Have we ever proved any non-solvable case of reciprocity without the Langlands program ?

Question

The reciprocity of the title is the following not completely well-posed problem:

Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe" (in some sense) the set of prime $p$ such that $P \mod p$ has a given reduction (for example is irreducible, or on the opposite is a product of $n$ linear factors, etc.).

As is well-known, when one root (hence all roots) of $P$ happens to be a cyclotomic integer (in particular in the very special case where $P$ is quadratic), then the problem has a precise solution, called Artin's reciprocity law, which is the core of class field theory. The sets of primes $p$ such that $P$ has a given reduction are then given by congruences modulo fixed integers satisfied by $p$, that is those sets are union of "set primes in artithemtic sequence". The Galois group of $P$ is abelian in this case.

As the opposite of the complexity spectrum, there are polynomial $P(X)$ of degree $n \geq 5$, whose Galois group is big, say $S_n$, or $A_n$ in particular not solvable. In this case, it is my understanding that our only hope to find a reciprocity law in general is by making huge progress in the Langlands program.

But is it possible, in some special case (for polynomials $P$ of Galois group $S_n$ say satisfying somme assumtions, or for an explicit family of polynomials of variable degree $n \geq 5$, or even for just one explicit polynomials) where some kind of reciprocity law has been worked out (even only partially) by some method that does not involve the Langlands program (i.e. modular forms, automorphic forms, etc.) ?

Answer 1

This is a very interesting question and would make an excellent topic for a doctoral thesis in the history of mathematics. I will interpret the question as

Which pre-Langlands results, problems, and theories --- apart from what is easily deducible from the theory of $\;\mathrm{GL}_1$ (from Gauß to Tate) --- can now be considered a part of the Langlands programme ?

There is nothing original in my answer : everything is gleaned from the writings of Langlands, Serre and Weil. I may have misrepresented some of their words, and in any case our future doctoral candidate will have to delve deeper into the original sources.

Fricke & Klein (1912) observe that the modular curve $X_0(11)$ of level $\Gamma_0(11)$ is defined by the equation $\sigma^2=1-20\tau+56\tau^2-44\tau^3$.

Hasse (193?) asks a doctoral student (Pierre Humbert) to prove that the $L$-function of an elliptic curve $E$ over $\mathbf{Q}$ (defined as the product over various primes $p$ of the $\zeta$-function of $E$ modulo $p$) is entire and satisfies a functional equation. Humbert sagely decides to work on quadratic forms with Siegel instead.

Weil (1951) asks in his report Sur la théorie du corps de classes for a galoisian interpretation of the whole idèle class group of a number field (as opposed to the quotient of the said group by the connected component of the identity), analogous to the galoisian interpretation in the function field case. See https://mathoverflow.net/questions/41318 in this regard.

Weil (1952) shows that certain elliptic curves with complex multiplications (such as $y^2=x^4+1$) are modular.

Deuring (1953--1957) proves (following a suggestion by Weil) that all elliptic curves with complex multiplications are modular.

Eichler (1954) proves that the $L$-function of $X_0(N)$ is essentially the product of Hecke $L$-functions attached to cuspidal eigenforms of weight $2$ and level $N$. This was generalised by Shimura (1958) and completed by Igusa (1959).

Taniyama (1955) asks at the Tokyo-Nikko conference a somewhat imprecise question which some interpret as implying that one can prove Hasse's conjecture for $E$ by showing that $E$ is modular.

Shimura (1966) explicitly determines the reciprocity law for the splitting of rational primes in the number field obtained by adjoining the $l$-torsion ($l$ prime) of the Fricke curve $X_0(11)$ in terms of the coefficient $c_l$ of $q^l$ in the modular form $$ q\prod_{n>0}(1-q^n)^2(1-q^{11n})^2 $$ (but only for $l<100$ for which he could check that the mod-$l$ representation is surjective).

Weil (1967) proves that if an elliptic curve over $\mathbf{Q}$ is modular, then it has to be modular of level equal to its conductor, and assigns the Übungsaufgabe to the interested reader to show that every elliptic curve over $\mathbf{Q}$ is indeed modular.

Around this time Langlands wrote a letter to Weil and changed the world.

Answer 2

This question gives me the chance to advertise a result contained in http://arxiv.org/abs/1201.2124 which characterizes primes which are completely split in torsion fields extensions $K(E[N])/K$ of elliptic curves over number fields. Sorry for being self-referential.

Let $K$ be a number field, $E$ an elliptic curve over $K$, and $N$ an integer $>0$. For a finite prime $\mathfrak{p}$ of $K$ with residue field $k_\mathfrak{p}$, denote by $a_\mathfrak{p}$ the trace of $E \text{ mod } \mathfrak{p}$, and by $\Delta_\mathfrak{p}$ the discriminant $a_\mathfrak{p}^2-4|k_\mathfrak{p}|$ of the characteristic polynomial $x^2-a_\mathfrak{p}x+|k_\mathfrak{p}|$.

$\textbf{Theorem}.$ There exists a universal family of polynomials $\{\mathcal{P}_D(x)\}_{D\leq 0}$ satisfying the following property. Let $\mathfrak{p}$ be a prime of good reduction for $E$ which does not divide $N$, and for which $E\text{ mod }\mathfrak{p}$ is not special* if $N=2$. Then $\mathfrak{p}$ splits completely in $K(E[N])/K$ if and only if both conditions below are satisfied:

i) $N^2$ divides $\Delta_\mathfrak{p}$, and $\mathcal{P}_{\Delta_\mathfrak{p}/N^2}(\;j_E\;)\equiv 0\text{ mod }\mathfrak{p}$;

ii) $a_\mathfrak{p}\equiv 2 +\dfrac{\Delta_\mathfrak{p}}{N}\text{ mod }N^*$;

where $N^*=N$ if $N$ is odd, and $N^*=2N$ otherwise.

---

*this condition, not explained here, avoids only finitely many $\mathfrak{p}$.

If $D$ is a negative discriminant, the polynomial $\mathcal{P}_D(x)$ is monic with integer coefficients. Its roots are the $j$-invariants of complex elliptic curves with CM by an order containing the imaginary quadratic order of discriminant $D$. Moreover $\mathcal{P}_0(x)=0$ and $\mathcal{P}_D=1$ if $D$ is not a discriminant.

The proof of the result is via local methods and relies on the fact that if the ring of $k_\mathfrak{p}$-endomorphisms of $E\text{ mod }\mathfrak{p}$ is a quadratic order, then the action of $\text{Frob}_\mathfrak{p}$ on $E[N](\bar K)$ is equivalent to the action of $\text{Frob}_\mathfrak{p}$ on $\tilde E_\mathfrak{p}[N](\bar K)$, where $\tilde E_\mathfrak{p}$ is the Deuring lifting of $E\text{ mod }\mathfrak{p}$. The evaluation of the polynomials $\mathcal{P}_D(x)$ at the $j$-invariant $j_E$ of $E$ enters in condition i) in order to identify the correct lifting of $E\text{ mod }\mathfrak{p}$ (for this to work in the supersingular case one has to make some observations).

The Theorem was well known if $\mathfrak{p}$ is an ordinary prime for $E$. The fact that the above formulation remains true for supersingular primes (infinitely many when $K$ is real) is perhaps the novelty.

Since the methods used in the proof are rather antique, I realize that the result might be not so interesting to experts. But at least its statement gives an idea of how a reciprocity law in a non-solvable context might look like.

Adelmann in his book "The Decomposition of Primes in Torsion Point Fields" treats the same problem. He employs modular polynomials to characterize complete split primes.