examples of etale fundamental groups

$\pi_1 X = 1$ for
- ${\mathbf{A}}^1_{/ {{\mathbf{C}}}} $
- ${\mathbf{A}}^1_{/ {{\mathbf{Z}}}} $
- ${\mathbf{P}}^1_{/ {{\mathbf{Z}}}} $
- $\operatorname{Spec}{\mathbf{Z}}$
- $\operatorname{Spec}{\mathbf{Z}}[i]$
- $\operatorname{Spec}k$ for $k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } $.
- $\operatorname{Spec}{\mathbf{C}}[x,y]/\left\langle{y^2-x^3}\right\rangle$,
$\pi_1 X = \widehat{{\mathbf{Z}}}$ for $X = \cdots$
- ${\mathbf{G}}m{}{/ {k}} $
- $\operatorname{Spec}{ {\mathbf{Z}}_{\widehat{p}} }$
- $\operatorname{Spec}{\mathbf{Z}}[x]/\left\langle{x^6-1}\right\rangle$,
- ${\mathbf{C}}{\left(\left( t \right)\right) }$
- $\operatorname{Spec}{\mathbf{C}}[x,y]/\left\langle{y^2-x^3-x^2}\right\rangle$,
- $\operatorname{Spec}k$ for $k= { \mathbf{F} }_q$ any finite field.
$\pi_1 X = C_2$ for
- $X = \operatorname{Spec}{\mathbf{Z}}[\sqrt{-3}], \operatorname{Spec}{\mathbf{Z}}[\sqrt{-5}], \operatorname{Spec}{\mathbf{R}}$
For $X = {\mathbf{G}}_{m}{}_{/ {k}} = \operatorname{Spec}k[x,x^{-1}]$, \begin{align*}\pi_1^\text{ét}(X) = \widehat{{\mathbf{Z}}},\end{align*} the profinite integers.
For $E\in \mathrm{Ell} _{/ {{\mathbf{Q}}}} $ an elliptic curve, \begin{align*}\pi_1^\text{ét}(E) = TE \coloneqq\cocolim_{n\in {\mathbf{Z}}_{\geq 0}} E[n]\end{align*} where ${\mathbf{Z}}_{\geq 0}$ 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 $\pi_1 X_{ { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu } } \cong \widehat{{\mathbf{Z}}}{ {}^{ \scriptscriptstyle\oplus^{2} } }$
$\pi_1 X = \prod_{m\in \operatorname{Spec}A} \widehat{{\mathbf{Z}}}$ if $A$ is a finite ring.
$\pi_1 \operatorname{Spec}k = { \mathsf{Gal}} (k^{ {}^{ \operatorname{sep} } }/k)$.
$\pi_1 \operatorname{Spec}{\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{n} } \right] } = { \mathsf{Gal}} ({\mathbf{Q}}^{{\scriptscriptstyle\mathrm{un}}, n}/ {\mathbf{Q}})$ where ${\mathbf{Q}}^{{\scriptscriptstyle\mathrm{un}}, n}$ is the maximal extension of ${\mathbf{Q}}$ unramified away from $n$.
$\pi_1 X = \mathrm{Prof}(\pi_1 X({\mathbf{C}}))$, the profinite completion of the usual fundamental group when $X$ is a finite type scheme over ${\mathbf{C}}$.
$\pi_1 X = \mathrm{Prof}(\mathsf{Free}_2)$ for $X = {\mathbf{P}}^1{\mathbf{C}}\setminus\left\{{ 0,1,\infty }\right\}$.
$\pi_1 X$ for $X={\mathbf{P}}^1_{/ {{\mathbf{Q}}}} \setminus\left\{{ 0,1,\infty }\right\}$ is unknown but fits into a SES \begin{align*}1\to \mathrm{Prof}(\mathsf{Free}_2) \to \pi_1 {\mathbf{P}}^1_{/ {{\mathbf{Q}}}} \setminus\left\{{ 0,1,\infty }\right\} \to { \mathsf{Gal}} ({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbf{Q} \mkern-1.5mu}\mkern 1.5mu }/{\mathbf{Q}}) \to 1\end{align*}

General properties

$\pi_1 \operatorname{Spec}k = \pi_1 {\mathbf{P}}^1_{/ {k}} $ for $k$ any field.