I-ladic.tex

\chapter{\texorpdfstring{$\ell$}{ℓ}-adic representations}%
\label{ch:i}

\section{The notion of an \texorpdfstring{$\ell$}{ℓ}-adic representation}
\subsection{Definition}\label{sec:I_11}
Let $K$ be a field,
\dpage
and let $\sepcl K$ be a separable algebraic closure of $K$.  Let $G =
\Gal(\sepcl K / K)$ be the Galois group of the extension $\sepcl K/K$.
The group $G$, with the Krull topology, is compact and totally disconnected.
Let $\ell$ be a prime number, and let $V$ be a finite-dimensional vector space
over the field $\Q_\ell$ of $\ell$-adic numbers.  The full linear group
$\Aut(V)$ is an $\ell$-adic Lie group, its topology being induced by the
natural topology of $\End(V)$; if $n = \dim(V)$, we have $\Aut(V) \cong \GL(n,
\Q_\ell)$.

\begin{mydef}
An $\ell$-adic representation of $G$ (or, by abuse of language, of $K$) is a
continuous homomorphism $\rho \colon G \to \Aut(V)$.
\end{mydef}

\begin{obs}
\begin{enumerate}
\item\label{rmk:I_11_1}
	A \emph{lattice} of $V$ is a sub-$\Z_\ell$-module $T$ which is free of
	finite rank, and generate $V$ over $\Q_\ell$, so that $V$ can be
	identified with $T \otimes_{\Z_\ell} \Q_\ell$.  Notice that there
	exists a lattice of $V$ which is stable under $G$. This follows from
	the fact that $G$ is compact.

	Indeed,
	\dpage
	let $L$ be any lattice of $V$, and let $H$ be the set of
	elements $g \in G$ such that $\rho(g)L = L$. This is an open
	subgroup of $G$, and $G/H$ is finite. The lattice $T$
	generated by the lattices $\rho(g)L$, $g \in G/H$, is stable
	under $G$.

	Notice that $L$ may be identified with the projective limit of
	the free $(\Z/\ell\Z)$-modules $T/\ell^m T$, on which $G$
	acts; the vector space $V$ may be reconstructed from $T$ by $V
	= T \otimes_{\Z_\ell} \Q_\ell$.

\item If $\rho$ is an $\ell$-adic representation of $G$, the group $G
	= \Img(\rho)$ is a closed subgroup of $\Aut(V)$, and hence, by
	the $\ell$-adic analogue of Cartan's theorem (cf.\ 
	\cite[5-42]{28}) $G$ is itself an $\ell$-adic Lie group.
	Its Lie algebra $\mathfrak{g} = \Lie(G)$ is a subalgebra of
	$\End(V) = \Lie(\Aut(V))$. The Lie algebra $\mathfrak{g}$ is
	easily seen to be invariant under extensions of finite type of
	the ground field $K$ (cf.\ \cite{24}, 1.2).
\end{enumerate}
\end{obs}

\subsubsection*{Exercises}
\begin{enumerate}
	\item\label{ex:basis_triangular_grp}
		Let $V$ be a vector space of dimension 2 over a field $k$ and
		let $H$ be a subgroup of $\Aut(V)$. Assume that $\det(1-h) = 0$
		for all $h \in H$. Show the existence of a basis of $V$ with
		respect to which $H$ is contained either in the subgroup $
		\begin{psmallmatrix}
			1 & * \\
			0 & *
		\end{psmallmatrix}
		$ or in the subgroup $
		\begin{psmallmatrix}
			1 & 0 \\
			* & *
		\end{psmallmatrix}
		$ of $\Aut(V)$.
	\item\label{ex:I11_ex2}
		Let $\rho \colon G \to \Aut(V_\ell)$ be an $\ell$-adic
		representation of $G$, where $V_\ell$ is a $\Q_\ell$-vector
		space of dimension 2. Assume $\det(1-\rho(s))= 0 \mod\ell$ for
		all $s \in G$. Let $T$ be a lattice of $V_\ell$ stable by $G$.
		Show the existence of a lattice $T'$ of $V_\ell$ with the
		following two properties:
	\begin{enumerate}
		\item $T'$ is stable by $G$.
		\item Either $T'$ is a sublattice of index $\ell$ of $T$ and
			\dpage
			$G$ acts trivially on $T/T'$ or $T$ is a sublattice of
			index $\ell$ of $T'$ and $G$ acts trivially on $T/T'$.
			\label{errata:t't}

			(Apply exercise~\ref{ex:basis_triangular_grp} above to
			$k = \F_\ell$ and $V = T/\ell T$.)
	\end{enumerate}

	\item Let $\rho$ be a semi-simple $\ell$-adic representation of $G$ and
		let $U$ be an invariant subgroup of $G$.  Assume that, for all
		$x \in U$, $\rho(x)$ is unipotent (all its eigenvalues are
		equal to 1). Show that $\rho(x) = 1$ for all $x \in U$. (Show
		that the restriction of $\rho$ to $U$ is semi-simple and use
		Kolchin's theorem to bring it to triangular form.)
	\item Let $\rho \colon G \to \Aut(V_\ell)$ be an $\ell$-adic
		representation of $G$, and $T$ a lattice of $V_\ell$ stable
		under $G$. Show the equivalence of the following properties:
	\begin{enumerate}
		\item The representation of $G$ in the $\F_\ell$-vector space
			$T/\ell T$ is irreducible.
		\item The only lattices of $V_\ell$ stable under $G$ are the
			$\ell^n T$, with $n \in \Z$.
	\end{enumerate}
\end{enumerate}

\subsection{Examples}\label{sec:I_12}
\subsubsection{Roots of unity}\label{sec:I_121}
Let $\ell \ne \char(K)$. The group $G = \Gal(\sepcl K / K)$ acts on the group
$\mu_m$ of $\ell^m$-th roots of unity, and hence also on $T_\ell(\mu) =
\invlim_{m\in\N} \mu_m$. The $\Q_\ell$-vector space $V_\ell(\mu) = T_\ell(\mu)
\otimes_{\Z_\ell} \Q_\ell$ is of dimension 1, and the homomorphism $\chi_\ell
\colon G \to \Aut(V_\ell) = \Q_\ell^\times$ defined by the action of $G$ on
$V_\ell$ is a 1-dimensional $\ell$-adic representation of $G$. The character
$\chi_\ell$ takes its values in the group of units $U$ of $\Z_\ell$; by
definition
$$ g(z) = z^{\chi_\ell(g)} \quad \text{if } g \in G, \; z^{\ell^m} = 1. $$

\subsubsection{Elliptic curves}\label{sec:I_122}
Let $\ell \ne \char(K)$. Let $E$ be an elliptic
curve defined over $K$ with a given rational point $o$. One knows that
\dpage
there is a unique structure of group variety on $E$ with $o$ as neutral
element. Let $E_m$ be the kernel of multiplication by $\ell^m$ in $E(\sepcl
K)$, and let
\todo[pinktask]{¿Deberíamos modernizar la notación a $E[\ell^m]$?}
\[
	T_\ell(E) = \invlim_m E_m, \qquad V_\ell(E) = T_\ell(E)
	\otimes_{\Z_\ell} \Q_\ell.
\]
The Tate module $T_\ell(E)$ is a free $\Z_\ell$-module on which $G =
\Gal(\sepcl K / K)$ acts (cf.\ \cite{12}, chap.\ VII). The corresponding
homomorphism $\pi_\ell \colon G \to \Aut(V_\ell(E))$ is an $\ell$-adic
representation of $G$. The group $G_\ell = \Img(\pi_\ell)$ is a closed
subgroup of $\Aut(T_\ell(E))$, a 4-dimensional Lie group isomorphic to $\GL(2,
\Z_\ell)$. (In chapter~\ref{ch:iv}, we will determine the Lie algebra of
$G_\ell$, under the assumption that $K$ is a number field.)

Since we can identify $E$ with its dual (in the sense of the duality of abelian
varieties) the symbol $(x,y)$ (cf.\ \cite{12}, \textit{loc.\ cit.}) defines
canonical isomorphisms
\[
	\textstyle\bigwedge^2 T_\ell(E) = T_\ell(\mu), \qquad
	\bigwedge^2 V_\ell(E) = V_\ell(\mu).
\]
Hence $\det(\pi_\ell)$ is the character $\chi_\ell$ defined in example 1.

\subsubsection{Abelian varieties}\label{sec:I_123}
Let $A$ be an abelian variety over $K$ of dimension $d$. If $\ell \ne
\char(K)$, we define $T_\ell(A)$, $V_\ell(A)$ in the same way as in example 2.
The group $T_\ell(A)$ is a free $\Z_\ell$-module of rank $2d$ (cf.\ \cite{12},
\textit{loc.\ cit.}) on which $G = \Gal(\sepcl K/K)$ acts.

\subsubsection{Cohomology representations}\label{sec:I_124}
Let $X$ be an algebraic variety defined over the field $K$, and let $\sepcl X =
X \times_K \sepcl K$ be the corresponding variety over $\sepcl K$. Let $\ell
\ne \char(K)$, and let $i$ be an integer. Using the étale cohomology of
\citeauthor{3}~\cite{3} we let
\[
	H^i(\sepcl X, \Z_\ell) = \invlim_n H^i\big( (\sepcl X)_{\text{ét}},
	\Z/\ell^n\Z \big),
	\qquad H^i_\ell(\sepcl X) = H^i(\sepcl X, \Z_\ell) \otimes_{\Z_\ell}
	\Q_\ell.
\]
\dpage
The group $H^i_\ell(\sepcl X)$ is a vector space over $\Q_\ell$ on which $G =
\Gal(\sepcl K/K)$ acts (via the action of $G$ on $\sepcl X$). It is finite
dimensional, at least if $\char(K) = 0$ or if $X$ is proper. We thus get an
$\ell$-adic representation of $G$ associated to $H^i_\ell(\sepcl X)$; by taking
duals we also get homology $\ell$-adic representations.
Examples 1, 2, 3 are particular cases of homology $\ell$-adic representations
where $i = 1$ and $X$ is respectively the multiplicative group $\GG_m$, the
elliptic curve $E$, and the abelian variety $A$.

\subsubsection*{Exercise}
\begin{enumerate}[(a)]
	\item Show that there is an elliptic curve $E$, defined over $K_0 =
		\Q(T)$, with $j$-invariant equal to $T$.
	\item\label{exr:elliptic_I12_b}
		Show that for such a curve, over $K = \C(T)$, one has $G_\ell =
		\SL(T_\ell(E))$ (cf.\ \citeauthor{10}~\cite{10} for an
		algebraic proof).
	\item Using \ref{exr:elliptic_I12_b}, show that, over $K_0$, we have
		$G_\ell = \GL(T_\ell(E))$.
	\item Show that for any closed subgroup $H$ of $\GL(2, \Z_\ell)$ there
		is an elliptic curve (defined over some field) for which
		$G_\ell = H$.
\end{enumerate}

\section{\texorpdfstring{$\ell$}{ℓ}-adic representations of number fields}
\subsection{Preliminaries}%
\label{sec:I_21}
(For the basic notions concerning number fields, see for instance
\citeauthor{6}~\cite{6}, \citeauthor{13}~\cite{13} or
\citeauthor{44}~\cite{44}.)
Let $K$ be a number field (i.e.\ a finite extension of $\Q$). Denote by $M_K^0$
the set of all finite places of $K$, i.e.\, the set of all normalized discrete
valuations of $K$ (or, alternatively, the set of prime ideals in the ring
$\mathcal{O}_K$ of integers of $K$).
The \strong{residue field} $k_v$ of a place $v \in M_K^0$
is a finite field with $\numnorm(v) = p_v^{\deg(v)}$ elements, where
\dpage
$p_v = \char(k_v)$ and $\deg(v)$ is the degree of $k_v$ over $\F_{p_v}$. The
ramification index $e_v$ of $v$ is $v(p_v)$.

Let $L/K$ be a finite Galois extension with Galois group $G$,
and let $w \in M_L^0$.
The subgroup $D_w$ of $G$ consisting of those $g \in G$ for which $gw = w$ is
the \strong{decomposition group} of $w$. The restriction
of $w$ to $K$ is an integral multiple of an element $v \in M_K^0$; by abuse
of language, we also say that $v$ is the restriction of $w$ to $K$, and we
write $w \mid v$ (\textquote{$w$ divides $v$}). Let $L$ (resp.\ $K$) be the
completion of $L$ (resp.\ $K$) with respect to $w$ (resp.\ $v$). We have
$D_w = \Gal(L_w/K_v)$. The group $D_w$ is mapped homomorphically onto
the Galois group $\Gal(\lambda_w/k_v)$ of the corresponding residue extension
$\lambda_w/k_v$. The kernel of $G \to \Gal(\lambda_w/k_v)$ is the inertia group
$I_w$ of $w$. The quotient group $D_w/I_w$ is a finite cyclic group generated
by the \strong{Frobenius element} $F_w$; we have $F(\lambda) =
\lambda^{\numnorm(v)}$ for all $\lambda \in \lambda_w$.
The valuation $w$ (resp.\ $v$) is called \strong{unramified} if $I_w = \{ 1
\}$. Almost all places of $K$ are unramified.

If $L$ is an arbitrary algebraic extension of $\Q$, one defines $M_K^0$ to be
the projective limit of the sets $M_{L_\alpha}^0$, where $L_\alpha$ ranges
over the finite sub-extensions of $L/\Q$. Then, if $L/K$ is an 
arbitrary Galois extension of the number field $K$, and $w \in M_L^0$, one
defines $D_w$, $I_w$, $F_w$ as before. If $v$ is an unramified place of $K$,
and $w$ is a place of $L$ extending $v$, we denote by $F_v$ the conjugacy
class of $F_w$ in $G = \Gal(L/K)$.

\begin{mydef}
Let $\rho \colon \Gal(\algcl K/K) \to \Aut(V)$ be an $\ell$-adic representation
of $K$, and let $v \in M_K^0$. We say that $\rho$ is unramified at $v$ if
$\rho(I_w) = \{ 1 \}$ for any valuation $w$ of $\algcl K$ extending $v$.
\end{mydef}

If the representation $\rho$ is unramified at $v$, then the
\dpage
restriction of $\rho$ to $D_w$ factors through $D_w/I_w$ for any $w\mid v$;
hence $\rho(F_w) \in \Aut(V)$ is defined; we call $\rho(F_w)$ the
\strong{Frobenius} of $w$ in the representation $\rho$, and we denote it by
$F_{w, \rho}$. The conjugacy class of $F_{w, \rho}$ in $\Aut(V)$ depends only
on $v$; it is denoted by $F_{v, \rho}$. If $L/K$ is the extension of $K$
corresponding to $H = \Ker(\rho)$, then $\rho$ is unramified at $v$ if and only
if $v$ is unramified in $L/K$.

\subsection{\v Cebotarev's density theorem}\label{sec:I_22}
Let $P$ be a subset of $M_K^0$. For each integer $n$, let $a_n(P)$
be the number of $v \in P$ such that $\numnorm v \le n$. If $a$ is a real number,
one says that $P$ \strong{has density} $a$ if
\[
	\lim \frac{a_n(P)}{a_n(M_K^0)} = a \qquad \text{when}\quad n \to \infty.
\]
Note that $a_n(M_K^0) \sim n/\log(n)$, by the prime number theorem (cf.\
Appendix, or \cite{13}, chap.~VIII), so that the above relation may be
rewritten:
\[
	a_n(P) = a \cdot \frac{n}{\log(n)} + o \left(\frac{n}{\log(n)}\right).
\]
\begin{ex}
A finite set has density $0$. The set of $v \in M_K^0$ of degree $1$ (i.e.\ such
that $\numnorm v$ is prime) has density $1$. The set of ordinary prime numbers
whose first digit (in the decimal system, say) is $1$ has no density.
\end{ex}

\begin{thm}\label{thm-chebotarev}
Let $L$ be a finite Galois extension of the number field $K$, with Galois group
$G$. Let $X$ be a subset of $G$, stable by
\dpage
conjugation. Let $P_X$ has density equal to $\Card(X)/\Card(G)$. 
\end{thm}
For the proof, see \cite{7}, \cite{1}, or the Appendix.

\begin{cor}
For every $g \in G$, there exist infinitely many unramified places $w \in
M_K^0$ such that $F_w = g$.
\end{cor}

For infinite extensions, we have:
\begin{cor}
Let $L$ be a Galois extension of $K$, which is unramified outside a finite set
$S$.
\begin{enumerate}[a)]
\item The Frobenius elements of the unramified places of $L$ are dense in
	$\Gal(L/K)$.
\item Let $K$ be a subset of $\Gal(L/K)$, stable by conjugation. Assume that
	the boundary of $X$ has measure zero with respect to the Haar measure
	$\mu$ of $X$, and normalize $\mu$ such that its total mass is $1$. Then
	the set of places $v \not\in S$ such that $F_v \subset X$ has a density
	equal to $\mu(X)$.
\end{enumerate}
\end{cor}

Assertion (b) follows from the theorem, by writing $L$ as an increasing union
of finite Galois extensions and passing to the limit (one may also use Prop.~1
of the Appendix). Assertion (a) follows from (b) applied to a suitable
neighborhood of a given class of $\Gal(L/K)$.

\subsubsection*{Exercise}
Let $G$ be an $\ell$-adic Lie group and let $X$ be an analytic subset of $G$
(i.e.\ a set defined by the vanishing of a family of analytic functions on $G$).
Show that the boundary of $X$ has measure zero
\dpage
with respect to the Haar measure of $G$.

\subsection{Rational \texorpdfstring{$\ell$}{ℓ}-adic representations}
\label{sec:I_23}
Let $\rho$ be an $\ell$-adic representation of the number field $K$. If $v \in
M_K^0$, and if $v$ is unramified with respect to $\rho$, we let $P_{v,\rho}(T)$
denote the polynomial $\det(1 - F_{v,\rho} T)$.

\begin{mydef}
The $\ell$-adic representation $\rho$ is said to be
\strong{rational}\index{Rational (representation)} (resp.\ 
\strong{integral}\index{Integral (representation)}) if there exists a finite
subset $S$ of $M_K^0$ such that
\begin{enumerate}[(a)]
	\item Any element of $M_K^0 \setminus S$ is unramified with respect to 
	$\rho$.
	\item If $v \not\in S$, the coefficients of $P_{v,\rho}(T)$ belong to
		$\Q$ (resp.\ to $\Z$).
\end{enumerate}
\end{mydef}

\begin{obs}\label{rmk:I_23_1}
	Let $K'/K$ be a finite extension. An $\ell$-adic representation $\rho$
	of $K$ defines (by restriction) an $\ell$-adic representation
	$\rho_{/K'}$ de $K'$. If $\rho$ is rational (resp.\ integral), then the
	same is true for $\rho_{/K'}$; this follows from the fact that the
	Frobenius elements relative to $K'$ are powers of those relative to
	$K$.
\end{obs}

\begin{ex}
The $\ell$-adic representations of $K$ given in examples~\ref{sec:I_121},
\ref{sec:I_122}, \ref{sec:I_123} of section \ref{sec:I_12} are rational (even
\emph{integral}) representation.
In example~\ref{sec:I_121}, one can take for $S$ the set $S_{\ell}$ of
elements $v$ of $M_K^0$ with $\rho_v = \ell$; In examples~\ref{sec:I_122},
\ref{sec:I_123}, one can take for $S$ the union of $S_\ell$ and the set
$S_A$ where $A$ has ``bad reduction''; the fact that the corresponding
Frobenius has an integral characteristic polynomial (which is independent of
$\ell$) is a consequence of Weil's results on endomorphisms of abelian
varieties (cf.\ \cite{40} and \cite{12}, chap.~VII).
\dpage
The rationality of the cohomology representation is a well-known open question.
\todo[color=pink!40]{Ver si sigue siendo una pregunta abierta.}
\end{ex}

\begin{mydef}
Let $\ell'$ be a prime, $\rho'$ an $\ell'$-adic representation of $K$, and assume that $\rho$, $\rho'$ are rational. Then $\rho$, $\rho'$ are said to be \strong{compatible} if there exists a finite subset $S$ of $M_K^0$ such that $\rho$ and $\rho'$ are unramified outside of $S$ and $P_{v,\rho}(T) = P_{v,\rho'}(T)$ for $v \in M_K^0 \setminus S$.
\end{mydef}
(In other words, the characteristic polynomials of the Frobenius elements arte
the same for $\rho$ and $\rho'$, at least for almost all $v$'s.)

If $\rho\colon \Gal(\algcl{K}/K) \to \Aut(V)$ is rational $\ell$-adic
representation of $K$, then $V$ has a composition series
\[
	V = V_0 \supset V_1 \supset \cdots \supset V_q = 0
\]
of $\rho$-invariants subspaces with $V_i / V_{i+1}$ ($0 \leq i \leq q - 1$)
\emph{simple} (i.e.\ \emph{irreducible}). The $\ell$-adic representation
$\rho'$ of $K$ defined by $V' = \sum_{i=0}^{q-1} V_i / V_{i + 1}$ is
semi-simple, rational, and compatible with $\rho$; it is the
``semi-simplification''\break of $V$. 

\begin{thm}
Let $\rho$ be a rational $\ell$-adic representation of $K$, let $\ell'$ be a prime. Then there exists at most one (up to isomorphism) $\ell'$-adic rational representation $\rho'$ of $K$ which is semi-simple and compatible with $\rho$.
\end{thm}

(Hence there exists a unique (up to isomorphism) rational semi-simple $\ell$-adic representation compatible with $\rho$.)

\begin{proof}
Let $\rho_1'$, $\rho_2'$ be semi-simple $\ell$-adic representations of $K$ 
\dpage
which are rational and compatible with $\rho$.

We first prove that $\Tr(\rho_1'(g)) = \Tr(\rho_2'(g))$ for all $g \in G$. Let $H = G/(\Ker(\rho_1') \cap \Ker(\rho_2'))$; the representations $\rho_1'$, $\rho_2'$ may be regarded as representations of $H$, and it suffices to show that $\Tr(\rho_1'(h)) = \Tr(\rho_2'(h))$ for all $h \in H$. Let $K' \subset \algcl{K}$ be the fixed field of $H$. Then by the compatibility of $\rho_1'$, $\rho_2'$ there is a finite subset $S$ of $M_K^0$ such that for all $v \in M_K^0 \setminus S$, $w \in M_K'^0$, $w \mid v$, we have $\Tr(\rho_1'(F_w)) = \Tr(\rho_2'(F_w))$. But, by cor. \ref{thm-chebotarev} to \v{C}ebotarev's theorem (cf.\ \ref{sec:I_22}) the $F_w$ are dense in $H$. Hence $\Tr(\rho_1'(h)) = \Tr(\rho_2'(h))$ for all $h \in H$ since $\Tr \circ \rho_1'$, $\Tr \circ \rho_2'$ are continuous.

The theorem now follows from the following result applied to the group ring $\Lambda = \Q_{\ell}[H]$.
\end{proof}

\begin{lem}
Let $k$ be a field of characteristic zero, let $\Lambda$ be a $k$-algebra, and let $\rho_1$, $\rho_2$ be two finite-dimensional linear representations of $\Lambda$. If $\rho_1$, $\rho_2$ are semi-simple and have the same trace ($\Tr \circ \rho_1 = \Tr \circ \rho_2)$, then they are isomorphic.
\end{lem}

For the proof see Bourbaki, Alg., ch. 8, \S12, n°1, prop.~3.
\todo[color=pink!40]{Cómo citar esto}

\begin{mydef}
For each prime $\ell$ let $\rho_\ell$ be a rational $\ell$-adic representation of $K$. The system $(\rho_\ell)$ is said \strong{to be compatible}\index{Compatible (system $(\rho_\ell)$)} if $\rho_\ell$, $\rho_\ell'$ are compatible for any two primes $\ell$, $\ell'$. The system $(\rho_\ell)$ is said \strong{to be strictly compatible}\index{Strictly compatible (system $(\rho_\ell))$} if there exists a finite subset $S$ of $M_K^0$ such that:
\begin{enumerate}[(a)]
	\item Let $S_\ell = \{ v \mid \rho_v = \ell \}$. Then, for every $v \not\in S \cup S_\ell$, $\rho_\ell$ is unramified at $v$ and $P_{v,\rho_\ell}(T)$ has rational coefficients.
\item $P_{v,\rho_\ell}(T) = P_{v,\rho_{\ell'}}(T)$ if $v \not\in S \cup S_\ell \cup S_{\ell'}$.
\end{enumerate}
\end{mydef}
\dpage
When a system $(\rho_\ell)$ is strictly compatible, there is a smallest finite set $S$ having properties (a) and (b) above. We call it the \strong{exceptional set}\index{Exceptional set (of a system)} of the system.

\begin{ex}
The systems of $\ell$-adic representations given in examples \ref{sec:I_121},
\ref{sec:I_122}, \ref{sec:I_123} of section \ref{sec:I_12} are strictly
compatible. The exceptional set of the first one is empty. The exceptional set
of example~\ref{sec:I_122} (resp.\ \ref{sec:I_123}) is the set of
places where the elliptic curve (resp.\ the abelian variety) has ``bad
reduction'', cf.\ \cite{32}.
\end{ex}

\subsubsection*{Questions}
\begin{enumerate}
	\item Let $\rho$ be a rational $\ell$-adic representation. Is true that $P_{v,\rho}$ has coefficients for all $v$ such that $\rho$ is unramified at $v$?

A somewhat similar question is:

	Is any compatible system strictly compatible?
	\item Can any rational $\ell$-adic representation be obtained (by tensor products, direct sums, etc.) from ones coming from $\ell$-adic cohomology?
	\item Given a rational $\ell$-adic representation $\rho$ of $K$, and a prime $\ell'$, does there exist a rational $\ell'$-adic representation $\rho'$ of $K$ compatible with\break $\rho$? $\rightarrow$ [no: easy counter-examples].
	\item Let $\rho$, $\rho'$ be rational $\ell$, $\ell'$-adic representations of $K$ which are compatible and semi-simple.
	\begin{enumerate}[(i)]
		\item If $\rho$ is abelian (i.e.\ , if $\Img(\rho)$ is abelian), is it true that $\rho'$ is abelian? (We shall see in chapter~\ref{ch:iii} that this is true at least if $\rho$ is ``locally algebraic''.) $\rightarrow$ [yes: this follows from \cite{36}.]
		\item Is it true that $\Img(\rho)$ and $\Img(\rho')$ are Lie groups of the
\dpage		
		same dimension? More optimistically, is it true that there exists a Lie algebra $\mathfrak{g}$ over $\Q$ such that $\Lie(\Img(\rho)) = \mathfrak{g} \otimes_\Q \Q_{\ell}$ and $\Lie(\Img(\rho')) = \mathfrak{g} \otimes_\Q \Q_{\ell'}$?
	\end{enumerate}
	\item Let $X$ be a non-singular projective variety defined over $K$, and
	let $i$ be an integer. Is the $i$-th cohomology representation $H_\ell^i(\sepcl{X})$ semi-simple? Does its Lie algebra contain the homotheties if $i\geq1$? (When $i=1$, an affirmative answer to either one of these questions would imply a positive solution for the ``conguence subgroup problem'' on abelian varieties, cf.\ \cite{24}, \S3.) $\rightarrow$ [yes: for $i=1$: see \cite{48} and also \cite{75}.]
\end{enumerate}

\begin{obs}
The concept of an $\ell$-adic representation can be generalized by replacing
the prime $\ell$ by a place $\lambda$ of a number field  $E$. A $\lambda$-adic
representation is then a continuous homomorphism $\Gal(\sepcl{K}/K) \to
\Aut(V)$, where $V$ is a finite-dimensional vector space over the local field
$E_\lambda$. The concepts of rational $\lambda$-adic representation, compatible
representations, etc., can be defined in a way similar to the $\ell$-adic case.
\end{obs}

\subsubsection*{Exercise}
\begin{enumerate}
	\item Let $\rho$ and $\rho'$ be two rational, semi-simple, compatible representations. Show that, if $\Img(\rho)$ is finite, the same is true for $\Img(\rho')$ and that $\Ker(\rho) = \Ker(\rho')$. (Apply exer. 3 of \ref{sec:I_11} to $\rho'$ and to $U = \Ker(\rho)$.) Generalize this to $\lambda$-adic representations (with respect to a number field $E$).
	\item Let $\rho$ (resp. $\rho'$) be a rational $\ell$-adic (resp. $\ell'$-adic) representation of $K$, of degree $n$. Assume $\rho$ and $\rho'$ are compatible. If $s \in G = \Gal(\algcl{K}/K)$, let $\sigma_i(s)$ (resp. $\sigma_i'(s)$) be the
\dpage 
	$i$-th coefficient of the characteristic polynomial of $\rho(s)$ (resp. $\rho'(s)$). Let $P(X_0,\hdots,X_n)$ be a polynomial with rational coefficients, and let $X_P$ (resp. $X'_P$) be the set of $s \in G$ such that $P(\sigma_0(s),\hdots,\sigma_n(s)) = 0$ (resp. $P(\sigma_0'(s),\hdots,\sigma_n'(s)) = 0$).
	\begin{enumerate}
		\item Show that the boundaries of $X_P$ and $X_P'$ have measure zero for the Haar measure $\mu$ of $G$ (use Exer. of \ref{sec:I_22}).
		\item Assume that $\mu$ is normalized, i.e.\ $\mu(G) = 1$. Let $T_P$ be the set of $v \in M_K^0$ at which $\rho$ is unramified, and for which the coefficients $\sigma_0,\hdots,\sigma_n$ of characteristic polynomial of $F_{v,\rho}$ satisfy the equation $P(\sigma_0,\hdots,\sigma_n) = 0$. Show that $T_P$ has density equal to $\mu(X_P)$.
		\item Show that $\mu(X_P) = \mu(X_P')$.
	\end{enumerate}
\end{enumerate}

\subsection{Representations with values in a linear algebraic group}
\label{sec:I_24}
Let $H$ be a linear algebraic group defined over a field $K$. If
$k'$ is a commutative $k$-algebra, let $H(k')$ denote the group of points
of $H$ with values in $k'$. Let $A$ denote the coordinate ring (or
``affine ring'') of $H$. An element $f \in A$ is said to be \strong{central} if
$f(xy) = f(yx)$ for any $x, y \in H(k')$ and any commutative $k$-algebra
$k'$. If $x \in H(k')$ we say that the conjugacy class of $x$ in $H$ is
\strong{rational over $k$} if $f(x) \in k$ for any central element $f$ of $A$.

\begin{mydef}
Let $H$ be a linear algebraic group over $\Q$, and let
$K$ be a field. A continuous homomorphism $\rho \colon \Gal(\sepcl K/K) \to H(\Q_\ell)$\index{$\ell$-adic representation of a field}
is called an $\ell$-adic representation of $K$ with values in $H$.
\end{mydef}
(Note that $H(\Q_\ell)$ is, in a natural way, a topological group and even
an $\ell$-adic Lie group.)

If $K$ is a number field, one defines in an obvious way what it
\dpage
means for $\rho$ to be unramified at a place $v \in M_K^0$; if $w\mid v$, one
defines the Frobenius element $F_{w, \rho} \in H(\Q_\ell)$ and its conjugacy
class $F_{v, \rho}$. We say, as before, that $\rho$ is \strong{rational} if
\begin{enumerate}[(a)]
\item\label{ax:rational_ladic_a}
	there is a finite set $S$ of $M_K^0$ such that $\rho$ is unramified
	outside $S$,
\item\label{ax:rational_ladic_b}
	if $v \notin S$, the conjugacy class $F_{v, \rho}$ is rational over $\Q$.
\end{enumerate}
Two rational representations $\rho$, $\rho'$ (for primes $\ell$, $\ell'$) are said to
be \strong{compatible} if there exists a finite subset $S$ of $M_K^0$ such that $\rho$
and $\rho'$ are unramified outside $S$ and such that for any central 
element $f \in A$ and any $v \in M_K^0 \setminus S$ we have $f(F_{v, \rho}) =
f(F_{v, \rho})$. One defines in the same way the notions of \strong{compatible}
and \strong{strictly compatible systems} of rational representations.

\begin{obs}
\begin{enumerate}
\item If the algebraic group $H$ is abelian, then condition
	\ref{ax:rational_ladic_b} above means that $F_{v, \rho}$ (which is now
	an element of $H(\Q_\ell)$) is rational over $\Q$, i.e.\ belongs to
	$H(\Q)$.
\item Let $V_0$ be a finite-dimensional vector space over $\Q$, and
	let $\GL_{V_0}$ be the linear algebraic group over $\Q$ whose group of
	points in any commutative $\Q$-algebra $k$ is $\Aut(V_0 \otimes_\Q k)$; in 
	particular, if $V_\ell = V_0 \otimes_\Q \Q_\ell$, then
	$\GL_{V_0}(\Q_\ell) = \Aut(V_\ell)$. If $\varphi \colon H \to
	\GL_{V_0}$ is a homomorphism of linear algebraic groups over $\Q$, call
	$\varphi_\ell$ the induced homomorphism of $H(\Q_\ell)$ into
	$\GL_{V_0}(\Q_\ell) = \Aut(V_\ell)$. If $\rho$ is an $\ell$-adic
	representation of $\Gal(\algcl K/K)$ into $H(\Q_\ell)$, one gets by
	composition a linear $\ell$-adic representation $\varphi_\ell \circ
	\rho \colon \Gal(\sepcl K/K) \to \Aut(V_\ell)$. Using the fact that the
	coefficients of the characteristic polynomial are central functions,
	one sees that
	\dpage
	$\varphi_\ell \circ \rho$ is \emph{rational} if $\rho$ is rational ($K$
	a number field).  Of course, compatible representations in $H$ give
	compatible linear representations. We will use this method of
	constructing compatible representations in the case where $H$ is
	abelian (see ch.~\ref{ch:ii}, \ref{sec:II_25}).
\end{enumerate}
\end{obs}

\subsection{\texorpdfstring{$L$}{L}-functions attached to rational 
representations}

Let $K$ be a number field and let $\rho=(\rho_\ell)$ be a strictly compatible 
system of rational $\ell$-adic representations, with exceptional set $S$. If $v 
\not\in S$, denote by $P_{v,\rho}(T)$ the rational polynomial does not depend 
on the choice of $\ell$. Let $s$ be a complex number. 

One has:
\begin{align*}
	P_{v,\rho}(\numnorm v)^{-s} 
	&= \det(1 - F_{v,\rho} / (\numnorm v)^{s})\\
	&= \prod_i (1 - \lambda_{i,v} / (\numnorm v)^{s}),
\end{align*}

where the $\lambda_{i,v}$'s are the eigenvalues of $F_{v,\rho}$ (note that the 
$\lambda_{i,v}$'s are algebraic numbers and hence may be identified with 
complex numbers). Put:
\[
L_\rho(s) = \prod_{v \not\in S} \frac{1}{P_{v,\rho}((\numnorm v)^{-s})}.
\]

This is a \emph{formal} Dirichlet series $\sum_{n=1}^{\infty} a_n / n^s$, with 
coefficients in $\Q$.

In all known cases, there exists a constant $k$ such that $\left|{\lambda_{i,v} 
\leq (\numnorm v)^k}\right|$, and this implies that $L_\rho$ is convergent in 
some half plane $\Re(s) > C$; one conjectures it extends to a meromorphic 
function in the whole plane.
\dpage
When $\rho$ comes from $\ell$-adic cohomology, 
there are some further conjectures on the zeros and poles of $L_\rho$, cf.\ Tate 
\cite{36}; these, as indicated by Tate, may be applied to get equidistribution 
properties of Frobenius elements, cf.\ Appendix~\ref{sec:I_A}.

\begin{obs}
\begin{enumerate}
	\item\label{rmk:I_24_1} One can also associate $L$-functions to $E$-\emph{rational} systems of $\lambda$-adic representations (\ref{rmk:I_23_1}, Remark), where $E$ is a number field, once an embedding of $E$ into $\C$ has been chosen.
\todo[pinktask]{No cacho si el C es los complejos o qué weá}
	\item We have given a definition of the local factors of $L_\rho$ only at 
	the places $v \not\in S$. One can give a more sophisticated definition in 
	which local factors are defined for all places, even (with suitable 
	hypotheses) for primes at infinity (gamma factors); this is necessary when 
	one wants to study functional equations. We don't go into this here. 
	$\rightarrow$ [see \cite{51}, \cite{74}.]
	
	\item Let $\phi(s) = \sum a_n / n^s $ be a Dirichlet series. Using the 
	theorem in~\ref{sec:I_23}, one sees that there is (up to isomorphism) at 
	most one semi-simple system $\rho = (\rho_\ell)$ over $\Q$ such that 
	$L_\rho = \phi$. Whether there does exist one (for a given $\phi$) is often a quite interesting question. For instance, is it so for Ramanujan's $\phi(s) = \sum_{n = 1}^{\infty} \tau(n)/n^s$, where $\tau(n)$ is defined by the identity
	\[
	x \prod_{n=1}^{\infty}(1 - x^n)^24 = \sum_{n=1}^{\infty}\tau(n)x^n ?
	\]
	There is considerable numerical evidence for this, based on the congruence 
	properties of $\tau$ (Swinnerton-Dyer, unpublished); of course,
	\todo[pinktask]{¿Seguirá así?}
	such a $\rho$ would be of dimension $2$, and its exceptional set $S$ would be empty. $\rightarrow$ [proved by Deligne: see \cite{49}, \cite{50}, \cite{65},\dots]
\end{enumerate}
\end{obs}

More generally, there seems to be a close connection between
\dpage
modular forms, such as $\sum \tau(n) x^n$, and rational (or algebraic) $\ell$-adic representations; see for instance \citeauthor{33}~\cite{33} and \citeauthor{45}~\cite{45}. $\rightarrow$ [see also \cite{49}, \cite{51}, \cite{65}, \cite{66}, \cite{68}, \cite{84}.]

\begin{ex}
\begin{enumerate}[1.]
	\item If $G$ acts through a \emph{finite} group, $L_\rho$ is an Artin (non abelian) $L$-series, at least up to a finite number of factors (cf.\ \cite{1}). All Artin $L$-series are gotten in this way, provided of course one uses $E$-rational representations (cf.\ Remark~\ref{rmk:I_24_1}) and not merely rational ones.
	\item If $\rho$ is the system associated with an elliptic curve $E$ (cf.\ \ref{sec:I_12}), the corresponding $L$-function gives the non-trivial part of zeta function of $E$. The symmetric powers of $\rho$ give the zeta functions of the products $E \times \hdots \times E$, cf.\ \citeauthor{36}~\cite{36}.
\end{enumerate}
\end{ex}

\begin{subappendices}
\section{Equipartition and \texorpdfstring{$L$}{L}-functions}
\label{sec:I_A}

\subsection{Equipartition}
Let $X$ be a compact topological space and $C(X)$ the Banach
space of continuous, complex-valued, functions on $X$, with its usual norm
$\|f\| = \sup_{x \in X} |f(x)|$. For each $x \in X$ let $\delta_x$ be the Dirac
measure associated to $x$; if $f \in C(X)$, we have $\delta_x(f) = f(x)$.

Let $(x_n)_{n\ge 1}$ be a sequence of points of $X$. For $n \ge 1$, let
\[
	\mu_n = \frac{ \delta_{x_1} + \cdots + \delta_{x_n} }{n}
\]
and
\dpage
let $\mu$ be a Radon measure on $X$ (i.e.\ a continuous linear form on $C(X)$,
cf.\ Bourbaki, Int., chap.~III, \S 1). The sequence $(x_n)$ is said to be
\strong{$\mu$-equidistributed},\index{Equidistribution}
or \emph{$\mu$-uniformly distributed}, if $\mu_n \to \mu$ \emph{weakly} as $n
\to \infty$, i.e.\ if $\mu_n(f) \to \mu(f)$ as $n \to \infty$ for any $f \in
C(X)$. Note that this implies that $\mu$ is positive and of total mass 1.  Note
also that $\mu_n(f) \to \mu(f)$ means that
\[
	\mu(f) = \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^{n} f(x_i).
\]

\begin{lem}\label{lem:IA_1}
Let $(\varphi_\alpha)$ be a family of continuous functions on $X$ with the
property that their linear combinations are dense in $C(X)$. Suppose that, for
all $\alpha$, the sequence $( \mu_n(\varphi_\alpha) )_{n>1}$ has a limit.  Then
the sequence $(x_n)$ is equidistributed with respect to some measure $\mu$ it
is the unique measure such that $\mu(\varphi_\alpha) = \lim_{n\to\infty}
\mu_n(\varphi_\alpha)$ for all $\alpha$.
\end{lem}

If $f \in C(X)$, an argument using equicontinuity shows that the sequence
$(\mu_n(f))$ has a limit $\mu(f)$, which is continuous and linear in $f$; hence
the lemma.

\begin{prop}
Suppose that $(x_n)$ is $\mu$-equidistributed. Let $U$ be a subset of $X$ whose
boundary has $\mu$-measure zero, and, for all $n$, let $n_U$ be the number of
$m \le n$ such that $x_m \in U$. Then $\lim_{n\to\infty} (n_U/n) = \mu(U)$.
\end{prop}
Let $\Int U$ be the interior of $U$. We have $\mu(\Int U) = \mu(U)$.  Let
$\varepsilon > 0$.  By the definition of $\mu(\Int U)$ there is a continuous
function $\varphi \in C(X)$, $0 \le \varphi \le 1$, with $\varphi = 0$ on $X
\setminus \Int U$ and $\mu(\varphi) \ge \mu(U) - \varepsilon$.  Since
$\mu_n(\varphi) \le n_U/n$ we have
\[
	\liminf_{n\to\infty} \frac{n_U}{n} \ge \lim_{n\to\infty} \mu_n(\varphi)
	= \mu(\varphi) \ge \mu(U) - \varepsilon,
\]
\dpage
from which we obtain $\liminf n_U/n \ge \mu(U)$. The same argument
applied to $X \setminus U$ shows that
$$ \liminf_{n\to\infty} \frac{n - n_U}{n} \ge \mu(X \setminus U). $$
Hence $\limsup_n n_U/n \le \mu(U) \le \liminf n_U/n$, which implies the
proposition.

\begin{ex}
\begin{enumerate}[label=\arabic*., ref=\arabic*]
\item Let $X = [0,1]$, and let $\mu$ be the Lebesgue measure. A sequence
	$(x_n)$ of points of $X$ is $\mu$-equidistributed if and only if for
	each interval $[a, b]$, of length $d > 0$ in $[0,1]$ the number of $m
	\le n$ such that $x_m \in [a, b]$ is equivalent to $dn$ as $n \to
	\infty$.
\item\label{ex:IA1_2}
	Let $G$ be a compact group and let $X$ be the space of conjugacy classes
	of $G$ (i.e.\ the quotient space of $G$ by the equivalence relation
	induced by inner automorphisms of $G$). Let $\mu$ be a measure on $G$;
	its image of $G \to X$ is a measure on $X$, which we also denote by
	$\mu$. We then have:
\end{enumerate}
\end{ex}

\begin{prop}\label{prop:IA_2}
The sequence $(x_n)$ of elements of $X$ is $\mu$-equidistributed if and only if
for any irreducible character $\chi$ of $G$ we have
$$ \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^{n} \chi(x_i) = \mu(\chi). $$
\end{prop}

The map $C(X) \to C(G)$ is an isomorphism of $C(X)$ onto the
space of central functions on $G$; by the Peter-Weyl theorem, the
\dpage
irreducible characters $\chi$ of $G$ generate a dense subspace of $C(X)$.
Hence the proposition follows from lemma~\ref{lem:IA_1}.

\begin{corp}
Let $\mu$ be the Haar measure of $G$ with $\mu(G) = 1$.
Then a sequence $(x_n)$ of elements of $X$ is $\mu$-equidistributed if and
only if for any irreducible character $\chi$ of $G$, $\chi \ne 1$ we have
$$ \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^{n} \chi(x_i) = 0. $$
\end{corp}
This follows from Prop.~\ref{prop:IA_2} and the following facts:
\begin{align*}
	\mu(\chi) &= 0 \qquad \text{if $\chi$ is irreducible $\ne 1$} \\
	\mu(1) &= 1.
\end{align*}

\begin{corp}[\citeauthor{46}~\cite{46}]
Let $G = \R/\Z$, and let $\mu$ be the normalized Haar measure on $G$. Then
$(x_n)$ is $\mu$-equidistributed if and only if for any integer $m \ne 0$ we
have
$$ \sum_{n\le N} e^{2\pi mi x_n} = o(N) \qquad (N \to \infty). $$
\end{corp}
For the proof, it suffices to remark that the irreducible characters of $\R/\Z$
are the mappings $x \mapsto e^{2\pi mi x}$ ($m \in \Z$).

\subsection{The connection with \texorpdfstring{$L$}{L}-functions}
\label{sec:I_A2}
Let $G$ and $X$ be as in Example~\ref{ex:IA1_2} above: $G$ a compact group and
$X$ the space of its conjugacy classes. Let $x_v$, $v \in M$, be a family of
elements of $X$, indexed by a denumerable set $M$, and let $v \mapsto \numnorm
v$ be a function on $M$ with values in the set of integers $\ge 2$.
\dpage
We make the following \emph{hypotheses:}
\begin{enumerate}[(1), series=Lfunc_hyp]
\item
	The infinite product $\prod_{v\in M} \frac{1}{1 - (\numnorm v)^{-s}}$
	converges for every $s \in \C$ with $\Re(s) > 1$, and extends to a
	meromorphic function on $\Re(s) > 1$ having neither zero nor pole
	except for a simple pole at $s = 1$.
\item Let $\rho$ be an irreducible representation of $G$, with character
	$\chi$, and put
	\[
		L(s, \rho) = \prod_{v\in M} \frac{1}{\det(1 -
		\rho(x_v)(\numnorm v)^{-s})}.
	\]
	Then this product converges for $\Re(s) > 1$, and extends to a
	meromorphic function on $\Re(s) > 1$ having neither zero nor pole
	except possibly for $s = 1$.
\end{enumerate}
The \emph{order} of $L(s, \rho)$ at $s = 1$ will be denoted by $-c_\chi$. Hence,
if $L(s,\rho)$ has a pole (resp.\ a zero) of order $m$ at $s = 1$, one has
$c_\chi = m$ (resp.\ $c_\chi = -m$).

Under these assumptions, we have:
\begin{thm}\label{thm:I_A2_1}
\begin{enumerate}[(a)]
\item The number of $v \in M$ with $\numnorm v \le n$ is equivalent to $n/\log
	n$ (as $n \to \infty$).
\item For any irreducible character $\chi$ of $G$, we have
	$$ \sum_{\numnorm v\le n} \chi(x_v) = c_\chi \, \frac{n}{\log n} +
	o(n/\log n), \qquad (n \to \infty). $$
\end{enumerate}
\end{thm}
The theorem results, by a standard argument, from the theorem of
Wiener-Ikehara, cf.\ \ref{sec:I_A3} below.
Suppose now that the function $v \mapsto \numnorm v$ has the following
property:

\begin{enumerate}[resume*=Lfunc_hyp]
\item\dpage
	There exists a constant $C$ such that, for every $n \in \Z$, the number
	of $v \in M$ with $\numnorm v = n$ is $\le C$.
\end{enumerate}
One may then arrange the elements of $M$ as a sequence
$(v_i)_{i\ge 1}$. so that $i \le j$ implies $\numnorm v_i \le \numnorm v_j$ (in
general, this is possible in many ways). It then makes sense to speak about the
equidistribution of the sequence of $x_v$'s; using (3), one shows easily that
this does not depend on the chosen ordering of $M$. Applying
theorem~\ref{thm:I_A2_1} and proposition~\ref{prop:IA_2}, we obtain:
\begin{thm}
The elements $x_v$ ($v \in M$) are equidistributed in $X$
with respect to a measure $\mu$ such that for any irreducible character
$\chi$ of $G$ we have
$$ \mu(\chi) = c_\chi. $$
\end{thm}
\begin{cor}
	The elements $x_v$ ($v \in M$) are equidistributed in $X$
	normalized Haar measure of $G$ if and only if $c_\chi = 0$ for every
	irreducible character $\chi \ne 1$ of $G$, i.e.\, if and only if the
	$L$-functions relative to the non trivial irreducible characters of $G$
	are holomorphic and non zero at $s = 1$.
\end{cor}

\begin{ex}
\begin{enumerate}[series=ex_IA3]
\item Let $G$ be the Galois group of a \emph{finite} Galois extension
	$L/K$ of the number field $K$, let $M$ be the set of unramified places
	of $K$, let $x_v$ be the Frobenius conjugacy class defined by $v \in M$,
	and let $\numnorm v$ be the norm of $v$, cf.\ \S\ref{sec:I_21}.

	Properties (1), (2), (3) are satisfied with $c_\chi = 0$ for all
	irreducible $\chi \ne 1$. This is trivial for (3). For (1), one remarks
	that $L(s,l)$ is the zeta function of $K$ (up to a finite number of
	terms), hence has a simple pole at $s = 1$ and is holomorphic on the
	\dpage
	rest of the line $\Re(s) = 1$, cf.\ for instance
	\citeauthor{13}~\cite{13}, chap.\ VII; for a proof of (2), cf.\
	\citeauthor{1}~\cite[121]{1}.  Hence theorem~\ref{thm:I_A2_2} gives the
	equidistribution of the Frobenius elements, i.e.\ the \v Cebotarev
	density theorem, cf.\~\ref{sec:I_22}.

\item Let $C$ be the idèle class group of a number field $K$, and let $\rho$ be
	a continuous homomorphism of $C$ into a compact abelian Lie group $G$.
	An easy argument (cf.\ ch.~\ref{ch:iii}, \ref{sec:III_22}) shows that
	$\rho$ is almost everywhere unramified (i.e.\, if $U_v$ denotes the
	group of units at $v$, then $\rho(U_v) = 1$ for almost all $v$). Choose
	$\pi_v \in K$ with $v(\pi_v) = 1$. If $\rho$ is unramified at $v$, then
	$\rho(\pi_v)$ depends only on $v$, and we set $x_v = \rho(\pi_v)$. We
	make the following \emph{assumption:}
	\begin{displayquote}
		\slshape
		\textbf{(*)}
		The homomorphism $\rho$ maps the group $C$ of idèles of
		volume 1 onto $G$.
	\end{displayquote}
	(Recall that the \strong{volume} of an idèle $\vec a = (a_v)$ is
	defined as the product of the normalized absolute values of its
	components $a_v$, cf.\ \citeauthor{13}~\cite{13} or
	\citeauthor{44}~\cite{44}.)

	Then, the elements $x_v$ are \emph{uniformly distributed} in $G$ with
	respect to the normalized Haar measure. This follows from
	theorem~\ref{thm:I_A2_1} and the fact that the $L$-functions relative
	to the irreducible characters $\chi$ of $G$ are Hecke $L$-functions
	with Grössencharakters; these $L$-functions are holomorphic and
	non-zero for $\Re(s) \ge 1$ if $\chi \ne 1$, see \cite{13}, chap.\ VII.
\end{enumerate}
\end{ex}

\begin{obs}
This example (essentially due to Hecke) is given in Lang
(\emph{loc.\ cit.}, ch.~VIII, \S 5) except that Lang has replaced the condition
(*) by the condition ``$\rho$ is surjective'', which is insufficient. This
led him to affirm that, for example, the sequence $(\log p)_p$ (and also
the sequence $(\log n)_n$) is uniformly distributed modulo 1; however,
\dpage
one knows that this sequence is not uniformly distributed for any
measure on $\R/\Z$ (cf.\ \citeauthor{22}~\cite[179-180]{22}).
\end{obs}

\begin{enumerate}[resume*=ex_IA3]
\item (Conjectural example).
	Let $E$ be an elliptic curve defined over a number field $K$ and let
	$M$ be the set of finite places $v$ of $K$ such that $E$ has good
	reduction at $v$, cf.\ \ref{sec:I_12} and chap.~\ref{ch:iv}.  Let $v
	\in M$, let $\ell \ne p_v$ and let $F_v$ be the Frobenius conjugacy
	class of $v$ in $\Aut(T_\ell(E))$. The eigenvalues of $F_v$ are
	algebraic numbers; when embedded into $\C$ they give conjugate complex
	numbers $\pi_v$, $\bar{\pi}_v$ with $|\pi_v| = (\numnorm v)^{1/2}$.  We
	may write then
	\[
		\pi_v = (\numnorm v)^{1/2} e^{i \phi_v}; \quad
		\bar{\pi}_v = (\numnorm v)^{1/2} e^{-i \phi_v} \qquad
		\text{with } 0 \le \phi_v \le \pi.
	\]
	On the other hand, let $G = \SU(2)$ be the Lie group of $2 \times 2$
	unitary matrices with determinant 1. Any element of the space $X$ of
	conjugacy classes of $G$ contains a unique matrix of the form
	\[
		\begin{pmatrix}
			e^{i \phi} & 0 \\
			0 & e^{-i \phi}
		\end{pmatrix}, \qquad 0 \le \phi \le \pi.
	\]
	The image in $X$ of the Haar measure of $G$ is known to be
	$\frac{2}{\pi}\sin^2 \phi \, \ud\phi$.  The irreducible
	representations of $G$ are the $m$-th symmetric powers $\rho_m$ of the
	natural representation $\rho_1$ of degree 2.

	Take now for $x_v$ the element of $X$ corresponding to the angle
	$\phi = \phi_v$ defined above. The corresponding $L$ function,
	relative to $\rho_m$, is:
	\[
		L_{\rho_m}(s) = \prod_{v} \prod_{a=0}^{a=m} \frac{1}{ 1 -
		e^{i(m - 2a)\phi_v} (\numnorm v)^{-s} }.
	\]
	If we put:
	\[
		L_m^1(s) = \prod_{v} \prod_{a=0}^{a=m} \frac{1}{ 1 -
		\pi_v^{m-a} \bar{\pi}_v^a (\numnorm v)^{-s} }
	\]
	\dpage
	we have
	\[
		L_{\rho_m}(s) = L_m^1(s - m/2).
	\]
	The function $L$ has been considered by \citeauthor{36}~\cite{36}. He
	conjectures that $L_m^1$, for $m \ge 1$, is holomorphic and non zero
	for $\Re(s) \ge 1 + m/2$, provided that $E$ has no complex
	multiplication. Granting this conjecture, the corollary to theorem 2
	would yield the uniform distribution of the $x_v$'s, or, equivalently,
	that the angles $\phi_v$ of the Frobenius elements are uniformly
	distributed in $[0, \pi]$ with respect to the measure
	$\frac{2}{\pi}\sin^2 \phi \, \ud\phi$ (``conjecture of
	Sato-Tate'').

	One can expect analogous results to be true for other $\ell$-adic
	representations.
\end{enumerate}

\subsection{Proof of theorem~\ref{thm:I_A2_1}}
\label{sec:I_A3}
The logarithmic derivative of $L$ is
\[
	\frac{L'(s)}{L(s)} = -\sum_{\substack{v\ge 1 \\ m\ge 1}}
	\frac{\chi(x_v^m) \log(\numnorm v)}{(\numnorm v)^{ms}},
\]
where $x_v^m$ is the conjugacy class consisting of the $m$-th powers of
elements in the class $x_v$. One sees this by writing $L$ as the product
\[
	\prod_{j, v} \frac{1}{1 - \lambda_v^{(j)}( \numnorm v )^{-s}}
\]
where
\dpage
the $\lambda_v^{(j)}$ are the eigenvalues of $x_v$ in the given representation.
Now the series
\[
	\sum_{\substack{v\ge 1 \\ m\ge 1}} \frac{\log(\numnorm v)}{ |(\numnorm
	v)^{ms}| },
\]
converges for $\Re(s) > 1/2$.  Indeed it suffices to show that
\[
	\sum_v \frac{\log(\numnorm v)}{ (\numnorm v)^\sigma } < \infty
\]
if $\sigma > 1$; but this series is majorized by
\[
	\text{(Constant)} \times \sum_{v} \frac{1}{(\numnorm v)^{\sigma +
	\varepsilon}}, \qquad (\varepsilon > 0).
\]
On the other hand, the convergence for $\sigma > 1$ of the product
\[
	\prod_{v} \frac{1}{1 - (\numnorm v)^{-\sigma}}
\]
shows that
\[
	\sum_{v} \frac{1}{(\numnorm v)^\sigma} < \infty
\]
for $\sigma > 1$; hence our assertion. One can therefore write
\[
	\frac{L'(s)}{L(s)} = -\sum_{v} \frac{\chi(x_v) \log(\numnorm
	v)}{(\numnorm v)^s} + \phi(s)
\] 
where $\phi(s)$ is holomorphic for $\Re(s) > \frac{1}{2}$. Moreover, by
hypothesis,
\dpage
$L'/L$ can be extended to a meromorphic function on $\Re(s) \ge 1$
which is holomorphic except possibly for a simple pole at $s = 1$ with residue
$-c_\chi$. One may then apply the Wiener-Ikehara theorem (cf.\ \cite[123]{13}):
\begin{thm}
	Let $F(s) = \sum_{n=1}^\infty a_n/n^s$ be a Dirichlet series with
	complex coefficients. Suppose there exists a Dirichlet series $F(s) =
	\sum_n a_n^+/n^s$ with positive real coefficients such that
\begin{enumerate}[(a)]
	\item $|a_n| \le a_n^+$ for all $n$;
	\item The series $F^+$ converges for $\Re(s) > 1$;
	\item The function $F$ (resp.\ $F^+$) can be extended to a meromorphic
		function on $\Re(s)\ge 1$ having no poles except (resp.\ except
		possibly) for a simple pole at $s=1$ with residue $c_+ > 0$
		(resp.\ $c$).
\end{enumerate}
	Then
	\[
		\sum_{m\le n} a_n = cn + o(n) \qquad (n \to \infty),
	\]
	(where $c = 0$ if $F$ is holomorphic at $s = 1$).
\end{thm}
One applies this theorem to
\[
	F(s) = -\sum_{v} \frac{\chi(x_v) \log(\numnorm v)}{(\numnorm v)^s},
\]
and we take for $F^+$ the series
\[
	d \sum_{v} \frac{\log(\numnorm v)}{(\numnorm v)^s},
\]
where $d$ is the degree of the given representation $\rho$; this is possible
\dpage
since $\chi(x_v)$ is a sum of $d$ complex numbers of absolute value 1,
hence $|\chi(x_v)| \le d$; moreover, the series
\[
	\sum_{v} \frac{\log(\numnorm v)}{(\numnorm v)^s}
\]
differs from the logarithmic derivative of
\[
	\prod_{v} \frac{1}{1 - (\numnorm v)^{-s}}
\]
by a function which is holomorphic for $\Re(s) > 1/2$ as we saw above.
Hence by the Wiener-Ikehara theorem we have
\[
	\sum_{\numnorm v \le n} \chi(x_v) \log(\numnorm v) = c_\chi n + o(n)
	\qquad (n \to \infty).
\]
Consequently, by the Abel summation trick (cf.\ \cite[124]{13}, Prop.~1),
\[
	\sum_{\numnorm v \le n} \chi(x_v) = c_\chi \frac{n}{\log n} + o(n/\log
	n) \qquad (n \to \infty).
\]
and in particular,
\[
	\sum_{\numnorm v \le n} 1 = \frac{n}{\log n} + o(n/\log n) \qquad (n
	\to \infty).
\]
Hence,
\[
	\frac{ \sum_{\numnorm v \le n} \chi(x_v) }{ \sum_{\numnorm v \le n} 1 }
	\longrightarrow c_\chi \qquad \text{as } n\to \infty,
\]
and we may apply proposition~\ref{prop:I_A2_2} to conclude the proof.
\hfill
q.e.d.

\end{subappendices}