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 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 $S(\mathbb{Q}(\zeta_l))$. "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's recent Bourbaki exposé 1017 arXiv: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...