Meaning of Hasse-Arf theorem

Question

I am reading about the Hasse-Arf theorem in Serre's 'Local Fields' and I have a hard time understanding what exactly it means for the upper numbering to have jumps only at integers. It seems like a quite arbitrary result on indices. What is a good way to think about this theorem? Are there illuminating consequences or special cases?

Answer 1

The question on the "interest" of a notion or a result can present many aspects. In the specific case here, fix a base local field $K$ in the sense of Serre's book. Then:

1) For a galois extension $L/K$ with group $G$ (not necessarily finite), two filtrations can be defined on $G$, the lower filtration $G_u$ (for real $u\ge -1$), and the upper filtration $G^v=G_{\psi(v)}$. As stressed by Serre somewhere in his chap. IV, the lower numbering is adapted to subgroups, in the sense that $H_u=H\cap G_u$, whereas the upper numbering is adapted to quotients, $(G/H)^v=G^vH/H$. In a remark after the proof of prop. 14 of IV, 3, Serre even gives a unified numerotation $G(t)$.

2) The Hasse-Arf thm. states that, if $G$ is abelian, then a jump $v$ in the upper filtration must be an integer; in other terms, if $G_u \neq G_{u+1}$, then $\phi (u)$ is an integer. This gives non trivial "congruential" information on the lower jumps. For instance, just try to give a direct proof in the simple example of a finite cyclic $p$ - extension, and you'll see that this is far from being obvious.

3) More important are applications to local CFT, see chap. XV. If $G$ is abelian (not necessarily finite), the reciprocity homomorphism $\theta: K^* \to G$ sends the filtration ${U_K}^v$ of the unit group $U_K$ onto the filtration $G^v$. More precisely, if $G$ is abelian finite, $\theta$ induces an iso. $K^*/N(L^*) \cong G$, where $N$ is the norm of $L/K$, and if moreover $L/K$ is totally ramified, an iso $\theta_n : {U_K}^n/{U_K}^{n+1}N({U_L}^{\psi (n)}) \cong G^n/G^{n+1}$ for any integer $n$. A specific application of the Hasse-Arf thm. concerns the so called explicit reciprocity laws. Independently of CFT, using polynomials of a certain type, one gets a normic iso. $N_n : {U_L}^{\psi (n)}/{U_L}^{\psi (n+1)} \to {U_K}^n/{U_K}^{n+1}$ (V, 6), as well as an iso. $\delta_n : {U_K}^n/{U_K}^{n+1}N({U_L}^{\psi (n)})\cong G_{\psi(n)}/G_{\psi(n) +1}$ (XV, 2). By definition $G_{\psi(n)}=G^n$, and by the Hasse-Arf thm. $G_{\psi(n) +1}=G^{n+1}$. It follows that $\delta_n$ and $\theta_n$ have the same target, and it can be shown that actually $\theta_n (x)=\delta_n (x^{-1})$, hence an explicit reciprocity law.

I refrain from evoking other applications such as the conductor of a galois extension, which would bring us too far ./.