---
date: 2022-04-16 21:40
modification date: Saturday 16th April 2022 21:40:33
title: "examples of etale fundamental groups"
aliases: [examples of etale fundamental groups]
---

Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [profinite completion](Unsorted/profinite%20completion.md)

---

# examples of etale fundamental groups

- $\pi_1 X = 1$ for 
	- $\AA^1\slice \CC$
	- $\AA^1\slice \ZZ$
	- $\PP^1\slice \ZZ$
	- $\spec \ZZ$
	- $\spec \ZZ[i]$
	- $\spec k$ for $k=\kbar$.
	- $\spec \CC[x,y]/\gens{y^2-x^3}$, 
- $\pi_1 X = \hat{\ZZ}$ for $X = \cdots$
	- $\GG_m{}\slice k$
	- $\spec \ZZpadic$ 
	- $\spec \ZZ[x]/\gens{x^6-1}$,
	- $\CC\fls{t}$
	- $\spec \CC[x,y]/\gens{y^2-x^3-x^2}$, 
	- $\spec k$ for $k= \FF_q$ any finite field.
- $\pi_1 X = C_2$ for 
	- $X = \spec \ZZ[\sqrt{-3}], \spec \ZZ[\sqrt{-5}], \spec \RR$
- For $X = \GG_{m}{}\slice k = \spec k[x,x\inv]$, $$\pi_1^\et(X) = \hat\ZZ,$$the [profinite integers](Unsorted/profinite%20integers.md).
- For $E\in \Ell\slice \QQ$ an [elliptic curve](elliptic%20curve.md), 
  $$\pi_1^\et(E) = TE \da \cocolim_{n\in \ZZ_{\geq 0}} E[n]$$  where $\ZZ_{\geq 0}$ is ordered by divisibility.
	  - 1If $X \rightarrow \operatorname{Spec}(k)$ is an elliptic curve with identity $\mathcal{O}: \operatorname{Spec}(k) \rightarrow X$ and $\operatorname{char}(k)=0$, then $\pi_1 X_{\kbar} \cong \hat{\ZZ}\sumpower{2}$
- $\pi_1 X = \prod_{m\in \spec A} \hat{\ZZ}$ if $A$ is a finite ring.
- $\pi_1 \spec k = \Gal(k^\sep/k)$.
- $\pi_1 \spec \ZZ\invert{n} = \Gal(\QQ^{\unram, n}/ \QQ)$  where $\QQ^{\unram, n}$ is the maximal extension of $\QQ$ [unramified](unramified.md) away from $n$.
- $\pi_1 X = \mathrm{Prof}(\pi_1 X(\CC))$, the profinite completion of the usual fundamental group when $X$ is a [finite type](finite%20type.md) scheme over $\CC$.
- $\pi_1 X = \mathrm{Prof}(\Free_2)$ for $X = \PP^1\CC\smts{0,1,\infty}$.
- $\pi_1 X$ for $X=\PP^1\slice{\QQ}\smts{0,1,\infty}$ is unknown but fits into a SES $$1\to \mathrm{Prof}(\Free_2) \to \pi_1 \PP^1\slice{\QQ}\smts{0,1,\infty} \to \Gal(\QQbar/\QQ) \to 1$$

# General properties

- $\pi_1 \spec k = \pi_1 \PP^1\slice k$ for $k$ any field.