On Jannsen's conjecture for Hecke characters of imaginary quadratic fields
Abstract
Description
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.
To appear in Proceedings of the Primeras Jornadas de Teoria de Numeros
To appear in Proceedings of the Primeras Jornadas de Teoria de Numeros