# How would Hilbert and Weber think about the Langlands programme?

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
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
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
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
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

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