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 theory of valuations

In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains.^[1]^[2]

The structure of the set of extensions is known better when L/K is Galois.

Decomposition group and inertia group

Let (K, v) be a valued field and let L be a finite Galois extension of K. Let S_v be the set of equivalence classes of extensions of v to L and let G be the Galois group of L over K. Then G acts on S_v by σ[w] = [w ∘ σ] (i.e. w is a representative of the equivalence class [w] ∈ S_v and [w] is sent to the equivalence class of the composition of w with the automorphism σ : L → L; this is independent of the choice of w in [w]). In fact, this action is transitive.

Given a fixed extension w of v to L, the decomposition group of w is the stabilizer subgroup G_w of [w], i.e. it is the subgroup of G consisting of all elements that fix the equivalence class [w] ∈ S_v.

Let m_w denote the maximal ideal of w inside the valuation ring R_w of w. The inertia group of w is the subgroup I_w of G_w consisting of elements σ such that σx ≡ x (mod m_w) for all x in R_w. In other words, I_w consists of the elements of the decomposition group that act trivially on the residue field of w. It is a normal subgroup of G_w.

The reduced ramification index e(w/v) is independent of w and is denoted e(v). Similarly, the relative degree f(w/v) is also independent of w and is denoted f(v).

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 ${\mathcal {O}}_{K}$ is the ring of integers of $K$ .^[3] (This is stronger than the primitive element theorem.) Then, for each integer $i\geq -1$ , 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) $w(s(x)-x)\geq i+1$ for all $x\in {\mathcal {O}}_{L}$
(iii) $w(s(\alpha )-\alpha )\geq i+1.$

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 $L,K$ .^[4]
$G_{0}=1\Leftrightarrow L/K$ 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 $i\geq 0$ .

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 $i_{G}$ .^[5] $i_{G}$ satisfies the following: for $s,t\in G$ ,

$i_{G}(s)\geq i+1\Leftrightarrow s\in G_{i}.$
$i_{G}(tst^{-1})=i_{G}(s).$
$i_{G}(st)\geq \min\{i_{G}(s),i_{G}(t)\}.$

Fix a uniformizer $\pi$ of $L$ . Then $s\mapsto s(\pi )/\pi$ induces the injection $G_{i}/G_{i+1}\to U_{L,i}/U_{L,i+1},i\geq 0$ 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.^[6]) It follows from this^[7]

$G_{0}/G_{1}$ 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:^[8]

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 $\sigma \in G$ , $i_{G/H}(\sigma )={1 \over e_{L/K}}\sum _{s\mapsto \sigma }i_{G}(s)$ .^[9]

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 $s\in G_{i},t\in G_{j},i,j\geq 1$ , then $sts^{-1}t^{-1}\in G_{i+j+1}$ .^[10] In the terminology of Lazard, this can be understood to mean the Lie algebra $\operatorname {gr} (G_{1})=\sum _{i\geq 1}G_{i}/G_{i+1}$ 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:^[11]

G_{s}=\operatorname {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 $x_{1}$ are $x_{2}={\sqrt {2-{\sqrt {2}}}}$ , $x_{3}=-x_{1}$ , $x_{4}=-x_{2}$ .

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 $x_{1}-x_{3}=2x_{1}$ , 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 ${\mathfrak {D}}_{K/Q_{2}}=\pi ^{11}$

$x_{1}$ satisfies X⁴ − 4X² + 2, which has discriminant 2048 = 2¹¹.

Ramification groups in upper numbering

If $u$ is a real number $\geq -1$ , let $G_{u}$ denote $G_{i}$ where i the least integer $\geq u$ . In other words, $s\in G_{u}\Leftrightarrow i_{G}(s)\geq u+1.$ Define $\phi$ by^[12]

\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 $-1<t\leq 0$ .^[13] Then $\phi (u)=u$ for $-1\leq u\leq 0$ . 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:^[14] 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

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 $G^{u}H/H=(G/H)^{u}$ .^[15]^[16] 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.^[17]

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^[18]

U_{K}^{n}/(U_{K}^{n}\cap N_{L/K}(L^{*}))\ .

Notes

^ Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. Vol. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001.
^ Zariski, Oscar; Samuel, Pierre (1976) [1960]. Commutative algebra, Volume II. Graduate Texts in Mathematics. Vol. 29. New York, Heidelberg: Springer-Verlag. Chapter VI. ISBN 978-0-387-90171-8. Zbl 0322.13001.
^ 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. Vol. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001.
Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 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. (eds.). 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. Vol. 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.

[1] Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. Vol. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001.

[2] Zariski, Oscar; Samuel, Pierre (1976) [1960]. Commutative algebra, Volume II. Graduate Texts in Mathematics. Vol. 29. New York, Heidelberg: Springer-Verlag. Chapter VI. ISBN 978-0-387-90171-8. Zbl 0322.13001.

[N178-3] Neukirch (1999) p.178

[4] since $G/G_{0}$ is canonically isomorphic to the decomposition group.

[S7962-5] Serre (1979) p.62

[6] Conrad

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

[S64-8] Serre (1979) 4.1 Prop.4, p.64

[S63-9] Serre (1979) 4.1. Prop.3, p.63

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

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

[S67156-12] Serre (1967) p.156

[N179-13] Neukirch (1999) p.179

[S67155-14] Serre (1967) p.155

[N180-15] Neukirch (1999) p.180

[S75-16] Serre (1979) p.75

[N355-17] Neukirch (1999) p.355

[Sn3031-18] Snaith (1994) pp.30-31

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