# Reference for the odd dihedral case of Artin's conjecture

The example that Matt Emerton cited here prompted me to become interested in how one proves that odd two dimensional dihedral Galois representation are modular. This is the first case of the strong Artin conjecture for two dimensional representations and I feel like understanding it would be helpful in getting some sense for why Galois representations are modular. Emerton mentioned that the theorem was proved by Hecke; according to Gelbart's review of the Serre/Deligne paper on Galois representations attached to weight 1 modular forms; the dihedral case follows from Hecke's work on theta series attached to binary quadratic forms.

Chandan Singh Dalawat give some more detail on the example that Emerton gives on pp. 5-6 of his article titled Splitting Primes, citing an article of Serre for still more detail. I have some glimmerings of how this goes in the case under discussion; in that case one needs to show that the Artin L-function is 1/2 of the difference of two theta series; presumably one uses class field theory for the splitting field viewed as a cubic extension of the quadratic subfield. The two quadratic forms used to define the relevant theta series correspond to the nonprincipal ideal classes of the quadratic subfield. But I don't see exactly how it should go.

I've seen references to

J.-P. Serre, Modular forms of weight 1 and Galois representations. In: Algebraic
Number Fields (1977), pp. 193–268 = Œuvres/Collected Papers III, Springer-
Verlag, Berlin, 1986, pp. 292–367.


but given that the result goes back to Hecke it seems like there should be expositions along classical lines from an earlier time (1930's-1960's). and I haven't been able to find them. Does anyone know such a reference?

• If $\rho$ odd two-dimensional Artin rep. of dihedral type, easy calc shows $\rho = \mathrm{Ind}_{G_K}^{G_{\mathbf{Q}}}\chi$ for $K$ imag quad and $\chi$ a char of $G_K$ with finite image into $\mathbf{C}^{\times}$. CFT says $\chi$ a char. on frac. ideals of $\mathcal{O}_K$ prime to some ideal $\mathfrak{f}_{\chi}$, mod princ. ideals gen. by elts. $\equiv 1 \; \mathrm{mod} \; \mathfrak{f}_\chi$. Form $\theta_{\chi}(z)=\sum_{\mathfrak{a} \subset \mathcal{O}} \chi(\mathfrak{a})e^{2 \pi i N(\mathfrak{a})z}$, split into arith. progs mod $\mathfrak{f}$, get theta series of bin quad forms. – David Hansen Jun 13 '11 at 3:58
• Jonah: The coefficients match because $\mathrm{tr} \rho(\mathrm{Frob}_p)=\chi(\mathrm{Frob}_{\mathfrak{p}})+\chi(\mathrm{Frob}_{\mathfrak{\overline{p}}})$ for $p$ split, and $\rho(\mathrm{Frob}_p)=0$ for $p$ inert. Oddness is key in the matching of $\det{\rho}$ with the nebentypus character of the twisted theta series, which has weight one and thus an odd nebentypus character. I would say that the motivation for forming the twisted theta series is...it works! :) (For even dihedral reps, you form a similar theta series which is actually a Maass form; this was Maass's original construction.) – David Hansen Jun 13 '11 at 5:27
• Also, the modularity of theta series of positive-definite quadratic forms is presented very clearly in Iwaniec's "classical topics" book. – David Hansen Jun 13 '11 at 5:30
• Jonah: The statement about the traces follows from literally writing out the character of an induced representation, using Frobenius's formula, in conjunction with the knowledge that $1 \to G_{K} \to G_{\mathbf{Q}} \to \mathbf{Z}/2\mathbf{Z} \to 1$. – David Hansen Jun 13 '11 at 17:21
• Kimball: Hecke had no general converse theorem; his work dealt only with the level one case. You need to wait until Weil for a flexible converse theorem. One must be careful when associating Hecke's name with dihedral Artin reps - clearly he thought about theta functions, but he was an analytic guy and I doubt an Artin representation ever crossed his mind. It is fine to say "in principle, modularity of dihedral Artin reps. goes back to Hecke", but not "Hecke proved that dihedral Artin reps. are modular." – David Hansen Jun 13 '11 at 17:27

## 1 Answer

There's a beautiful history behind this.

Basically, Artin and Hecke were working on different sides of this "dihedral modularity conjecture" at the same time (the 20s) and at the same place (Hamburg), but apparently they never discussed this aspect of their research.

So they had the tools to prove this instance of what would become the Langlands program by 1927, but they didn't know it!

There is a brief account of this in Tate's paper "The general reciprocity law" (note that he was Artin's doctoral student), and a more extended historical survey of Artin and Hecke's work during that time on Cogdell's article "On Artin L-functions".

I think this is how the proof would have looked like back in 1927 (although in modern notation, and not in german!)

Arithmetic side (Artin)

Let $\rho:\mathrm{Gal}(L/K)\to \mathrm{GL}_2(\mathbb{C})$ be 2-dimensional dihedral complex representation. From representation theory we know that $\rho$ is monomial, that is, induced from a 1-dimensional representation.

Artin had proved in 1923 that his L-functions behave well under representation theoretic operations and, in particular, induction. Therefore, there is an L-function $L(\varrho,s)=L(\rho,s)$, with $\varrho$ one-dimensional (abelian).

From Artin reciprocity (1927) we have that $L(\varrho,s)=L(\chi,s)$, with $L(\chi,s)$ a Hecke L-function.

The last step is Hecke's proof from 1917 that abelian L-functions are meromorphic for non-trivial characters. Since $\varrho \neq 1$, the original L-function $L(\rho,s)$ is meromorphic on the complex plane, and we have proved the Artin conjecture for dihedral representations.

Automorphic side (Hecke)

Hecke had been studying theta series, and in particular in 1927 he constructed a cusp form $f_\theta$ of weight $1$ as a linear combination of $\theta$-series of binary quadratic forms attached to $K$.

He had alredy proved the basic properties of the L-functions of arbitrary modular forms and Hecke characters, so he knew their functional equation.

In the case of his $f_\theta$ the gamma-factor was very simple, just $\Gamma(s)$.

So, according to Tate, he listed all the Hecke L-functions that shared that same gamma-factor. After weeding out the one coming from Eisenstein series (which in turn correspond to cylic (reducible) two-dimensional representations), he was left with a correspondence $L(\chi,s)=L(f_\theta,s)$.

Arithmetic side revisited

This would have been an easy step for either one of them, if they had known what the other one was up to.

A quick inspection of the gamma-factor of the Artin L-function shows that the only representations for which it equals $\Gamma(s)$ are the ones odd and two-dimensional. Since the other two-dimensional odd representations are irreducible (except the cyclic, which we have alredy mentioned correspond to Eisenstein series), we have showed:

$$L(\rho,s)=L(f_\theta,s)$$

Jacquet-Langlands proof

The first actual proof of the result follows from the converse theorem for $\mathrm{GL}_2$ in "Automorphic forms on GL(2)" (1971). But I don't think they mention the dihedral case in particular. Langlands does, saying that it is implicit in the works of Hecke and Maass, in his 1975 book "Base change for GL(2)".

A different proof follows from the results by Deligne and Serre in "Formes modulaires de poids 1" (1974).

I'm not sure of what relevance Maass' work has in this case. The same goes for some attributions to Brauer, since his induction theorem isn't really needed here.

To answer the actual question, no, there's no direct reference for this result before 1971. That said, technically Artin's 1927 paper implies this case of the (weak) Artin conjecture, and we now know (by a result of Booker, 2003) that this "weak" case implies the strong Artin conjecture.