2003/07/04 by Alessandra Bertapelle, Bertapelle, Alessandra, Maurizio Candilera +3
Mathematics · #14F20 #14L05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.AG #math.NT #msc:14F20 #msc:14L05
paper · pdf · doi:10.48550/arxiv.math/0307059
22 pages
arxiv created 2003/07/04 · openalex publication_date 2003/07/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \cal N\colon M(Gk)→ M(Gk) on Dieudonné modules such that F\cal NV=\cal N and \cal 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 \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.