Conjugacy classes of series in positive characteristic and Witt vectors

Let k be the algebraic closure of Fp and K be the local field of formal power series with coefficients in k. The aim of this paper is the description of the set Yn of conjugacy classes of series of order p for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic p which are invertible and of finite order p for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series by means of Witt vectors of finite length. We develop some tools which permit us to construct a bijection between a set An of Witt vectors and a set Xn of pairs constituted by a cyclic totally ramified extension L/K of degree p and a generator of its Galois group. We are able to define for any element of An a sequence of ramification breaks. We also describe another bijection between Yn and the orbits of An under a certain group action. Ramification breaks of a series belonging to Yn can be recovered from the components of a corresponding vector in An.