Notes on class field theory

Let $L/K$ be a Galois extension of finite extensions of ${\mathbb{Q}}_p$ with Galois group $G\text{.}$ For each integer $i\ge -1\text{,}$ let $G_i$ be the set of $g\in G$ satisfying the equivalent conditions of Lemma 4.4.2. The $G_i$ form a decreasing sequence of subgroups of $G\text{;}$ these together form the lower numbering ramification filtration on $G\text{.}$ In particular, $G_{-1}=G$ and $G_0$ equals the inertia subgroup of $G\text{.}$

For convenience later, we extend the definition of the filtration $G_i$ to arbitrary real values $i\ge -1$ by setting $G_i=G_{\lceil i\rceil }.$

From the definition, we see that the formation of the lower numbering filtration is compatible with subgroups: if $H=\mathrm{Gal}(L/M)$ is a subgroup of $G\text{,}$ then $H_i=H\cap G_i$ for all $i\ge -1\text{.}$ However, it is not at all clear what happens when we pass from $G$ to a quotient.

For $i\ge 0\text{,}$ let $U_L^i$ be the subgroup of ${\mathfrak{o}}_L^{\ast }$ consisting of elements $\alpha$ for which $v_L(\alpha -1)\ge i\text{.}$ The group $U_L^0/U_L^1$ is naturally isomorphic to the group of units of the residue field ${\mathfrak{o}}_L/{\pi }_L\text{.}$ For $i>0\text{,}$ the group $U_L^i/U_L^{i+1}$ carries the structure of a one-dimensional vector space over ${\mathfrak{o}}_L/{\pi }_L\text{;}$ for any choice of the uniformizer ${\pi }_L$ we may use the class of ${\pi }_L^i$ as the basis element, but there is no distinguished choice without this breaking of symmetry.

By Lemma 4.4.4, for $i\ge 0$ we may view $G_i$ as the maximal subgroup of $G$ carrying $U_L^0$ into itself. In particular, the quotient $G_i/G_{i+1}$ is naturally isomorphic to a subgroup of $U_L^i/U_L^{i+1}\text{.}$

This gives us the following structural properties of $G\text{.}$ First, the group $G_{-1}/G_0$ is isomorphic to the residue field extension, which is cyclic. Next, $G_0/G_1$ is isomorphic to a subgroup of $U_L^0/U_L^1\text{,}$ and so is cyclic of order prime to $p\text{.}$ Finally, for $i\ge 1\text{,}$ $G_i/G_{i+1}$ is a subgroup of $U_L^i/U_L^{i+1}\text{,}$ and so is an elementary abelian $p$-group. In particular, $G$ is a solvable group, as noted in Remark 4.2.3.

Let $L/K$ be a Galois extension of finite extensions of ${\mathbb{Q}}_p$ with Galois group $G\text{.}$ We define the breaks in the ramification filtration for the lower numbering (respectively, the upper numbering) as the values of $i$ for which $G_i\ne G_j$ for all $j>i$ (resp. $G^i\ne G^j$ for all $j>i$).

By definition, the breaks for the lower numbering are integers, while the breaks for the upper numbering are only guaranteed to be rational numbers. In fact, it is possible to exhibit examples where the breaks for the upper numbering are not integers (see Exercise 2 and Exercise 3). However, in the next section we will see that this cannot occur for abelian extensions.