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With $\mathbb{Q}(\zeta_k)$ a cyclotomic field, $\chi_1,\ldots,\chi_{\phi(k)}$ the Dirichlet characters modulo $k$ and $\tilde{\chi}_1,\ldots,\tilde{\chi}_{\phi(k)}$ the underlying primitive Dirichlet characters, let $$\Phi(n) = \begin{pmatrix}\tilde{\chi}_1(n) & &\\ & \ldots & \\ & & \tilde{\chi}_{\phi(k)}(n) \end{pmatrix}$$ Then $$\det\left(\sum_{n=1}^\infty \Phi(n) n^{-s}\right)= \prod_{j=1}^{\phi(k)}L(s,\tilde{\chi}_j)=\zeta_{\mathbb{Q}(\zeta_k)}(s) = \sum_{I \subset \mathcal{O}_{\mathbb{Q}(\zeta_k)}} N(I)^{-s}$$

Artin reciprocity says the same holds for any abelian extension $L/K$, with Dirichlet characters replaced by Hecke characters.

Then one can ask what would be a natural generalization of this function $\Phi : \mathbb{Z}\to \text{GL}_{\phi(k)}(\mathbb{C})$ in the context of Artin L-functions and the decomposition of $L(s,\rho)$ as a product or quotient of Hecke L-functions ?

What about the associated theta function $\displaystyle\Theta(z)=\sum_{n=-\infty}^\infty \Phi(n) e^{2i \pi n^2 z}$ in the context of automorphic forms (or is it $\det(\sum_{n=-\infty}^\infty \Phi(n) e^{2i \pi n^2 z})$) ?

Does it make sense to look at certain $n$-dimensional representations of ideal groups of number fields as generalization of Hecke characters (that is $1$-dimensional representation of $\mathcal{I}_{K,\mathfrak{f}}$ that is a product of finitely many local characters on $\mathcal{P}_{K,\mathfrak{f}}$) ?

Some thoughts to address Kimball's perfect comment about the fact ideal groups are abelian while we are interested in non-abelian Galois extensions :

To construct the Artin L-functions, what we need is $\displaystyle\mathfrak{Z}_{L/K}(s) =\sum_{g \in G} \quad [g] \sum_{\mathfrak{p}^k\in \mathcal{O}_K, \text{Frob}_\mathfrak{p}^k = g} \frac{N(\mathfrak{p})^{-sk}}{k}$ where $L/K$ is Galois, $G = \text{Gal}(L/K)$ and $[g]$ the basis of $\mathbb{C}[G]$.

As $G$ is non-abelian, we can't look at $\exp(\mathfrak{Z}_{L/K}(s) )$ and obtain a nice Dirichlet series as above.

What I wrote above gives instead a way to construct $\displaystyle Z_{L/K}(s) =\sum_{g' \in G^{ab}, \in G, g \sim g'} [g'] \sum_{\mathfrak{p}^k\in \mathcal{O}_K, \text{Frob}_\mathfrak{p}^k = g} \frac{N(\mathfrak{p})^{-sk}}{k}$ where $G^{ab}$ is the abelianization of $G$, so that $\exp(Z_{L/K}(s))$ can be constructed as above in term of (abelian) representations of (the abelian group) $\mathcal{I}_K$, or in term of Hecke characters.

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  • $\begingroup$ I don't understand the last question. Idele class characters are generalizations of ideal class characters, but since these are abelian you don't really get anything new by looking at n-dimensional representations. (You do however, if you work with noncommutative division algebras.) $\endgroup$
    – Kimball
    Oct 7, 2017 at 3:49
  • $\begingroup$ @Kimball Exactly. And abelian representation of an ideal class group become an abelian representation of the Galois group of the Hilbert class field which becomes non-abelian when induced to the Galois group over $\mathbb{Q}$. Does it mean we could see the non-abelian thing from the beginning, on the arithmetic side instead of the Galois side ? $\endgroup$
    – reuns
    Oct 7, 2017 at 4:08
  • $\begingroup$ @Kimball I added some thoughts in my question. Do you have some intuition on non-abelian class field theory, and what we can do in the abelian/ideal group framework and how to "solve" the non-abelian problem ? $\endgroup$
    – reuns
    Oct 8, 2017 at 2:44

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I don't really understand what you're looking for, but I'm pretty sure this isn't it. If $\rho$ runs over the irreducible Artin representations $\rho_1, \ldots, \rho_k$ of $L/K$, we have $$\zeta_L(s)/\zeta_K(s) = \prod_\rho L(s, \rho) = \det diag(L(s,\rho_1), \ldots, L(s,\rho_k)).$$ So the most direct analogue of your function $\Phi$ in the abelian case (say where $K=\mathbb Q$) would be $$\Phi(n) = diag(a_n(\rho_1), \ldots, a_n(\rho_k)),$$ where $L(s,\rho) = \sum a_n(\rho) n^{-s}$. If $p$ is an unramified prime, $a_p(\rho)$ should just be the trace of Frobenius at $p$, but the description is more complicated for general $n$.

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  • $\begingroup$ Hi, if $K$ is a number field, $\mathcal{I}_K$ its ideal class group, then say a subgroup (or subset ?) $H \subset \mathcal{I}_K$ is arithmetic if $\sum_{I\in \mathcal{O}_K \cap I} N(I)^{-s}$ has a meaning in term of Hecke L-functions, more generally in term of Artin L-functions and automorphic forms. Do you have an intuition on what are the arithmetic subsets of $\mathcal{I}_K$, and how they are related to arithmetic subsets of $\mathcal{I}_L,\mathcal{I}_F$ for larger and smaller fields $L \supset K \supset F$ ? $\endgroup$
    – reuns
    Oct 8, 2017 at 21:09
  • $\begingroup$ @reuns I would guess that what you call "arithmetic" sets of ideal classes will be something like the set of ideal classes with prescribed ramification type (at least for prime ideal representatives). If so, this should tell you something about how they behave along field extensions. $\endgroup$
    – Kimball
    Oct 9, 2017 at 4:39
  • $\begingroup$ oops I meant $\mathcal{I}_K$ is the ideal group not ideal class group $\endgroup$
    – reuns
    Oct 9, 2017 at 23:01

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