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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?

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Interesting choice of title! –  quid Feb 28 '12 at 19:20
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Those who begin explanations to a general mathematical audience with "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... –  Tom Leinster Feb 28 '12 at 19:30
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How familiar are you with the story when $n = 2$? There are relatively down-to-Earth things you can say here about the relationship between prime splitting in certain number fields and the Fourier coefficients of certain modular forms. Matthew Emerton gives a nice example here: mathoverflow.net/questions/11747/… –  Qiaochu Yuan Feb 28 '12 at 20:14
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Michio Kuga of Stony Brook would summarize Japanese fairy tales and other traditional stories as "Many people die." I wasn't there, but on at least one occasion, he did say "Many people try to find non-abelian class field theory. Many people die." –  Will Jagy Feb 28 '12 at 20:32
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Are there any finite galoisian extensions $K|\mathbf{Q}$ which are ramified only at $2$ (resp. $3$) and such that the group $\mathrm{Gal}(K|\mathbf{Q})$ is not solvable ? The answer to this concrete, elementary question requires some deep results in the Langlands programme; see $$ $$ mathoverflow.net/questions/78422/… –  Chandan Singh Dalawat Feb 29 '12 at 5:40
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up vote 10 down vote accepted

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.

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Thank you very much. I just glanced over your article and the one by Taylor to which you included links; later I will read them more thoroughly. They do look like what I have been looking for. –  Robert Kucharczyk Mar 4 '12 at 19:45
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