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 $T$.
1 Answer
$\begingroup$
$\endgroup$
1
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
-
$\begingroup$ Another useful source is a paper of Watkins. magma.maths.usyd.edu.au/~watkins/papers/hecke.pdf If I understand the situation correctly, in his terms you have been given a modulus $I\Omega$ and an $\infty$-type and you want to determine if they are "coherent" (section 4, page 7), which is whether all the units that are 1 mod $I\Omega$ are mapped to 1 by the $\infty$-type (which is a product over embeddings). $\endgroup$– HELLOCommented Apr 12, 2015 at 22:26