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 $K$. (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 $L,K$.
• $G_{0}=1\Leftrightarrow L/K$ 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 $i_{G}$. $i_{G}$ 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.) It follows from this

• $G_{0}/G_{1}$ 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:

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

If $H$ is a normal subgroup of $G$, then, for $\sigma \in G$, $i_{G/H}(\sigma )={1 \over e_{L/K}}\sum _{s\mapsto \sigma }i_{G}(s)$.

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 $sts^{-1}t^{-1}\in G_{i+j+1}$. 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:

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

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

Example: a quartic extension

Let K be the extension of 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).

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

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

Ramification groups in upper numbering

If $u$ is a real number $\geq -1$, 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

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

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

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

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

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. 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 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.