Are there partially algebraic Hecke characters?

Question

$\newcommand{\C}{\mathbb{C}} \newcommand{\U}{\mathbb{U}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}}$ Let $F$ be a number field.

Let $\chi\colon \mathbb{A}_F^\times/F^\times \to \C^\times$ be a Hecke character with local components $\chi_v$ for each place $v$ of $F$. When $v$ is an archimedean place, then there are $m_v\in\Z$, $\sigma_v,\varphi_v\in\R$ such that for all $x\in F_v^\times$, $$ \chi_v(x) = \left(\frac{x}{|x|}\right)^{m_v}|x|^{\sigma_v+i\varphi_v}. $$ Recall that $\chi$ is algebraic if for every archimedean place $v$ the local component $\chi_v$ is a rational function, i.e. if there exists integers $p_v,q_v\in\Z$ such that $$ \chi_v(x) = x^{p_v}\bar{x}^{q_v}. $$ We understand well when algebraic characters exists: up to a finite order character, they factor through the norm to the maximal CM subfield of $F$ (or to $\Q$ if there is no CM subfield).

For an algebraic character, all $\varphi_v = 0$, and up to finite index and multiplication by a power of the norm this characterises algebraic characters.

Now let $\Sigma$ be a subset of the set of archimedean places of $F$.

Does there exist Hecke characters as above such that $\varphi_v=0$ if and only if $v\in\Sigma$?

If $|\Sigma|\le 1$ then you can obtain such characters by multiplying any Hecke character by a suitable power of the norm.

If $\Sigma$ is the set of all archimedean places, we know the answer by the usual theory of algebraic characters.

Otherwise, since I do not see an obvious reason why such characters should exist, the "obvious guess" is that they do not. Is it true?

I was able to rule out a couple more cases in an ad-hoc way, and it seems to quickly involve transcendence problems.

Answer 1

Yes, such partially algebraic characters exist. This is proved in Section 5.6 (Section 5.5 in the published version) of my paper with Pascal Molin Computing groups of Hecke characters.

Let me repeat the construction here: assume $F$ is a quadratic extension of another number field $F_0$, let $\sigma$ be the nontrivial automorphism of $F/F_0$, and let $R$ be the set of complex places of $F$ that restrict to a real place of $F_0$. Then every $\chi$ in the $-1$ eigenspace of $\sigma$ modulo torsion satisfies $\varphi_v = 0$ for all $v\in R$. If $F$ is not CM then it is easy to prove with the Artin-Weil theorem that for some of these characters there will also exist a complex place $v$ such that $\varphi_v\neq 0$.

However, I don't know if every partially algebraic Hecke character has to come from this construction. Contrary to the case of algebraic characters, it does not seem to follow from Galois theory alone.