Ramification group

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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 $i\geq -1$ , 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 $G_{1}/G_{0}$ 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 $i\geq 0$ .

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 $i_{G}$ .^[3] $i_{G}$ 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 $L$ . $s\mapsto s(\pi )/\pi$ then 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.^[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]

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

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 $s\in G_{i},t\in G_{j},i,j\geq 1$ , 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 $\operatorname {gr}(G_{1})=\sum _{{i\geq 1}}G_{i}/G_{{i+1}}$ is abelian.

Example

Let K be 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 $π$ ²; (2)= $π$ ⁴.

Now x₁-x₃=2x₁, which is in $π$ ⁵.

and x₁-x₂= ${\sqrt {4-2{\sqrt {2}}\ }}$ , 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).

$w({\mathfrak {D}}_{{K/Q}})$ = 3+3+3+1+1 = 11. so that the different ${\mathfrak {D}}_{{K/Q}}$ = $π$ ¹¹.

x₁ 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^[9]

$\phi (u)=\int _{0}^{u}{dt \over (G_{0}:G_{t})}$

where, by convention, $(G_{0}:G_{t})$ is equal to $(G_{{-1}}:G_{0})^{{-1}}$ if $t=-1$ and is equal to $1$ for $-1<t\leq 0$ .^[10] 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:^[11] 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 $G^{u}H/H=(G/H)^{u}$ .^[12]^[13] 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.^[14]

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^[15]

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

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 (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, 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.

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