Let L/K be a finite Galois extension of complete discrete valued fields of characteristic p. Assume that the induced residue field extension $k_L/k_K$ is separable. For an integer n ≥ 0, let $W_n{(}_L)$ denote the ring of Witt vectors of length n with coefficients in ${}_L$. We show that the proabelian group ${H^1(G,W_n{(}_L))}_{n\in \mathbb{N}}$ is zero. This is an equicharacteristic analogue of Hesselholt’s conjecture, which was proved before when the discrete valued fields are of mixed characteristic.

In this paper we describe how to perform computations with Witt vectors of length $3$ in an efficient way and give a formula that allows us to compute the third coordinate of the Greenberg transform of a polynomial directly. We apply these results to obtain information on the third coordinate of the $j$-invariant of the canonical lifting as a function on the $j$-invariant of the ordinary elliptic curve in characteristic $p$.

Let $k$ be the algebraic closure of ${𝔽}_p$ 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 ${𝒴}_n$ of conjugacy classes of series of order $p^n$ 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 $p^n$ for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series by means...

We 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.

On montre que tout anneau local régulier complet muni d’une valuation de rang $1$ peut être plongé, en tant qu’anneau valué, dans un anneau de séries de Puiseux généralisées.