I'm needing to find out if there exists an algebraic Hecke character for a number field F, $\phi: \mathbb{A}\_F \rightarrow \mathbb{C}$, for a fixed infinite part $\phi\_\infty$ and a fixed component $\phi\_p$ (for just one finite prime $p$). Maybe it would help if I can find a classification of algebraic Hecke characters. In fact the original problem is about an algebraic Hecke character in the a torus $T$.



Algebraic Hecke characters with a given infinite part
A standard reference for algebraic Hecke characters is Chapter Zero of Schappacher's book "Periods of Hecke characters", <http://link.springer.com/book/10.1007%2FBFb0082094>


