# Generalization of a $\det$ series for $\zeta_{\mathbb{Q}(\zeta_k)}$

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.

• 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.) Oct 7, 2017 at 3:49
• @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 ? Oct 7, 2017 at 4:08
• @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 ? Oct 8, 2017 at 2:44

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 $n$.
• 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 $L \supset K \supset F$ ? Oct 8, 2017 at 21:09
• oops I meant $\mathcal{I}_K$ is the ideal group not ideal class group Oct 9, 2017 at 23:01