without choice)
Consider the absolute Galois group $G_{\mathbb{Q}} := \mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$.
As I understand it, the only torsion elements have order $2$ (by Artin-Schreier), and they are ...
2
votes
0
answers
246
views
Is the absolute Galois group $\text{Gal}(\bar K/K)$ isomorphic to $\text{Gal}(K(S)/K)$?
Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, maximal ideal $\mathfrak{m}$ and uniformizer $\pi$. Let $\bar K$ be the algebraic closure of $K$ and $\bar{\...
2
votes
0
answers
264
views
Absolute Galois group of Q and stratification of moduli space of curves
This is slightly related, but distinct from, a question I asked earlier.
The moduli space of ribbon graphs with metric (with all vertices having degree at least 3) is isomorphic to the moduli space of ...
2
votes
0
answers
107
views
Which fields have no extensions of degree divisble by a fixed prime?
Let $p$ be a prime. What are the most general examples of a field $K$ such that for any finite extension $L/K$ the degree $[L:K]$ is prime to $p$?
Certainly, there are algebraically closed examples ...
1
vote
0
answers
195
views
Is semi-simplicity of Galois representations local?
Let $\rho:G_{\mathbb{Q}}\rightarrow \text{Gl}(V)$ be a finite dimensional $\ell$-adic Galois representation. Then for each prime, by pre-composing $\rho$ with the natural inclusion $G_{\mathbb{Q}_p}\...
1
vote
0
answers
321
views
Extending systems of l-adic representations to other l
I'm asking this not because I have an idea how one might approach it, but because it seems natural and inherently interesting.
Let $K$ be a number field, $G_K$ its absolute Galois group, and $\ell\...
0
votes
0
answers
77
views
Elliptic curves and images of decompositions group exceptional?
Given an elliptic curve $E$ and its mod $p$ Galois representation $\bar{\rho}_{E,p}$, I am wondering what are the possibilities for $\bar{\rho}_{E,p}(G_l)$, where $G_l:=$Gal($\overline{\mathbb{Q}_l}/\...