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How do I understand Hecke characters? For example, is there a bijection between Hecke characters and something? For example, if a Hecke character factors through a ray class group, by Artin reciprocity, we would have a 1-1 correspondence with characters of finite abelian Galois groups with restricted ramification.

I'm aware of Understanding Hecke Characters as Extension of Dirichlet Characters, but this does not answer my question. Motivation for Hecke characters is a bit more helpful. For example, what is the interpretation of infinity types?

  • To answer your first question, class field theory gives a bijection between Hecke characters over $K$ (i.e. characters of $\mathbb A_K^\times/K^\times$) and one dimensional $\ell$-adic Galois representations $\mathrm{Gal}(\overline K/K)\to \overline{\mathbb Q}_\ell^\times$. The infinity type gets encoded in a slightly complicated way as whats going on at $G_K$ restricted to the decomposition group at primes above $\ell$. So you do get a correspondence with characters of Galois groups of abelian, but not necessarily finite, extensions. Is that the type of thing you're looking for? – Mathmo123 Sep 02 '22 at 10:13
  • @Mathmo123: Yes, that's in my spirit. Does this hold for all $\ell$ when fixing one Hecke character? I'm asking because the image of the decomposition group at $\ell$ can be infinite, but not that at other primes. So in fact, you get a compatible system of $\ell$-adic Galois representations? Do you have a reference (not too technical, but yet precise) explaining the interpretation of the infinity types? –  Sep 02 '22 at 12:38
  • My previous comment was slightly imprecise. The correspondence with $\ell$-adic representations only holds for algebraic Hecke characters: namely, characters whose infinity types are suitably integral. In that case, you do get a compatible system. Take a look at these notes for more detail. – Mathmo123 Sep 02 '22 at 12:48
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    Essentially, the point is that, by class field theory, there is a natural isomorphism $K^\times\backslash\mathbb A_K^\times/\overline{(K_\infty^\times)^0}\cong G_K^{ab}$. So as long as the infinity type (encoded in $K_\infty^\times$) plays nice, there is a tight connection between Hecke characters and Galois characters. – Mathmo123 Sep 02 '22 at 12:53
  • @Mathmo123: Thanks, that and this note sheds much more light on my question. –  Sep 04 '22 at 07:57

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