Questions tagged [absolute-galois-group]
The absolute-galois-group tag has no usage guidance.
56
questions
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262 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 ...
2
votes
0answers
190 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 ...
1
vote
0answers
92 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
1answer
173 views
$G_{\mathbb Q}$ and primes of $\overline{\mathbb{Z}}$
I know that if $K/\mathbb Q$ is a finite Galois extension (i.e. a Galois number field), then for any prime $(p)\subseteq \mathbb Z$, the Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ acts ...
8
votes
1answer
252 views
Commutator subgroup of the absolute Galois group - a closed subgroup
Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible that the commutator subgroup of the absolute Galois group of $K$ (considered as an abstract group) is a closed subgroup? This property ...
11
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0answers
208 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 ...
1
vote
2answers
267 views
Meaning of epimorphism from full Galois group to some group
My problem has two parts: let $\;G:=\operatorname{Gal}(\overline{\Bbb Q}/\Bbb Q)\;$ be the full Galois group of the rationals and $\;K\;$ be some finite group, then:
(1) Does having an epimorphism (...
7
votes
0answers
212 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 ...
2
votes
1answer
170 views
maximal pro-l-quotients of absolute Galois groups
Let $K$ be a field, preferably a function field of a variety $X$ over $\overline{\mathbb{F}}_p$. I am looking for an answer or existing literature on the following question:
What is known about the ...
6
votes
1answer
463 views
Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that ...
7
votes
1answer
394 views
Recovering the Zariski topology from the Zariski topology over an extension
Suppose $A$ is a $k$-algebra, with $k$ a field, and let $\ell$ be a field extension of $k$. Is there an easy way to see/recover $\mathrm{Spec}(A)$ in/from $\mathrm{Spec}(A \otimes_k \ell)$, using the ...
7
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1answer
413 views
absolute Galois group of the function field of a curve over $\mathbb{F}_p^{alg}$
In their 2008 paper "Torelli theorem for curves over finite fields" Bogomolov, Korotiaev and Tschinkel mention in the beginning of Section 9 that absolute Galois groups of curves over $\mathbb{F}_p^{...
7
votes
1answer
682 views
Teichmuller groupoids in Grothendieck's esquisse d'un programme
Grothendieck in his Esquisse d'un programme mentioned without any precise definition and construction that the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the whole "tower" ...
6
votes
1answer
422 views
Is co-restriction in Galois cohomology in fact the norm map via Kummer isomorphism?
Let $\mathrm{F}$ be a field that contains a root of unity of order $p$, where $p$ is a prime number. Fix an element $a$ such that $a \in \mathrm{F}$ and $\sqrt[p]{a} \notin \mathrm{F}$. Consider the ...
9
votes
1answer
648 views
Classify all the fields with abelian absolute Galois group
I'm wondering if anyone has classified all the fields $K$ such that $Gal(\bar{K}/K)$ is abelian?
The only examples I'm aware of are: finite fields, the real numbers $\mathbb{R}$ and $k((T))$ where $k$...
13
votes
1answer
773 views
What “should” be the absolute galois group of a field with one element
As far as I know there is many "suggestions" of what should be a "field with one element" $\mathbf{F}_{1}$.
My question is the following:
How we should think or what should be the "absolute Galois ...
3
votes
1answer
209 views
induced isomorphism in continuous cohomology
Suppose that we have a morphism between profinite groups $f: G_{1}\rightarrow G_{2}$ such that $f^{\ast}:H_{cont}^{\ast}(G_{2},A)\rightarrow H_{cont}^{\ast}(G_{1},A) $ is an isomorphism for any finite ...
28
votes
3answers
988 views
Consequences of Shafarevich conjecture
The Shafarevich conjecture states that the Galois group $\mathrm{Gal}({\overline{\mathbf{Q}}/\mathbf{Q}^{ab}})$ is a free profinite group, where $\mathbf{Q}^{ab}$ is the maximal abelian extension of $\...
1
vote
1answer
154 views
Another fix field of a certain galois group action
Let $E=\mathbb{F}_p(\!(u)\!)$, the Laurent series field over $\mathbb{F}_p$. Let $K/E$ be a finite normal separable extension. Consider the field $L=K(x \mid x^p-x-a=0 \text{ for some } a \in K)$. Let ...
3
votes
1answer
192 views
Fix field of a certain galois group action
Let $E= \mathbb{F}_p(\!(u)\!)$, $E^s$ a separable closure of $E$ and write $G_E= \mathrm{Gal}(E^s/E)$ for the absolute Galois group of $E$. Take a lift of the $u$-adic valuation on $E$ to $E^s$ and ...
7
votes
2answers
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Langlands program vs Shimura-Taniyama-Weil conjecture
Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves"
I hope I'm not distorting his phrase, can someone ...
1
vote
2answers
936 views
Is the absolute Galois group the same as the automorphism group? [closed]
Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of $\...
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votes
3answers
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Profinite groups as absolute Galois groups
It is a well-known result that all profinite groups arise as the Galois group of some field extension.
What profinite groups are the absolute Galois group
$\mathrm{Gal}(\overline{K}|K)$ of some ...
7
votes
1answer
339 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 ...
11
votes
1answer
701 views
Dessins d'enfants and absolute Galois group
I would like to know what is the recent progress about the group homomorphism
$$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$
$\mathrm{Gal}(\overline{\mathbf{...
2
votes
0answers
102 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 ...
5
votes
1answer
2k views
Unramified extension of number fields
Any finite field extension (in particular Galois extension) of $\mathbb{Q}$ is ramified. Is there an intuitive geometric explanation of this fact?
Suppose we have an number field $K$, is any Galois ...
6
votes
2answers
1k views
Frobenius elements in infinite extensions
Let $K$ be a number field, $\bar K$ an algebraic closure and $G$ the associated absolute Galois group. How can I define the Frobenius elements of $G$ or at least their conjugacy class?
I know how ...
0
votes
1answer
190 views
Fixed field of the Nebentypus of a newform for $\Gamma_1(N)$
Let $f=\sum_{n\geq 1}\in S_2(\Gamma_1(N),\varepsilon)$ be a normalized newform without CM and with Nebentypus $\varepsilon$. Let $L=\mathbb Q(a_n\colon n\in \mathbb N)$ be the number field generated ...
15
votes
2answers
661 views
Galois group for 0-dimensional motives
It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference or two to help me get some things straightened out.
One can ...
3
votes
0answers
966 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, ...
2
votes
2answers
532 views
The relationship between SL(2,Z) and Gal(Qbar,Q)
(caveat: I'm not a number-theorist or Langlands-programme-er, and I don't expect to understand all the answers to this question, but I figured they might be useful to someone besides me).
I've been ...
4
votes
1answer
139 views
Galois action on ultrapowers
Let $K$ be a char $0$ field with algebraic closure $\bar K$ and absolute Galois group $G$. Let $\mathcal U$ be an ultrafilter on $\mathbb N$ and $F=\bar K^\mathbb N/\mathcal U$ be the ultrapower of $\...
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0answers
295 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\...
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votes
2answers
917 views
Is it known if the absolute Galois group is “divisible”?
The definitions of a divisible group that I have seen all seem to assume abelian is an a priori property of the group. My question is as to whether or not it is known that--given a non-torsion element ...
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0answers
645 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 ...
8
votes
1answer
777 views
Absolute Galois group of the field of Puiseux series over $\overline{\mathbb{F}}_p$
Let $K$ be the field of Puiseux series with coefficients in $\overline{\mathbb{F}}_p$ (the algebraic closure of the field with $p$ elements).
What is the absolute Galois group of $K$?
Thank you to ...
2
votes
2answers
812 views
Place stabilizers for the absolute Galois Group
Fix an algebraic closure, $\overline{\mathbb{Q}}$ for the rationals and consider the set, $B_p$, of all places of $\overline{\mathbb{Q}}$ over a fixed (possibly infinite) prime, $p$, of $\mathbb{Q}$. ...
3
votes
2answers
863 views
Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?
If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
5
votes
2answers
1k views
non-continuous inverse Galois problem
Let $G=Gal(\bar{\mathbf{Q}}/\mathbf{Q})$ be the absolute Galois group over $\mathbf{Q}$.
Q1: Is it possible to find a (necessarily non-closed) normal subgroup $K\leq G$ such that
$G/K$ is free of ...
23
votes
3answers
3k views
Subgroups of GL(2,q)
I have been wondering about something for a while now, and the simplest incarnation of it is the following question:
Find a finite group that is not a subgroup of any $GL_2(q)$.
Here, $GL_2(q)$ is ...
1
vote
1answer
1k views
What does Gal(Q_p/Q) mean? [closed]
What does
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$ mean? ($p$ is a prime number.)
If it is defined as $\mathrm{Aut}(\overline{\mathbb{Q}}_{p}/\mathbb{Q})$, then does it have any property ...
38
votes
3answers
3k views
On what kind of objects do the Galois groups act?
I am neither number theorist nor algebraic geometer. I am wondering
whether Galois groups of number fields (say the absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which
...
14
votes
1answer
1k views
Is the etale fundamental group of Spec(Z)\{p_1,…,p_n} finitely presented?
(of course not, it's usually uncountable; I really mean is it the profinite completion of a finitely presented group).
By definition, $\pi_1^{\operatorname{et}}(\operatorname{Spec}(\mathbb Z)\...
21
votes
0answers
1k 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 ...
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0answers
760 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^...
22
votes
1answer
2k views
Are class numbers encoded in the absolute Galois group of ${\mathbb Q}$?
The absolute Galois group $G_{\mathbb Q}=\text{Gal}(\bar{\mathbb Q}/\mathbb Q)$, as a profinite group, encodes a lot of things: the whole lattice of number fields (closed subgroups of finite index), ...
11
votes
1answer
2k views
Maximal extension almost everywhere unramified and totally split at one place
Fix a finite set of primes $S$ and an additional prime $p$. Let $K$ be the maximal extension of $\mathbb{Q}$ that is unramified outside $S$ and $\infty$ and totally split at $p$. Is the extension $K$ ...
14
votes
1answer
1k views
What can we say about center of rational absolute Galois group?
Well the question is in the title.
I asked myself this question while thinking about something in Grothendieck-Teichmüller theory. I guess class field theory gives some insight into this, or I am ...
7
votes
2answers
2k views
Conjugacy classes in the absolute galois group
We consider $G_{\mathbb Q} = Gal(\mathbb {\bar Q}/\mathbb Q)$. The Frobenius elements corresponding to each prime are well-studied. But these are really not elements; these are only defined as some ...
