Questions tagged [anabelian-geometry]

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Galois rigidity for ℙ¹ with infinitely many punctures

A well-known result (due first to Nakamura I think) is that given a number field $K$, and a variety $U = \mathbb P^1 \setminus (\text{finitely many points})$ over $K$, the étale fundamental group of $...
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4 votes
0 answers
126 views

Constructing motivic representations through extensions of $\mathrm{SL}(2, \mathbb{Z})$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the Esquisse, Grothendieck explains how to find the representations of $G_{\mathbb{Q}}$ on Tate modules of Jacobians through its action on $\...
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9 votes
1 answer
2k views

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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3 votes
2 answers
441 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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14 votes
0 answers
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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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23 votes
3 answers
3k views

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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  • 2,371
5 votes
1 answer
482 views

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 ...
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  • 527
5 votes
1 answer
603 views

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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  • 1,560
14 votes
0 answers
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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 ...
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  • 1,560
4 votes
1 answer
192 views

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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14 votes
0 answers
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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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18 votes
1 answer
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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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30 votes
2 answers
4k views

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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  • 17k
29 votes
2 answers
12k 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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  • 1,009
7 votes
1 answer
564 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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15 votes
0 answers
447 views

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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75 votes
1 answer
13k views

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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13 votes
1 answer
665 views

Abelian $\ell$-adic representations in $\widehat{\mathrm{SL}(2,\mathbb{Z})}$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Gal{Gal}\newcommand{\Z}{\mathbb{Z}}$In Grothendieck's Esquisse he claims that the action of $$\Gal(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\...
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  • 3,477
16 votes
1 answer
3k views

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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13 votes
1 answer
872 views

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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7 votes
1 answer
961 views

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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  • 17k
3 votes
1 answer
448 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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  • 679
6 votes
2 answers
2k views

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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1 vote
1 answer
492 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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  • 1,835
14 votes
2 answers
2k views

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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  • 114k
23 votes
5 answers
5k 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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  • 1,009
10 votes
0 answers
422 views

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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  • 468
26 votes
1 answer
2k views

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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20 votes
2 answers
5k views

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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4 votes
1 answer
992 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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14 votes
1 answer
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 ...
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10 votes
2 answers
750 views

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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20 votes
2 answers
4k views

"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,...
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