All Questions
Tagged with fundamental-group arithmetic-geometry
15 questions
1
vote
0
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85
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Companions for positive characteristic arithmetic representations viewed as representations of the topological fundamental group?
Suppose $X / K$ is a variety over a finitely generated field over $\mathbb{Q}$. Fix an embedding $K \subset \mathbb{C}$ and let $\pi := \pi_1(X(\mathbb{C}), x)$ be the topological fundamental group. ...
- 3,730
3
votes
0
answers
96
views
Descent obstruction of an open curve in an elliptic curve
Let $E$ be an elliptic curve over a number field $k$, and for an extension $K/k$ we denote by $E_K$ the base change $E \times_k K$. By fixing an embedding $k \hookrightarrow \mathbb{C}$, the etale ...
- 895
47
votes
3
answers
5k
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"Cute" applications of the étale fundamental group
When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...
CommunityBot
- 1
11
votes
3
answers
1k
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Are "large enough" finite etale covers arithmetic?
Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-...
- 10.7k
8
votes
1
answer
339
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The direct product of the geometric fundamental group and the absolute Galois group
Given a geometrically connected variety $X$ over $\mathbb{Q}$ we have a short exact sequence
$$
1\to \pi_1(X_{\overline{\mathbb{Q}}})\to \pi_1(X)\to Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to 1.
$$
A ...
- 30.5k
4
votes
0
answers
195
views
Geometric fundamental group and algebraically closed residue field
my questions relates to the following talk of Tsuji:
https://www.youtube.com/watch?v=2brDj26phP0
At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...
- 455
5
votes
1
answer
584
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 ...
CommunityBot
- 1
3
votes
0
answers
135
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How does the fundamental group of $\mathbb G_{m,S}$ depend on the base scheme
Let $S$ be an integral noetherian regular scheme, and let $X =\mathbb{G}_{m,S}$.
How to compute $\pi_1^{et}(X)$?
Note. I am only interested in the part not coming trivially from the finite etale ...
- 39
11
votes
1
answer
386
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Is there a presentation to the kernel of the prime-to-$p$ fundamental short exact sequence of curves over finite fields?
Let $X$ be $\mathbb{P}^1_{\mathbb{F}_q}\smallsetminus \{a_1,...,a_r\}$, where $a_1,...,a_r$ are some $\mathbb{F}_q$-rational points. Let $\bar X :=X_{\bar{\mathbb{F}}_q}$. There is a short exact ...
- 149k
5
votes
1
answer
2k
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What is the arithmetic fundamental group?
Let $X$ be an irreducible variety defined over $\mathbf{F}_p$. What's the "arithmetic fundamental group" of $X$? How does this relate to the algebraic fundamental group of a scheme? What's a good ...
- 10.7k
9
votes
1
answer
627
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Do all varieties have only finitely many etale covers of fixed degree
I've been wondering about the following "finiteness statement" concerning etale covers for a while.
Let $K$ be a field of characteristic zero, not necessarily algebraically closed. A variety over $K$ ...
- 9,497
3
votes
1
answer
600
views
Grothendieck's section conjecture and base change: restricting sections
Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the ...
- 149k
4
votes
2
answers
349
views
Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group
A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. ...
- 96.4k
4
votes
1
answer
604
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A question about the Tannakian etale fundamental group of a curve
Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let $n\...
- 1,838
8
votes
2
answers
981
views
Covers of the projective line over Z and arithmetic Grauert-Remmert
This question is the two-dimensional analogue of Etale coverings of certain open subschemes in Spec O_K
There I asked if one could characterize in a way certain covers of $\textrm{Spec} \ O_K$. As ...
CommunityBot
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