Ramification des séries formelles

[摘要] Let $p$ be a prime number. Let $k$ be a finite field of characteristic $p$.The subset $X+X^2 k[[X]]$ of the ring $k[[X]]$ is a group under the substitutionlaw $circ $ sometimes called the Nottingham group of $k$; it is denoted by$mathcal{R}_k$. The ramification of one series $gammainmathcal{R}_k$ iscaracterized by its lower ramification numbers: $i_m(gamma)=ord_Xigl(gamma^{p^m} (X)/X - 1igr)$, as well as its upper ramification numbers:$$u_m (gamma) = i_0 (gamma) + frac{i_1 (gamma) - i_0(gamma)}{p} +cdots + frac{i_m (gamma) - i_{m-1} (gamma)}{p^m} , quad (m inmathbb{N}).$$By Sen's theorem, the $u_m(gamma)$ are integers. In this paper, we determinethe sequences of integers $(u_m)$ for which there exists $gammainmathcal{R}_k$such that $u_m(gamma)=u_m$ for all integer $m geq 0$.

[学科分类] 数学（综合）

[关键词] ramification;Nottingham group