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

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 $M_K$ 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 $\pi$ 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 ${\cal N}\colon M(G_k)\to M(G_k)$ on Dieudonn\'e modules such that $F{\cal N}V={\cal N}$ and ${\cal N}^2=0$. 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 ${\cal N}$ 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.