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Tate duality and wild ramification. (English) Zbl 0767.11057

This short paper gives a positive answer to the question of compatibility of the Tate pairing \(A(K)\times H^ 1(K,\widehat A)\to \mathbb{Q}/\mathbb{Z}\) with the natural filtrations on \(A(K)\) for an abelian variety \(A\) over a \({\mathfrak p}\)-adic field \(K\) and on \(H^ 1(K,\widehat A)\) where \(\widehat A\) is the dual abelian variety. It contains also some discussions whether this is true for general formal groups (here the answer is yet unknown).
One can add two remarks. First, its results can be extended in a proper way for the case of a perfect residue field via [I. Fesenko, Local class field theory: perfect residue field case (Preprint MPI für Mathematik, 1993)]. Second, the paper of M. I. Bashmakov and A. N. Kirillov [“Lutz filtration on formal groups”, Izv. Akad. Nauk SSSR, Ser. Mat. 39, 1227-1239 (1975; Zbl 0337.14032)] may provide the answer in the general case of formal groups.

MSC:

11S31 Class field theory; \(p\)-adic formal groups
14L05 Formal groups, \(p\)-divisible groups
11S15 Ramification and extension theory
14M99 Special varieties

Citations:

Zbl 0337.14032
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References:

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