Galoisian sets of prime numbers

Question

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...

Answer 1

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.

Answer 2

In order to add my bit to the already rich content on this site, here is a nice family of galoisian extensions $K_l|{\bf Q}$ with group ${\rm GL}_2({\bf F}_l)$ (indexed by primes $l\neq5$) for which a ``reciprocity law'' can be written down explicitly. I've come across this family recently while writing an expository article.

Let $E$ be the elliptic curve (over $\bf Q$) of conductor $11$ defined by $y^2+y=x^3-x^2$, with associated modular form $$ \eta_{1^2,11^2}=q\prod_{k>0}(1-q^k)^2(1-q^{11k})^2=\sum_{n>0}c_nq^n. $$ Let $K_l={\bf Q}(E[l])$, which is thus galoisian over $\bf Q$ and unramified at every prime $p\neq11,l$.

One can deduce from cor.1 on p.308 of Serre (Inventiones 1972) that for every prime $l\neq5$, the representation $$ \rho_{E,l}:{\rm Gal}(K_l|{\bf Q})\rightarrow{\rm GL}_2({\bf F}_l) $$ we get upon choosing an ${\bf F}_l$-basis of $E[l]$ is an isomorphism; cf. the online notes on Serre's conjecture by Ribet and Stein. Shimura did this for $l\in[9,97]$ (Crelle 1966).

Suppose henceforth that $l$ is a prime $\neq5$ and that $p$ is a prime $\neq11,l$. The characteristic polynomial of $\rho_{E,l}({\rm Frob}_p)\in{\rm GL}_2({\bf F}_l)$ is $$ T^2-\bar c_pT+\bar p\in{\bf F}_l[X]. $$ The prime $p$ splits completely in $K_l$ if and only if ${\rm Frob}_p=1$ in ${\rm Gal}(K_l|{\bf Q})$, which happens if and only if $$ \rho_{E,l}({\rm Frob}_p)=\pmatrix{1&0\cr0&1}. $$ If so, then $p,c_p\equiv1,2\pmod l$ but not conversely, for the matrix $\displaystyle\pmatrix{1&1\cr0&1}$ also has the characteristic polynomial $T^2-\bar2T+\bar1$. But these congruences on $p,c_p$ do rule out an awful lot of primes as not splitting completely in $K_l$.

In summary, we have the following ``reciprocity law" for $K_l$ : $$ \hbox{($p$ splits completely in $K_l$)} \quad\Leftrightarrow\quad E_p[l]\subset E_p({\bf F}_p), $$ where $E_p$ is the reduction of $E$ modulo $p$. Indeed, reduction modulo $p$ identifies $E[l]$ with $E_p[l]$ and the action of ${\rm Frob}_p$ on the former space with the action of the canonical generator $\varphi_p\in{\rm Gal}(\bar{\bf F}_p|{\bf F}_p)$ on the latter space. To say that $\varphi_p$ acts trivially on $E_p[l]$ is the same as saying that $E_p[l]$ is contained in the ${\bf F}_p$-rational points of $E_p$. The analogy with the multiplicative group $\mu$ is perfect: $$ \hbox{($p\neq l$ splits completely in ${\bf Q}(\mu[l])$)} \quad\Leftrightarrow\quad \mu_p[l]\subset \mu_p({\bf F}_p) $$ ($\Leftrightarrow l|p-1\Leftrightarrow p\equiv1\pmod l$), where $\mu_p$ is not the $p$-torsion of $\mu$ but the reduction of $\mu$ modulo $p$.

I requested Tim Dokchitser to compute the first ten $p$ which split completely in $K_7$, and his instantaneous response was 4831, 22051, 78583, 125441, 129641, 147617, 153287, 173573, 195581, and 199501.

It is true that all this (except the list of these ten primes) was known before Serre's conjecture was proved (2006--9) or even formulated (1973--87), but I find this example a very good illustration of the kind of reciprocity laws it provides.

I hope you enjoyed it as much as I did.