Monodromy of logarithmic Barsotti-Tate groups attached to 1-motives

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Let R be a complete discrete valuation ring with perfect residue field k of positive characteristic p and field of fractions K of characteristic 0. In this paper we consider a K-1-motive Mₖ as in [Ra] and its associated Barsotti-Tate group. This last does not in general extend to a Barsotti-Tate group over R. However, with some assumptions, it extends to a logarithmic Barsotti-Tate group over R. This follows from [Ra] and Kato's results on finite logarithmic group schemes. Once chosen a uniformizing parameter π of R, any logarithmic Barsotti-Tate group over R is described by two data (G,N) where G is a classical Barsotti-Tate group over R and N is a homomorphism of classical Barsotti-Tate groups. Moreover, if R=W(k), N induces a W(k)-homorphism N M(Gₖ)→ M(Gₖ) on Dieudonné modules such that F NV= N and N²=0. In the first part of the paper we recall these constructions and we show how to relate N with the ``geometric monodromy'' introduced by Raynaud. In the second part of the paper we give an explicit description of N in terms of additive extensions and integrals. In the last part of the paper we describe how to recover the logarithmic Barsotti-Tate group attached to a 1-motive from a concrete scheme endowed with a suitable logarithmic structure.