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 $G$ 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]

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

If $H$ is a normal subgroup of $G$, then, for $\sigma \in G$, $i_{G/H}(\sigma) = {1 \over e_{L/K}} \sum_{s \mapsto \sigma} i_G(s)$.^[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 ${\displaystyle K_{n}:=\mathbf {Q} _{p}(\zeta )/\mathbf {Q} _{p}}$, where $\zeta$ is a $p^{n}$-th primitive root of unity, can be described explicitly:^[9]

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

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

Example: a quartic extension

Let K be the extension of Q₂ generated by x₁=$\sqrt{2+\sqrt{2}\ }$. The conjugates of x₁ are x₂=$\sqrt{2-\sqrt{2}\ }$, 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 π²; (2)=π⁴.

Now x₁-x₃=2x₁, which is in π⁵.

and x₁-x₂=$\sqrt{4-2\sqrt{2}\ }$, which is in π³.

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).

${\displaystyle w({\mathfrak {D}}_{K/{Q_{2}}})}$ = 3+3+3+1+1 = 11. so that the different ${\displaystyle {\mathfrak {D}}_{K/{Q_{2}}}}$=π¹¹.

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

1. Neukirch (1999) p.178
2. since $G/G_0$ is canonically isomorphic to the decomposition group.
3. Serre (1979) p.62
4. Conrad
5. Use $U_{L, 0}/U_{L, 1} \simeq l^\times$ and $U_{L, i}/U_{L, i+1} \approx l^+$
6. Serre (1979) 4.1 Prop.4, p.64
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
10. Serre (1967) p.156
11. Neukirch (1999) p.179
12. Serre (1967) p.155
13. Neukirch (1999) p.180
14. Serre (1979) p.75
15. Neukirch (1999) p.355
16. Snaith (1994) pp.30-31

See also

• Ramification theory of valuations

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. Zbl 0956.11021. MR 1697859.
• 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 from the French by Marvin Jay Greenberg. 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.