The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say about such subsets, beyond what Class Field Theory has to say ?


I'll now introduce some terminology and recall some background.


Let $\mathbb{P}$ be the set of prime numbers. For every galoisian extension $K|\mathbb{Q}$, we have the subset $S(K)\subset\mathbb{P}$ consisting of those primes which split (completely) in $K$. The question is about characterising such subsets; we call them **galoisian** subsets.


If $T\subset\mathbb{P}$ is galoisian, there is a unique galoisian extension $K|\mathbb{Q}$ such that $T=S(K)$, cf. Neukirch (13.10). We say that $T$ is **abelian** if $K|\mathbb{Q}$ is abelian.


As discussed [here](https://mathoverflow.net/questions/11688/why-do-congruence-conditions-not-suffice-to-determine-which-primes-split-in-non-a) recently, a subset $T\subset\mathbb{P}$ is abelian if and only if it is defined by congruences. For example, the set of primes $\equiv1\pmod{l}$ [is the same as](https://mathoverflow.net/questions/10457/what-are-the-prime-ideals-in-rings-of-cyclotomic-integers/10473#10473) $S(\mathbb{Q}(\zeta\_l))$. "Being defined by congruences" can be made precise, and counts as a characterisation of abelian subsets of $\mathbb{P}$.


Neukirch says that Langlands' Philosophy provides a characterisation of all galoisian subsets of $\mathbb{P}$. Can this remark now be illustrated by some striking example ?


**Addendum** (28/02/2010) [Berger](http://www.umpa.ens-lyon.fr/~lberger/)'s recent Bourbaki exposé 1017 [arXiv:1002.4111](http://arxiv.org/abs/1002.4111) says that cases of the Fontaine-Mazur conjecture have been proved by Matthew Emerton as well. I didn't know this at the time of asking the question, and the unique answerer did not let on that he'd had something to do with Fontaine-Mazur...



Galoisian sets of prime numbers
I think it is easiest to illustrate the role of the Langlands program (i.e. non-abelian class field theory) in answering this question by giving an example.


E.g. consider the Hilbert class field $K$ of $F := {\mathbb Q}(\sqrt{-23})$; this is a degree 3 abelian extension of $F$, and an $S\_3$ extension of $\mathbb Q$. (It is the splitting field of the polynomial $x^3 - x - 1$.)


The 2-dimensional representation of $S\_3$ thus gives a representation
$\rho:Gal(K/{\mathbb Q}) \hookrightarrow GL\_2({\mathbb Q}).$  

A prime $p$ splits in $K$ if and only if $Frob\_p$ is the trivial conjugacy class
in $Gal(K{\mathbb Q})$, if and only if $\rho(Frob\_p)$ is the identity matrix, if and only
if trace $\rho(Frob\_p) = 2$. (EDIT: While $Frob\_p$ is a 2-cycle, resp. 3-cycle, if and only if $\rho(Frob\_p)$ has trace 0, resp. -1.)


Now we have the following reciprocity law for $\rho$: there is a modular form $f(q)$, in fact
a Hecke eigenform, of weight 1 and level 23, whose $p$th Hecke eigenvalue gives
the trace of $\rho(Frob\_p)$. (This is due to Hecke; the reason that Hecke could handle
this case is that $\rho$ embeds $Gal(K/{\mathbb Q})$ as a dihedral
subgroup of $GL\_2$, and so $\rho$ is in fact induced from an abelian character of the
index two subgroup $Gal(K/F)$.)


In this particular case, we have the following explicit formula:


$$f(q) = q \prod\_{n=1}^{\infty}(1-q^n)(1-q^{23 n}).$$


If we expand out this product as $f(q) = \sum\_{n = 1}^{\infty}a\_n q^n,$
then we find that $trace \rho(Frob\_p) = a\_p$ (for $p \neq 23$),
and in particular, $p$ splits completely in $K$ if and only if $a\_p = 2$.
(For example, you can check this way that the smallest split prime is $p = 59$;
this is related to the fact that $59 = 6^2 + 23 \cdot 1^2$.).
(EDIT: While $Frob\_p$ has order $2$, resp. 3, if and only if $a\_p =0$, resp. $-1$.)


So we obtain a description of the set of primes that split in $K$ in terms of
the modular form $f(q)$, or more precisely its Hecke eigenvalues (or what amounts
to the same thing, its $q$-expansion).


The Langlands program asserts that an analogous statement is true for any
Galois extension of number fields $E/F$ when one is given a continuous
representation $Gal(E/F) \hookrightarrow GL\\_n(\mathbb C).$ This is known
when $n = 2$ and either the image of $\rho$ is solvable (Langlands--Tunnell) or $F = \mathbb Q$ and $\rho(\text{complex conjugation})$ is non-scalar (Khare--Wintenberger--Kisin).
In most other contexts it remains open.


