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 detailed information on the ramification phenomena of the extension.

Ramification groups in lower numbering

Ramification groups are a refinement of the Galois group $G$ of a finite $L/K$ Galois extension of local fields. We shall write $w, \mathcal O_L, \mathfrak p$ for the valuation, the ring of integers and its maximal ideal for $L$ . As a consequence of Hensel's lemma, one can write $\mathcal O_L = \mathcal O_K[\alpha]$ for some $\alpha \in L$ where $O_K$ is the ring of integers of $K$ .^[1] (This is stronger than the primitive element theorem.) 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 $\mathcal O_L / \mathfrak p^{i+1}.$
(ii) $w(s(x) - x) \ge i+1$ for all $x \in \mathcal O_L$
(iii) $w(s(\alpha) - \alpha) \ge i+1.$

The group $G_i$ is called $i$ -th ramification group. They form a decreasing filtration,

G_{-1} = G \supset G_0 \supset G_1 \supset \dots \{*\}.

In fact, the $G_i$ are normal by (i) and trivial for sufficiently large $i$ by (iii). For the lowest indices, it is customary to call $G_{0}$ the inertia subgroup of $G$ because of its relation to splitting of prime ideals, while $G_{1}$ the wild inertia subgroup of $G$ . The quotient $G_0 / G_1$ is called the tame quotient.

The Galois group $G$ and its subgroups $G_i$ are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular,

$G/G_0 = \operatorname{Gal}(l/k),$ where $l, k$ are the (finite) residue fields of $L, K$ .^[2]
$G_0 = 1 \Leftrightarrow L/K$ is unramified.
$G_1 = 1 \Leftrightarrow L/K$ is 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 $i_G$ .^[3] $i_G$ satisfies the following: for $s, t \in G$ ,

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

Fix a uniformizer $\pi$ of $L$ . Then $s \mapsto s(\pi)/\pi$ induces the injection $G_i/G_{i+1} \to U_{L, i}/U_{L, i+1}, i \ge 0$ where $U_{L, 0} = \mathcal{O}_L^\times, U_{L, i} = 1 + \mathfrak{p}^i$ . (The map actually does not depend on the choice of the uniformizer.^[4]) It follows from this^[5]

$G_0/G_1$ is cyclic of order prime to $p$
$G_i/G_{i+1}$ is a product of cyclic groups of order $p$ .

In particular, $G_{1}$ is a p-group and $G_{0}$ 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:^[6]

w({\mathfrak {D}}_{L/K})=\sum _{s\neq 1}i_{G}(s)=\sum _{i=0}^{\infty }(|G_{i}|-1).

If $H$ is a normal subgroup of $G$ , then, for $\sigma \in G$ , $i_{G/H}(\sigma) = {1 \over e_{L/K}} \sum_{s \mapsto \sigma} i_G(s)$ .^[7]

Combining 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).

If $s \in G_i, t \in G_j, i, j \ge 1$ , then $sts^{-1}t^{-1} \in G_{i+j+1}$ .^[8] In the terminology of Lazard, this can be understood to mean the Lie algebra $\operatorname{gr}(G_1) = \sum_{i \ge 1} G_i/G_{i+1}$ is abelian.

Example: the cyclotomic extension

The ramification groups for a cyclotomic extension $K_{n}:=\mathbf {Q} _{p}(\zeta )/\mathbf {Q} _{p}$ , where $\zeta$ is a $p^{n}$ -th primitive root of unity, can be described explicitly:^[9]

G_{s}=Gal(K_{n}/K_{e}),

where e is chosen such that $p^{e-1}\leq s<p^{e}$ .

Example: a quartic extension

Let K be the extension of $Q 2$ generated by $x_{1}={\sqrt {2+{\sqrt {2}}\ }}$ . The conjugates of x₁ are x₂= $x_{2}={\sqrt {2-{\sqrt {2}}\ }},$ x₃ = −x₁, x₄ = −x₂.

A little computation shows that the quotient of any two of these is a unit. Hence they all generate the same ideal; call it $π$ . ${\sqrt {2}}$ generates $π$ ²; (2)= $π$ ⁴.

Now x₁ − x₃ = 2x₁, which is in $π$ ⁵.

and $x_{1}-x_{2}={\sqrt {4-2{\sqrt {2}}\,\,}},$ which is in $π$ ³.

Various methods show that the Galois group of K is $C_{4}$ , cyclic of order 4. Also:

G_{0}=G_{1}=G_{2}=C_{4}.

and $G_{3}=G_{4}=(13)(24).$

$w({\mathfrak {D}}_{K/Q_{2}})=3+3+3+1+1=11,$ so that the different ${\mathfrak {D}}_{K/Q_{2}}=\pi ^{11}$

x₁ satisfies x⁴ − 4x² + 2, which has discriminant 2048 = 2¹¹.

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, $s \in G_u \Leftrightarrow i_G(s) \ge u+1.$ Define $\phi$ by^[10]

\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 $-1 < t \le 0$ .^[11] 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:^[12] 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 the ramification groups in the lower numbering satisfy $G_u H/H = (G/H)_v$ (for $v = \phi_{L/F}(u)$ where $L/F$ is the subextension corresponding to $H$ ), and that the ramification groups in the upper numbering satisfy $G^u H/H = (G/H)^u$ .^[13]^[14] This allows one to define ramification groups in the upper numbering for infinite Galois extensions (such as the absolute Galois group of a local field) from the inverse system of ramification groups for finite subextensions.

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., $G_i = G_{i+1}$ whenever $\phi(i)$ is not an integer.^[15]

The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism. The image of $G^n(L/K)$ under the isomorphism

G(L/K)^{\mathrm{ab}} \leftrightarrow K^*/N_{L/K}(L^*)

is just^[16]

U^n_K / (U^n_K \cap N_{L/K}(L^*)) \ .

Notes

↑ Neukirch (1999) p.178
↑ since $G/G_0$ is canonically isomorphic to the decomposition group.
↑ Serre (1979) p.62
↑ Conrad
↑ Use $U_{L, 0}/U_{L, 1} \simeq l^\times$ and $U_{L, i}/U_{L, i+1} \approx l^+$
↑ Serre (1979) 4.1 Prop.4, p.64
↑ Serre (1979) 4.1. Prop.3, p.63
↑ Serre (1979) 4.2. Proposition 10.
↑ Serre, Corps locaux. Ch. IV, §4, Proposition 18
↑ Serre (1967) p.156
↑ Neukirch (1999) p.179
↑ Serre (1967) p.155
↑ Neukirch (1999) p.180
↑ Serre (1979) p.75
↑ Neukirch (1999) p.355
↑ Snaith (1994) pp.30-31

References

B. Conrad, Math 248A. Higher ramification groups
Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001.
Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.
Serre, Jean-Pierre (1967). "VI. Local class field theory". In Cassels, J.W.S.; Fröhlich, A. Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. London: Academic Press. pp. 128–161. Zbl 0153.07403.
Serre, Jean-Pierre (1979). Local Fields. Graduate Texts in Mathematics. 67. Translated by Greenberg, Marvin Jay. Berlin, New York: Springer-Verlag. ISBN 0-387-90424-7. MR 0554237. Zbl 0423.12016.
Snaith, Victor P. (1994). Galois module structure. Fields Institute monographs. Providence, RI: American Mathematical Society. ISBN 0-8218-0264-X. Zbl 0830.11042.

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[N178-1] Neukirch (1999) p.178

[2] since $G/G_0$ is canonically isomorphic to the decomposition group.

[S7962-3] Serre (1979) p.62

[4] Conrad

[5] Use $U_{L, 0}/U_{L, 1} \simeq l^\times$ and $U_{L, i}/U_{L, i+1} \approx l^+$

[S64-6] Serre (1979) 4.1 Prop.4, p.64

[S63-7] Serre (1979) 4.1. Prop.3, p.63

[8] Serre (1979) 4.2. Proposition 10.

[9] Serre, Corps locaux. Ch. IV, §4, Proposition 18

[S67156-10] Serre (1967) p.156

[N179-11] Neukirch (1999) p.179

[S67155-12] Serre (1967) p.155

[N180-13] Neukirch (1999) p.180

[S75-14] Serre (1979) p.75

[N355-15] Neukirch (1999) p.355

[Sn3031-16] Snaith (1994) pp.30-31