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9
votes
1answer
536 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, ...
5
votes
1answer
287 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 Zahlk├Ârper (1969) Kennzeichnung der ...
2
votes
1answer
231 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 $X_{\mathbb{Q}}$ ...
4
votes
1answer
415 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 ...
1
vote
1answer
211 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 ...
8
votes
2answers
1k 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 ...
5
votes
0answers
282 views

discretifications of the fundamental group functor

Grothendieck calls a "discretification" of a profinite group $\hat G$, a discrete group $G$ whose profinite completion is isomorphic to $\hat G$. Does Grothendieck also define a notion of the ...
16
votes
1answer
663 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 ...
12
votes
2answers
2k 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 ...
7
votes
2answers
541 views

Exotic automorphisms of the fundamental group of a curve?

A while back, J. 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 genus $g ...