Definition 64.27.1. A subgroup of the form $\text{Stab}(\overline y\in F_{\overline{x}}(Y))\subset \pi _1(X, \overline{x})$ is called open.
64.27 Fundamental groups
This material is discussed in more detail in the chapter on fundamental groups. See Fundamental Groups, Section 58.1. Let $X$ be a connected scheme and let $\overline{x}\to X$ be a geometric point. Consider the functor
\[ \begin{matrix} F_{\overline{x}}:
& \text{ finite étale } \atop \text{ schemes over } X
& \longrightarrow
& \text{ finite sets}
\\ & Y/X
& \longmapsto
& F_{\overline{x}}(Y) = \left\{ \text{ geom points }\overline y \atop \text{ of } Y \text{ lying over }\overline{x}\right\} = Y_{\overline{x}}
\end{matrix} \]
Set
\[ \pi _1(X, \overline{x}) = Aut(F_{\overline{x}}) = \text{ set of automorphisms of the functor }F_{\overline{x}} \]
Note that for every finite étale $Y \to X$ there is an action
\[ \pi _1(X, \overline{x}) \times F_{\overline{x}}(Y) \to F_{\overline{x}}(Y) \]
Theorem 64.27.2 (Grothendieck). Let $X$ be a connected scheme.
There is a topology on $\pi _1(X, \overline{x})$ such that the open subgroups form a fundamental system of open nbhds of $e\in \pi _1(X, \overline x)$.
With topology of (1) the group $\pi _1(X, \overline{x})$ is a profinite group.
The functor
is an equivalence of categories.
\[ \begin{matrix} \text{ schemes finite } \atop \text{ étale over }X
& \to
& \text{ finite discrete continuous } \atop \pi _1(X, \overline{x})\text{-sets}
\\ Y / X
& \mapsto
& F_{\overline{x}}(Y) \text{ with its natural action}
\end{matrix} \]
Proof. See [SGA1]. $\square$
Proposition 64.27.3. Let $X$ be an integral normal Noetherian scheme. Let $\overline y\to X$ be an algebraic geometric point lying over the generic point $\eta \in X$. Then ($\kappa (\eta )$, function field of $X$) where is the max sub-extension such that for every finite sub extension $M\supset L\supset \kappa (\eta )$ the normalization of $X$ in $L$ is finite étale over $X$.
\[ \pi _ x(X, \overline\eta ) = Gal(M/\kappa (\eta )) \]
\[ \kappa (\overline\eta )\supset M\supset \kappa (\eta ) = k(X) \]
Proof. Omitted. $\square$
Change of base point. For any $\overline{x}_1, \overline{x}_2$ geom. points of $X$ there exists an isom. of fibre functions
\[ \mathcal{F}_{\overline{x}_1} \cong \mathcal{F}_{\overline{x}_2} \]
(This is a path from $\overline{x}_1$ to $\overline{x}_2$.) Conjugation by this path gives isom
\[ \pi _1(X, \overline{x}_1) \cong \pi _1(X, \overline{x}_2) \]
well defined up to inner actions.
Functoriality. For any morphism $X_1\to X_2$ of connected schemes any $\overline{x}\in X_1$ there is a canonical map
\[ \pi _1(X_1, \overline{x}) \to \pi _1(X_2, \overline{x}) \]
(Why? because the fibre functor ...)
Base field. Let $X$ be a variety over a field $k$. Then we get
\[ \pi _1(X, \overline{x}) \to \pi _1(Spec(k), \overline{x}) =^{\text{prop}} Gal(k^{\text{sep}}/k) \]
This map is surjective if and only if $X$ is geometrically connected over $k$. So in the geometrically connected case we get s.e.s. of profinite groups
\[ 1 \to \pi _1(X_{\overline{k}}, \overline{x}) \to \pi _1(X, \overline{x}) \to Gal(k^{\text{sep}}/k) \to 1 \]
($\pi _1(X_{\overline{k}}, \overline{x})$: geometric fundamental group of $X$, $\pi _1(X, \overline{x})$: arithmetic fundamental group of $X$)
Comparison. If $X$ is a variety over $\mathbf{C}$ then
\[ \pi _1(X, \overline{x}) = \text{ profinite completion of } \pi _1(X(\mathbf{C})(\text{ usual topology}), x) \]
(have $x\in X(\mathbf{C})$)
Frobenii. $X$ variety over $k$, $\# k < \infty $. For any $x \in X$ closed point, let
\[ F_ x\in \pi _1(x, \overline{x}) = \text{Gal}(\kappa (x)^{\text{sep}}/\kappa (x)) \]
be the geometric frobenius. Let $\overline\eta $ be an alg. geom. gen. pt. Then
\[ \pi _1(X, \overline\eta ) \leftarrow ^{\cong } \pi _1(X, \overline{x}) {\text{functoriality} \atop \leftarrow } \pi _1(x, \overline{x}) \]
Easy fact:
\[ \begin{matrix} \pi _1(X, \overline\eta )
& \to ^{\deg } \pi _1(\mathop{\mathrm{Spec}}(k), \overline\eta ) *
& = Gal(k^{sep}/k)
\\ & & ||
\\ & & \widehat{\mathbf{Z}}\cdot F_{\mathop{\mathrm{Spec}}(k)}
\\ F_ x
& \mapsto
& \deg (x)\cdot F_{\mathop{\mathrm{Spec}}(k)}
\end{matrix} \]
Recall: $\deg (x) = [\kappa (x):k]$
Fundamental groups and lisse sheaves. Let $X$ be a connected scheme, $\overline{x}$ geom. pt. There are equivalences of categories
\[ \begin{matrix} (\Lambda \text{ finite ring})
& \text{fin. loc. const. sheaves of } \atop \Lambda \text{-modules of }X_{\acute{e}tale}
& \leftrightarrow
& \text{ finite (discrete) }\Lambda \text{-modules} \atop \text{ with continuous }\pi _1(X, \overline{x})\text{-action}
\\ (\ell \text{ a prime})
& \text{ lisse }\ell \text{-adic} \atop \text{ sheaves}
& \leftrightarrow
& \text{finitely generated }\mathbf{Z}_\ell \text{-modules }M\text{ with continuous} \atop \pi _1(X, \overline{x})\text{-action where we use } \ell \text{-adic topology on }M
\end{matrix} \]
In particular lisse $\mathbf{Q}_ l$-sheaves correspond to continuous homomorphisms
\[ \pi _1(X, \overline{x}) \to \text{GL}_ r(\mathbf{Q}_ l), \quad r\geq 0 \]
Notation: A module with action $(M, \rho )$ corresponds to the sheaf $\mathcal{F}_\rho $.
Trace formulas. $X$ variety over $k$, $\# k < \infty $.
$\Lambda $ finite ring $(\# \Lambda , \# k)=1$
\[ \rho : \pi _1(X, \overline{x})\to \text{GL}_ r(\Lambda ) \]continuous. For every $n\geq 1$ we have
\[ \sum _{d|n}d\left( \sum _{x\in |X|, \atop \deg (x)=d} \text{Tr}(\rho (F_ x^{n/d}))\right) = \text{Tr}\left( (\pi _ x^ n)^* |_{R\Gamma _ c(X_{\overline{k}}, \mathcal{F}_\rho )}\right) \]$l\neq char(k)$ prime, $\rho : \pi _1(X, \overline{x})\to \text{GL}_ r(\mathbf{Q}_ l)$. For any $n\geq 1$
\[ \sum _{d|n} d\left( \sum _{x\in |X| \atop \deg (x)=d} \text{Tr} \left( \rho (F_ x^{n/d}) \right) \right) = \sum _{i = 0}^{2\dim X} (-1)^ i \text{Tr}\left( \pi _ X^* |_{H_ c^ i(X_{\overline{k}}, \mathcal{F}_\rho )}\right) \]
Weil conjectures. (Deligne-Weil I, 1974) $X$ smooth proj. over $k$, $\# k = q$, then the eigenvalues of $\pi _ X^*$ on $H^ i(X_{\overline{k}}, \mathbf{Q}_ l)$ are algebraic integers $\alpha $ with $|\alpha |=q^{1/2}$.
Deligne's conjectures. (almost completely proved by Lafforgue + $\ldots $) Let $X$ be a normal variety over $k$ finite
\[ \rho : \pi _1(X, \overline{x}) \longrightarrow \text{GL}_ r(\mathbf{Q}_ l) \]
continuous. Assume: $\rho $ irreducible $\det (\rho )$ of finite order. Then
there exists a number field $E$ such that for all $x\in |X|$(closed points) the char. poly of $\rho (F_ x)$ has coefficients in $E$.
for any $x\in |X|$ the eigenvalues $\alpha _{x, i}$, $i = 1, \ldots , r$ of $\rho (F_ x)$ have complex absolute value $1$. (these are algebraic numbers not necessary integers)
for every finite place $\lambda $( not dividing $p$), of $E$ (maybe after enlarging $E$ a bit) there exists
\[ \rho \lambda : \pi _1(X, \overline{x}) \to \text{GL}_ r(E_\lambda ) \]compatible with $\rho $. (some char. polys of $F_ x$'s)
Theorem 64.27.4 (Deligne, Weil II). For a sheaf $\mathcal{F}_\rho $ with $\rho $ satisfying the conclusions of the conjecture above then the eigenvalues of $\pi _ X^*$ on $H_ c^ i(X_{\overline{k}}, \mathcal{F}_{\rho })$ are algebraic numbers $\alpha $ with absolute values Moreover, if $X$ smooth and proj. then $w = i$.
\[ |\alpha |=q^{w/2}, \text{ for }w\in \mathbf{Z},\ w\leq i \]
Proof. See [WeilII]. $\square$
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