# Questions tagged [anabelian-geometry]

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32
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### Neukirch's theorem on absolute Galois groups in English [duplicate]

Is there a paper or book available in English that proves the result of Neukirch on absolute Galois groups of number fields? I'm having a hell of a time with the German originals.

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**1**answer

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### Anabelian geometry ~ higher category theory

Note: I'm worried this question might be taken as controversial, because it relates to Shinichi Mochizuki’s work on the abc conjecture. However, my question has nothing to do with the correctness of ...

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353 views

### Down to earth, intuition behind a Anabelian group [closed]

An anabelian group is a group that is “far from being an abelian group” in a precise sense: It is a non-trivial group for which every finite index subgroup has trivial center.
I would like to know ...

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### Inter-Universal Teichmuller Theory and the Field with One Element

The idea of the "field with one element", or $\mathbb{F}_{1}$, is supposed to allow us to do for number fields what we can do for function fields. Hence this idea often comes up regarding problems ...

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**3**answers

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### How are motives related to anabelian geometry and Galois-Teichmuller theory?

In Recoltes et Semailles, Grothendieck remarks that the theory of motives is related to anabelian geometry and Galois-Teichmuller theory. My understanding of these subjects is not very solid at this ...

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### Reconstruction of hyperbolic curves using the fundamental group

In the paper Curves and their Fundamental Groups written by Gerd Faltings, Mochizuki's proof of Grothendick's conjecture on anabelian curves is explained.
In the proof, he shows that for two ...

**5**

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**1**answer

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### Relation - Anabelian geometry and Tate conjecture

A lot has been said about the relation between Anabelian algebraic geometry and Mordell conjecture.
I would like to know what is the relation between Anabelian algebraic geometry and Tate ...

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### reference - Grothendieck on Thurston's work

In his 'dernières' years Grothendieck gets "interested" in Thurston's work.
"[...] je me suis intéressé ces dernières années - la géométrie hyperbolique à la Thurston et ses relations au groupe de ...

**4**

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**1**answer

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### Conjugacy-invariance of sections of etale homotopy exact sequence

My questions arise on page xiv of Stix's Rational Points and Arithmetic of Fundamental Groups. Here is an excerpt:
Given a geometrically connected variety $X/k$, a fixed separable closure $\bar{k}/k$,...

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### Status of the “anabelian dream” ($\mathrm{dim} \leq 1$)

The anabelian conjectures for small dimensions have been known for quite some time. In full generality the results are:
Dimension 0. Finitely generated fields are anabelian (Pop)
Dimension 1. ...

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719 views

### Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$

I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)=\mathrm{Aut}\underline\pi^{temp}_{{\mathbb{C}}_p}$), ...

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### Grothendieck's “La longue Marche à travers la théorie de Galois”

It seems that Grothendieck's familly has given permission for the distribution of his unpublished works, so I hope it is ok to ask this.
Is there any way to obtain a copy (online or not) of "La ...

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**2**answers

9k views

### A road to inter-universal Teichmuller theory

What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian ...

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**1**answer

543 views

### Is $\mathcal M _{g,n}$ anabelian?

Are the moduli spaces $\mathcal M _{g,n}$ expected to be anabelian? Is there anything known in that direction?
Thank you very much for your help in any case!

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### Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?

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### What is an étale theta function?

Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, ...

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### abelian $\ell$-adic representations in $\widehat{SL(2,Z)}$

In Grothendieck's Esquisse he claims that the action of
$$\text{Gal}(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{SL(2,Z)})$$
obtained from the homotopy exact sequence of the étale ...

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### Why is the section conjecture important?

As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...

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### Applications of anabelian geometry to Galois representations?

One aspect of anabelian geometry is the study of the action of the absolute Galois group of a field $K$ on the etale fundamental group $\pi_1(X_\overline K)$, where $X$ is a (anabelian) variety and $...

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### Neukirch's papers and theorem

Have any of Neukirch's papers on anabelian geometry been translated? I'm mostly interested in:
Kennzeichnung der p-adischen und der endlichen algebraischen Zahlkörper (1969)
Kennzeichnung der ...

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**1**answer

385 views

### representation of algebraic fundamental group of projective line minus three point

everyone, I want to ask is there any result in the literature
similar to the following:
Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X_{\mathbb{Q}}$ ...

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**2**answers

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### Basics on anabelian geometry and Grothendieck's section conjecture

Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask ...

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483 views

### Field of definition of a finite etale cover of an anabelian curve

Let $X$ be an anabelian curve over a number field $K$ and let $p:Y\rightarrow X$ be a finite etale cover. Then is anything known (or has anything been conjectured) about the field of definition of $Y$?...

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### What conjectures in anabelian geometry are false?

Proving things suspected to be true in anabelian geometry is usually very hard. Maybe it is easier to disprove things suspected to be false?
In particular, I am interested in false generalizations of ...

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**5**answers

4k views

### Anabelian geometry study materials?

I want to study anabelian geometry, but unfortunately I'm having difficulties in finding some materials about it. If you could offer me some books/papers/articles I would be glad.

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### Discretifications of the fundamental group functor

Grothendieck calls a "discretification" of a profinite group $\widehat G$, a
discrete group $G$ whose profinite completion is isomorphic to $\widehat G$.
Does Grothendieck also define a notion of ...

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### Why does the Section Conjecture exclude curves of genus 1?

Let $X$ be an integral proper normal curve over a (perfect) field $F$, of genus $\geq 2$. One variant of Grothendieck's "section conjecture" states that the sections $G_F \rightarrow \pi_1(X)$ of the ...

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**2**answers

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### Why should the anabelian geometry conjectures be true?

I had probed friends of mine about Grothendieck's motivation for making the anabelian geometry conjectures, and they gave me the following explanation:
If $X$ is a hyperbolic curve over some field $K$...

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832 views

### Galois groups of number fields

It seems that it is conjectured, that the absolute Galois group of a number field determines already the number field up to isomorphism.
I would like to know if there is a profinite group G such that ...

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### 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 ...

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### Exotic automorphisms of the fundamental group of a curve?

A while back, Jordan S. Ellenberg brought the following problem to my attention.
If $G$ is a residually finite group, let $\widehat G$ be its profinite completion.
Let $S$ be a closed surface of ...

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### “Fermat's last theorem” and anabelian geometry?

Do I remember a remark in "Sketch of a program" or "Letter to Faltings" correctly, that acc. to Grothendieck anabelian geometry should not only enable finiteness proofs, but a proof of FLT too? If yes,...