Revistes Catalanes amb Accés Obert (RACO)

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

F. Bars


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 (Section 1). In Section 2 we define the $p$-adic realizations associated to Hecke characters over $K$. 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 (Section 3) and in the non-critical case (Section 4) for these realizations. 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, in Section 5 we prove that the conjecture follows from a general conjecture on Iwasawa theory for almost all Tate twists.

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