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Séminaire de Théorie des Nombres

Models of (Z/p^2Z)_K in unequal characteristic

Dajano Tossici

( MPIM - Bonn )

Salle de Conférences

le 13 février 2009 à 14:00

Let RR be a discrete valuation ring of unequal characteristic and let KK be its fraction field. The aim of this talk is to give the classification of models of (Z/p2Z)K(\mathbb{Z}/p^2\mathbb{Z})_ K, i.e. finite and flat group schemes over RR which are isomorphic to (Z/p2Z)K(\mathbb{Z}/p^2 \mathbb{Z})_K over KK. The main features of this classification are the following: 1) the parameters can be easily interpretated; 2) the description of the models is explict, i.e. it is given in terms of equations; 3) any model can be seen as the kernel of an exact sequence which coincides generically with the Kummer sequence. This sequence let us to generalize the Kummer Theory to describe torsors under these group schemes. The main tool which we use is the Sekiguchi-Suwa Theory, which we will briefly recall. If we will have enough time we will compare our work with the recent works of Breuil and Kisin about the classification of finite and flat goup schemes over a d.v.r.