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

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 (§1). In §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 §3 and in the non-critical case §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 §5 we prove that the conjecture follows from a general conjecture on Iwasawa theory for almost all Tate twists. 1 The Jannsen conjecture on local type Galois groups Jannsen’s conjecture [9] predicts the vanishing of a second Galois cohomology group for the p-adic realization of almost all Tate twists of a pure Chow motive. It also specifies the Tate twists where this cohomology group could not vanish. Without this specification the conjecture is a generalization of the classical weak Leopoldt conjecture. We refer to Jannsen’s original paper [9] and Perrin-Riou’s paper [14, Appendix B] for the relations with other conjectures and for general results. Let F be a number field with algebraic closure F . Let X be a smooth, projective variety of pure dimension d over F . Let p be a prime number, and S a finite set of places of F , containing all places above ∞ and p, and all primes where X has bad reduction. Let GS be the Galois group over F of the maximal S-ramified (unramified outside S) extension of F , that we call FS . Conjecture 1.1 (Jannsen). If X = X ×F F , then H(GS , H i et(X,Qp(n))) = 0 if { a) i+ 1 < n, or b) i+ 1 > 2n. This conjecture can be verified also from the étale cohomology with Zp or Qp/Zp coefficients. Lemma 1.2 (lemma 1 [9]). The following statements are equivalent: 1. H(GS , H i et(X,Qp(n))) = 0. Work partially supported by BFM2003-06092