Explanations to a general mathematical audience about the Langlands programme often advertise it as "non-abelian class field theory". They usually begin as follows: a modern style formulation of classical class field theory is to say that 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 $\operatorname{Gal}(K^{ab}|K)$. Pushing this even further, we see that we have a canonical identification of characters of the idele class group with characters of the absolute Galois group $\operatorname{Gal}(\bar{K}|K)$.


Then people usually go on to say that this should extend to a correspondence between a certain class of $n$-dimensional Galois representations and a certain class of representations of $\operatorname{GL}\_n(\mathbb{A}\_K)$ (where $\mathbb{A}\_K$ denotes the adeles of $K$), and very soon they have disappeared into (to me) far off realms.


While it should be clear from my description that I have no clue whatsoever concerning the Langlands programme, I know a little bit about global class field theory in its traditional formulation. That is, I understand it as a means to describe and classify abelian extensions of $K$ with prescribed ramifications, with the Artin map giving an isomorphism from a ray ideal class group of $K$ (say) to the Galois group of the corresponding ray class field over $K$.


So, my question is:



> 
> Do there exist results in the global Langlands programme which give us back some down-to-earth, may be ideal-theoretic, insights about number field extensions? And the same question for yet open questions in the global Langlands programme: would their answers give us some sort of "classical" information?
> 
> 
> 



How would Hilbert and Weber think about the Langlands programme?
This question deserves an expert answer such as [this one](https://mathoverflow.net/questions/11747/galoisian-sets-of-prime-numbers/12382#12382) by Emerton, but allow me to offer an outsider's perspective. The following remarks are taken from my expository article [arXiv:1007.4426](http://arxiv.org/abs/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](http://www.math.ias.edu/~rtaylor/shaw.pdf).


