Abstract
AbstractWe define két abelian schemes, két 1-motives and két log 1-motives and formulate duality theory for these objects. Then we show that tamely ramified strict 1-motives over a discrete valuation field can be extended uniquely to két log 1-motives over the corresponding discrete valuation ring. As an application, we present a proof to a result of Kato stated in [12, §4.3] without proof. To a tamely ramified strict 1-motive over a discrete valuation field, we associate a monodromy pairing and compare it with Raynaud’s geometric monodromy.
Highlights
The notion of 1-motive is introduced by Deligne in [3]
We show that a két log 1-motive can be regarded as a 2-term complex in the category of sheaves for the Kummer flat topology
First we give an equivalent description of két log 1-motives using Poincaré biextension, through which we present the dual theory of két log 1-motives
Summary
The notion of 1-motive is introduced by Deligne in [3]. A 1-motive over a base scheme is a two-term complex. Let be a discrete valuation field with ring of integers and a tamely ramified abelian variety over that has potentially good reduction. A két 1-motive over is a two-term complex = [ → − ] in the category of sheaves of abelian groups on (fs/ )ket, such that the degree −1 term is a két lattice and the degree 0 term is an extension of a két abelian scheme by a két torus on (fs/ )ket. A két log 1-motive over an fs log scheme is a 2-term complex = [ → − log] of sheaves of abelian groups on (fs/ )ket such that the degree -1 term is a két lattice over and is an extension of a két abelian scheme by a két torus on (fs/ )ket.
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