Higher ramification groups of Artin-Schreier extensions

Question

I am reading this paper http://www.numdam.org/article/JTNB_2005__17_2_689_0.pdf I have been trying to prove Proposition 2.1(3) about stabilizing of higher ramification groups. Can anyone give me a hint to prove this problem? Thank you.

Answer 1

So you have an Artin-Schreier extension $L=K(b)$ where $b^p-b=a$ where $v(a)=-m$, $m>0$ and $p\nmid m$. Then $v_K(b)=-m/p$, or $V_L(b)=-m$. The generator $\tau$ of the Galois group takes $b$ to $b+1$. To find the ramification groups under the lower numbering you want to find the largest $n$ such that $\tau$ acts trivially on the ring $O_L/P_L^n$.

Let's identify a generator of $P_L$. There are integers $r$, $s$ with $1=-rm+sp$. Choose $r$ positive with $0<r<p$. We can take $\pi_L=b^r\pi_K^s$. Then $$\frac{\tau(\pi_L)}{\pi_L}=\frac{(b+1)^r}{b^r}\equiv 1+r/b\pmod{b^{-2}}.$$ So $$\tau(\pi_L)-\pi_L\equiv r \pi_L/b\pmod{\pi_L/b^2}.$$ So $\tau$ acts trivially on $O_L/P_L^n$ iff $n\le m$ (Serre, Local Fields, Lemma IV.1). This determines the ramification jump for the lower numbering, and then one can determine it for the upper numbering too.