Ramification group

In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives a precisely information on the ramification phenomenon of the extension.

Ramification groups in lower numbering

Let $L/K$ be a finite Galois extension of local fields with group $G$ and finite residue fields $l, k$ . We shall write $w, \mathcal O_L, \mathfrak p$ for the valuation, the ring of integers and its maximal ideal for $L$ . It is known that one can write $\mathcal O_L = \mathcal O_K[\alpha]$ for some $\alpha \in L$ where $O_K$ is the ring of integers of $K$ . (This is stronger than the primitive element theorem and is a consequence of Hensel's lemma.) Then, for each integer $i \ge -1$ , we define $G_i$ to be the set of all $s \in G$ that satisfies the following equivalent conditions.

(i) $s$ operates trivially on $\mathcal O_L / \mathfrak p^{i%2B1}.$
(ii) $w(s(x) - x) \ge i%2B1$ for all $x \in \mathcal O_L$
(iii) $w(s(\alpha) - \alpha) \ge i%2B1.$

(i) shows that $G_i$ are normal and (ii) shows that $G_i = 1$ for sufficiently large $i$ . $G_i$ is then called the $i$ -th ramification group, and they form a finite decreasing filtration of $G$ with $G_{-1} = G$ . $G_0$ is called the inertia subgroup of $G$ . Note:

$G/G_0 = \operatorname{Gal}(l/k).$ ^[1]
$G_0 = 1 \Leftrightarrow L/K$ unramified.
$G_1 = 1 \Leftrightarrow L/K$ tamely ramified (i.e., the ramification index is prime to the residue characteristic.)

The study of ramification groups reduces to the totally ramified case since one has $G_i = (G_0)_i$ for $i \ge 0$ .

One also defines the function $i_G(s) = w(s(\alpha) - \alpha), s \in G$ . (ii) in the above shows $i_G$ is independent of choice of $\alpha$ and, moreover, the study of the filtration $G_i$ is essentially equivalent to that of $i_G$ .^[2] $i_G$ satisfies the following: for $s, t \in G$ ,

$i_G(s) \ge i %2B 1 \Leftrightarrow s \in G_i.$
$i_G(t s t^{-1}) = i_G(s).$
$i_G(st) \ge \min\{ i_G(s), i_G(t) \}.$

Fix a uniformizer $\pi$ of $L$ . $s \mapsto s(\pi)/\pi$ then induces the injection $G_i/G_{i%2B1} \to U_{L, i}/U_{L, i%2B1}, i \ge 0$ where $U_{L, 0} = \mathcal{O}_L^\times, U_{L, i} = 1 %2B \mathfrak{p}^i$ . It follows from this^[3]

$G_0/G_1$ is cyclic of order prime to $p$
$G_i/G_{i%2B1}$ is a product of cyclic groups of order $p$ .

In particular, $G_1$ is a p-group and $G$ is solvable.

The ramification groups can be used to compute the different $\mathfrak{D}_{L/K}$ of the extension $L/K$ and that of subextensions:

$w(\mathfrak{D}_{L/K}) = \sum_{s \ne 1} i_G(s) = \sum_0^\infty (|G_i| - 1).$

If $H$ is a normal subgroup of $G$ , then, for $\sigma \in G$ , $i_{G/H}(\sigma) = {1 \over e_{L/K}} \sum_{s \mapsto \sigma} i_G(s)$ .^[4]

Combing this with the above one obtains: for a subextension $F/K$ corresponding to $H$ ,

$v_F(\mathfrak{D}_{F/K}) = {1 \over e_{L/F}} \sum_{s \not\in H} i_G(s).$ ^[5]

If $s \in G_i, t \in G_j, i, j \ge 1$ , then $sts^{-1}t^{-1} \in G_{i%2Bj%2B1}$ .^[6] In the terminology of Lazard, this can be understood to mean the Lie algebra $\operatorname{gr}(G_1) = \sum_{i \ge 1} G_i/G_{i%2B1}$ is abelian.

Ramification groups in upper numbering

If $u$ is a real number $\ge -1$ , let $G_u$ denote $G_i$ where i the least integer $\ge u$ . In other words, $s \in G_u \Leftrightarrow i_G(s) \ge u%2B1.$ Define $\phi$ by

$\phi(u) = \int_0^u {dt \over (G_0�: G_t)}$

where, by convention, $(G_0�: G_t)$ is equal to $(G_{-1}�: G_0)^{-1}$ if $t = -1$ and is equal to $1$ for $-1 < t \le 0$ . Then $\phi(u) = u$ for $-1 \le u \le 0$ . It is immediate that $\phi$ is continuous and strictly increasing, and thus has the continuous inverse function $\psi$ defined on $[-1, \infty)$ . Define $G^v = G_{\psi(v)}$ . $G^v$ is then called the v-th ramification group in upper numbering. In other words, $G^{\phi(u)} = G_u$ . Note $G^{-1} = G, G^0 = G_0$ . The upper numbering is defined so as to be compatible with passage to quotients: if $H$ is normal in $G$ , then

$(G/H)^v = G^v H / H$ for all $v$

(whereas lower numbering is compatible with passage to subgroups.)

Herbrand's theorem states that $G_u H/H = (G/H)_v$ for $v = \phi_{L/F}(u)$ where $L/F$ is a subextension corresponding to $H$ .

The upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if $G$ is abelian, then the jumps in the filtration $G^v$ are integers; i.e., $G_i = G_{i%2B1}$ whenever $\phi(i)$ is not an integer.

References

^ since $G/G_0$ is canonically isomorphic to the decomposition group.
^ Serre, pg. 62
^ Use $U_{L, 0}/U_{L, 1} \simeq l^\times$ and $U_{L, i}/U_{L, i%2B1} \approx l^%2B$
^ Serre, 4.1. Proposition 3.
^ Serre, 4.1. Proposition 4.
^ Serre, 4.2. Proposition 10.

(1980), Local Fields, Berlin, New York: Springer-Verlag, ISBN 9780387904245, MR 0554237