# 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/Col lected 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?

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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 $\mathrm{tr} \rho(\mathrm{Frob}_p)=\chi(\mathrm{Frob}_{\mathfrak{p}})+\chi(\mathrm{Frob}_{\ma‌​thfrak{\overline{p}}})$ 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