Automorphisms of Formal Power Series Rings over a Valuation Ring

The aim of this paper is to report on recent work on liftings of groups of au-tomorphisms of a formal power series ring over a eld k of characteristic p to characteristic 0, where they are realised as groups of automorphisms of a formal power series ring over a suitable valuation ring R dominating the Witt vectors W(k): We show that the lifting requirement for a group of automorphisms places severe restrictions on the structure of the group as well as implying numerical con-gruences, which need to be satissed by the xed points of these automorphisms. In particular if the centre of the group isn't cyclic then it must be a p-group (an easy consequence observed by Matignon in Ma2]), and for p = 2 the dihedral groups D 2n ; with n odd, can never be lifted. We also draw attention to equivalent formulations of assertions on the existence of prescribed nite order automorphisms of formal power series rings over a valuation ring. Recently these questions have become important in the study of the arithmetic and geometry of curves over a p-adic eld, due to a local-global-principle for lifting galois covers of smooth curves from characteristic p to characteristic 0. More precisely the local-global-principle for liftings gives necessary and suucient conditions, whereby liftings of the inertia groups acting on the completions of the local rings at the points of a galois cover of smooth curves over k to smooth galois covers of the p-adic open disc over R; ensures a global lifting to a galois covering of smooth relative curves over R: The completed local rings are formal power series rings over k and R respectively, and so provide the setting. The most important question in this regard is the Oort-Sekiguchi Conjecture, which claims that the lifting problem can be solved whenever the group for the cover is cyclic. Here the only problem is caused by the p-part of the respective inertia groups, and the conjecture has been proved when these are ap or ap 2 cyclic groups with (a; p) = 1 (See O-S-S] and G-M1]).