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Let $K$ be a number field, $\chi : C_K \to \mathbb{C}^\ast$ a Hecke character (that is, a character of the idèle class group), and $L(\chi,s)$ the corresponding Hecke $L$-series. I wish to understand how one may construct a Galois extension $G = Gal(L/K)$ and a complex representation $\rho : G \to \mathbb{C}^\ast$ such that $\mathcal{L}(L/K,\rho,s) = L(\chi,s)$, where $\mathcal{L}$ here denotes the Artin L-function.

I know how this works when $\chi$ factors through a congruence subgroup mod $\mathfrak{m}$, or equivalently, if $\chi$ is a Dirichlet character mod $\mathfrak{m}$; namely, take $L = K^\mathfrak{m}$, the ray class field mod $\mathfrak{m}$, and use the Artin symbol to turn the given character into a Galois character.

But I am worried that trying to do the same thing for general $\chi$, replacing the Artin symbol with the map $\phi_K : C_K \to Gal(K^{ab}/K)$, will not work. Indeed, the Wikipedia article on 'Hecke Character' suggests that only the Dirichlet characters are accounted for by Class Field Theory (see the last paragraph in the section 'Definition using ideals'). This worries me.

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Not all Hecke characters correspond to Artin L-functions; only the algebraic Hecke characters that have trivial weight at infinity do. See this related question: mathoverflow.net/questions/66500/… –  Keerthi Madapusi Pera Jun 7 '11 at 23:02
Let $K$ be the rationals. Is $|.|$ a Hecke character for you? It's a character on $\mathbf{A}_K^\times$ and is trivial on $K^\times$. Its $L$-function would be something like $s\mapsto\sum_n n/n^s=\sum 1/n^{s-1}=\zeta(s-1)$. This doesn't satisfy the right sort of functional equation for an Artin $L$-function because the centre of symmetry is in the wrong place. So I don't think one can possibly expect an Artin rep in this case. –  Kevin Buzzard Jun 7 '11 at 23:03
By the way, you can't hope to recover irreducible Artin representations of dimensions > 2 via Hecke characters; you only get characters of the absolute Galois group. –  Keerthi Madapusi Pera Jun 7 '11 at 23:04
Darn, I see that I was naive to hope this. I guess Kevin's example also shows that there cannot be a common generalisation of Artin and Hecke L-series. –  Barinder Banwait Jun 7 '11 at 23:41
Stupid question, what is a Dirichlet character here as opposed to a Hecke character? EDIT: just read the wiki-entry, which uses the notion it is a finite order Hecke character. I had seen alternatively Dirichlet characters to be multiplicative functions on field elements (not ideals), so was confused –  Junkie Jun 7 '11 at 23:44
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up vote 7 down vote accepted

Dear Barinder,

Re. your comment "there cannot be a common generalization of Artin and Hecke $L$-series", to the contrary, there is such a common generalization, namely the $L$-series of a representation of the global Weil group. These will (conjecturally) have an analytic continuation and functional equation, and they include all Hecke $L$-series (Hecke characters, by which I mean idele class characters, are just one-dimensional reps. of the global Weil group), and all Artin $L$-series (which are reps. of the global Weil group which factor through the map to $G_K$).



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Thank you Matthew, and everyone else, for your very helpful comments. To learn more about this global Weil group, I shall first read the discussion at: mathoverflow.net/questions/12100/… –  Barinder Banwait Jun 8 '11 at 10:25
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