Slope content of a $p$-adic field (reviewed)

Let $G$ be the Galois group of the Galois closure $K^{gal}$ of a $p$-adic field $K$. Then $G$ has a filtration by higher ramification groups, which in turn are connected to the discriminants of subfields of $K^{gal}$. The slope content encodes information about this filtration giving the location and sizes of the jumps.

Let $G^i$ be the $i$th higher ramification group in upper numbering from Serre's Local Fields [10.1007/978-1-4757-5673-9]. They form a descending filtration of normal subgroups where $i$ is a continuous parameter, $i\geq -1$. Let $G^{(j)} = G^{j-1}$, and $G^{(j+)} = \cup_{\epsilon>0} G^{(j+\epsilon)}$. Then

• $G^{(0)}=G$.

• $G^{(1)}$ is the inertia subgroup of $G$, i.e., the kernel of the natural map from $G$ to the Galois group of the residue field.

• the fixed field of $G^{(1)}$ is the maximum unramified subfield of $K^{gal}/\mathbb{Q}_p$, which we denote by $K^{unram}$. The degree $[K^{unram}:\mathbb{Q}_p]$ is the unramified degree for $K^{gal}/\mathbb{Q}_p$. The extension $K^{gal}/K^{unram}$ is totally ramified.

• the group $G^{(1)}/G^{(1+)}$ is cyclic of order prime to $p$. Its order is the tame degree for $K^{gal}/\mathbb{Q}_p$ as it is the degree of the largest tamely totally ramified subextension.

• The group $G^{(1+)}$ is a $p$-group. Values of $s>1$ giving jumps in the filtration $G^{(s)}\neq G^{(s+)}$ are wild slopes. We repeat a slope $m$ times if $p^m =[G^{(s)} : G^{(s+)}]$, and then list the wild slopes in non-decreasing order.

The slope content then takes the form $[s_1,\ldots, s_k]_t^u$ where $t$ is the tame degree, $u$ is the degree of the maximum unramified subfield, and the $s_i$ give the list of the wild slopes. In particular, $[K^{gal}:\Q_p]=tup^k$. If $t$ or $u$ is equal to $1$, it is not printed.