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Note: I've revised the question just a little bit in the hope of making it easier.


Given an algebraic number field $F$, which we may as well take to be Galois over $\mathbb{Q}$, we denote by $S_F$ the set of rational primes that split in $F$. Sets of the form $S_F$ are called Galoisian. At some point, there was a discussion

Why do congruence conditions not suffice to determine which primes split in non-abelian extensions?

of the fact that the abelian Galoisian sets, that is, $S_F$ corresponding to $F$ abelian over $\mathbb{Q}$, are exactly the sets of primes defined by congruence conditions.

A while later, Matthew Emerton gave this nice answer

Galoisian sets of prime numbers

to a question of Chandan Singh Dalawat about non-abelian Galoisian sets.

I made a comment there I thought I would upgrade to a question. As Matthew points out, Neukirch's remark that the Langlands program provides a characterization of all Galoisian sets is probably meant as a metaphor for some other process. However, I couldn't help but hope that the characterization could be taken literally, at least for some special families. For example, we will refer to a number field $F$ as being of $GL_2$ type if it is the fixed field of $Ker(\rho)$, where $$\rho: Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{C})$$ is an irreducible two-dimensional Artin representation of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$*. Now call a set of primes a $GL_2$ Galoisian set if it is of the form $S_F$ for some extension $F$ of $GL_2$-type.

The question then is: can one use the Langlands program (or anything else) to give a sensible characterization of $GL_2$ Galoisian sets?

One could obviously change this question in any way that would make it more tractable. One could try to characterize, for example:

-Solvable $GL_2$ Galoisian sets, where the $GL_2$-field $F$ is further required to be solvable;

-Odd $GL_2$ Galoisian sets: $S_F$ where $F$ is the fixed field of a representation $\rho_f$ arising from a holomorphic modular form $f$ of weight one;

-Odd $GL_2$ Galoisan sets of conductor $N$, where we further require the form $f$ to have level $N$;

and so on. The last case probably admits a tautological answer of sorts, in that we can in principle list the finitely many forms (sorted by Dirichlet characters $\epsilon$), and then make some statement about the $p$'s where $$X^2-a_pX+\epsilon(p)=(X-1)^2.$$ Is it entirely unreasonable to hope for something more compact?


*The idea that we should simply organize fields in this manner corresponding to representations is perhaps a valuable perspective coming out of the Langlands program.


Added, 25 July:

Having thought about it a bit more, it occurs to me that this is yet another situation where Langlands urges us to go beyond a classical framework in seeking answers to non-abelian questions. For example, when we associate to an odd Artin represention

$$\rho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow Aut(V)$$

of dimension two an Artin $L$-function $$ L(\rho,s)=\sum a_n/n^s,$$ we can perfectly sensibly assert that the $a_n$'s follows a pattern. When asked what that pattern is, the answer, satisfying to some and mysterious to others, is that $$\sum a_nq^n$$ is a modular form. This is the kind of thing that comes out of Langlands.

Now, if we want to 'characterize,' say, odd $GL_2$ Galoisian sets, we can say the following: Enumerate the normalized holomorphic Hecke (new) eigenforms $f$ of weight one sorted by level $N$ and character $\epsilon$. For each such form, run over the prime numbers $p$ not dividing $N$, and take the number $a_p$ defined by the equation $$T_pf=a_pf.$$ for the $p$-th Hecke operator $T_p$. Now look at the set $S_f$ of primes $p$ such that $$(p,N)=1, \epsilon (p)=1, a_p=2.$$ These $S_f$'s are exactly the odd $GL_2$ Galoisian sets.

Perhaps it's unreasonable to want more from the Langlands' programme. Whether or not this is the final word on all such questions, well, that's a different matter.

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1 Answer 1

Here is an extremely naive answer to a case of my own question, which is surely obvious to experts. For me, even coming up with this silly version required quite a bit of conversation with Sugwoo Shin (who is of course blameless of any errors).

We give a description of the odd $GL_2$-Galoisian sets of level $N$ in terms of 'higher congruence conditions.' The point is to consider the $\mathbb{Q}$-Hecke algebra $H(N)$ determined by the Hecke operators acting on modular forms of weight 1, level $N$. The maximal ideals in $H(N)$ are in correspondence with Galois conjugacy classes of normalized new weight one eigenforms of level $N$. There is also a map $$p\mapsto T_p$$ from primes not dividing $N$ to $H(N)$.

Any given maximal ideal $m$ determines a Dirichlet character $\epsilon_m$, and one considers the set of primes $S(m)$ defined by the 'congruence conditions' $$(p,N)=1, \ \epsilon_m(p)=1, \ T_p\equiv 2 \ \ \mod \ m$$ These $S_m$ are exactly the odd $GL_2$ Galoisian sets of level $N$. With a bit more care, one should be able to give a similar description that doesn't refer to the level beforehand.

Of course this is no different from what I wrote before, but focussing on the Hecke algebra seems to allow a formulation that's rather analogous to the classical one. That is, one can forget about modular forms for a moment and examine sets of primes determined by congruences in the Hecke algebra.

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