  We present a collection of results on a conjecture of Jannsen about the
$p$-adic realizations associated to Hecke characters over an imaginary
quadratic field $K$ of class number 1.
  The conjecture is easy to check for Galois groups purely of local type. We
prove the conjecture under a geometric regularity condition for the imaginary
quadratic field $K$ at $p$, which is related to the property that a global
Galois group is purely of local type. Without this regularity assumption at
$p$, we present a review of the known situations in the critical case and in
the non-critical case for the realizations associated to Hecke characters over
$K$. We relate the conjecture to the non-vanishing of some concrete
non-critical values of the associated $p$-adic $L$-function of the Hecke
character.
  Finally, we prove that the conjecture follows from a general conjecture on
Iwasawa theory for almost all Tate twists.
