The purpose of this article is to give a cohomological formula for the unit-root part of the L-function associated to a Barsotti-Tate group G on a scheme S over a field of characteristic p when G extends to some compactification of S. This is an analogue of a part of a conjecture of Katz according to wich the L-function of an F-crystal should be expressed in terms of the p-adic etale sheaf corresponding to the unit-root part of the crystal. In order to carry out this project, we use the technics of [E-LS II] wich require in our case an extension of the Dieudonné crystalline theory ([B-B-M]) to “crystal of level mG” in the sense of Berthelot. We show that the unit-root L-function of the Dieudonné crystal associated to G can be expressed in terms of the syntomic cohomology of the Ext group of G by the constant sheaf.
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