Let Φ be an endomorphism of P 1 Q , the projective line over the algebraic closure of Q, of degree ≥ 2 defined over a number field K. Let v be a non-archimedean valuation of K. We say that Φ has critically good reduction at v if any pair of distinct ramification points of Φ do not collide under reduction modulo v and the same holds for any pair of branch points. We say that Φ has simple good reduction at v if the map Φ v , the reduction of Φ modulo v, has the same degree of Φ. We prove that if Φ has critically good reduction at v and the reduction map Φ v is separable, then Φ has simple good reduction at v.

Abstract. We discuss essential dimension of group schemes, with particular attention to infinitesimal group schemes. We prove that the essential dimension of a group scheme of finite type over a field k is at least equal to the difference between the dimension of its Lie algebra and its dimension. Furthermore, we show that the essential dimension of a trigonalizable group scheme of length p n over a field of characteristic p > 0 is at most n. We give several examples.

29 pagesInternational audienceWe present an account of the construction by S. Sekiguchi and N. Suwa of a cyclic isogeny of affine smooth group schemes unifying the Kummer and Artin-Schreier-Witt isogenies. We complete the construction over an arbitrary base ring. We extend the statements of some results in a form adapted to a further investigation of the models of the group schemes of roots of unity

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Let O K be a discrete valuation ring of mixed characteristics p0, pq, with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O K -models of the group scheme µ p n ,K of p n -th roots of unity, which we call Kummer group schemes. We carefully set out the general framework and algebraic properties of this construction. When k is perfect and O K is a complete totally ramified extension of the ring of Witt vectors W pkq, we provide a parallel study of the Breuil-Kisin modules of finite flat models of µ p n ,K , in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. We compute these objects for n ď 3. This leads us to conjecture that all finite flat models of µ p n ,K are Kummer group schemes.