On Jannsen's conjecture for Hecke characters of imaginary quadratic fields

Details On Jannsen's conjecture for Hecke characters of imaginary quadratic fields On Jannsen's conjecture for Hecke characters of imaginary quadratic fields

2007-03-29

arxiv.org/abs/math/0703882v1

Mathematics

Number Theory

To appear in Proceedings of the Primeras Jornadas de Teoria de Numeros

Scientific paper

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.

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