This question deserves an expert answer such as this one by Emerton, but allow me to offer an outsider's perspective. The following remarks are taken from my expository article arXiv:1007.4426.

First recall that the proportion of primes $p$ for which $T^2+1$ has
no roots (resp. two distinct roots) in $\mathbf{F}_p$ is $1/2$ (resp. $1/2$),
and that the proportion of $p$ for which $T^3-T-1$ has no roots
(resp. exactly one root, resp. three distinct roots) in $\mathbf{F}_p$ is
$1/3$ (resp. $1/2$, resp. $1/6$).

What is the analogue of the foregoing for the number of roots $N_p(f)$ of
$f=S^2+S-T^3+T^2$ in $\mathbf{F}_p$? A theorem of Hasse implies that $a_p=p-N_p(f)$ lies in the
interval $[-2\sqrt p,+2\sqrt p]$, so $a_p/2\sqrt p$ lies in $[-1,+1]$. What
is the proportion of primes $p$ for which $a_p/2\sqrt p$ lies in a given
interval $I\subset[-1,+1]$? It was predicted by Sato (on numerical grounds)
and Tate (on theoretical grounds), not just for this $f$ but for all
$f\in\mathbf{Z}[S,T]$ defining an "elliptic curve without complex multiplications",
that the proportion of such $p$ is equal to the area
$$
{2\over\pi}\int_{I}\sqrt{1-x^2}\;dx.
$$
of the portion of the unit semicircle projecting onto $I$. The Sato-Tate
conjecture for elliptic curves over $\mathbf{Q}$ was settled in 2008 by
Clozel, Harris, Shepherd-Barron and Taylor.

There is an analogue for "higher weights". Let $c_n$ (for $n>0$) be the
coefficient of $q^n$ in the formal product
$$
\eta_{1^{24}}=
q\prod_{k=1}^{+\infty}(1-q^{k})^{24}=0+1.q^1+\sum_{n>1}c_nq^n.
$$
In 1916, Ramanujan had made some deep conjectures about these
$c_n$; some of them, such as $c_{mm'}=c_mc_{m'}$ if $\gcd(m,m')=1$ and
$$
c_{p^r}=c_{p^{r-1}}c_p-p^{11}c_{p^{r-2}}
$$
for $r>1$ and primes $p$, which can be more succintly expressed as the
identity
$$
\sum_{n>0}c_nn^{-s}=\prod_p{1\over 1-c_p.p^{-s}+p^{11}.p^{-2s}}
$$
when the real part of $s$ is $>(12+1)/2$, were proved by Mordell in
1917. The last of Ramanujan's conjectures was proved by Deligne
only in the 1970s: for every prime $p$, the number
$t_p=c_p/2p^{11/2}$ lies in the interval $[-1,+1]$.

All these properties of the $c_n$ follow from the fact that the corresponding
function $F(\tau)=\sum_{n>0}c_ne^{2i\pi\tau.n}$ of a complex variable $\tau=x+iy$ ($y>0$) in
$\mathfrak{H}$ is a "primitive eigenform of weight $12$ and level $1$" (which
basically amounts to the identity $F(-1/\tau)=\tau^{12}F(\tau)$).

(Incidentally, Ramanujan had also conjectured some congruences satisfied by
the $c_p$ modulo $2^{11}$, $3^7$, $5^3$, $7$, $23$ and $691$, such as
$c_p\equiv1+p^{11}\pmod{691}$ for every prime $p$; they were at the origin of
Serre's modularity conjecture recently proved by Khare-Wintenberger and Kisin.)

We may therefore ask how these $t_p=c_p/2p^{11/2}$ are distributed: for
example are there as many primes $p$ with $t_p\in[-1,0]$ as with
$t_p\in[0,+1]$? Sato and Tate predicted in the 1960s that the precise
proportion of primes $p$ for which $t_p\in I$, for given interval $I\subset[-1,+1]$, is
$$
{2\over\pi}\int_{I}\sqrt{1-x^2}\;dx.
$$
This is expressed by saying that the $t_p=c_p/2p^{11/2}$ are
*equidistributed* in the interval $[-1,+1]$ with respect to the measure
$(2/\pi)\sqrt{1-x^2}\;dx$. Recently Barnet-Lamb, Geraghty, Harris and Taylor
have proved that such is indeed the case.

Their main theorem implies many such equidistribution results, including the
one recalled above for the elliptic curve $S^2+S-T^3+T^2=0$; for an
introduction to such *density theorems*, see
Taylor's review article Reciprocity laws and density theorems.

general mathematical audiencewith "for a global field K, the Artin map defines an isomorphism from the group of connected components of the idele class group to the Galois group Gal(K^ab|K)" should probably not be invited to speak to a general mathematical audience again... $\endgroup$ – Tom Leinster Feb 28 '12 at 19:303more comments