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.

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.

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 are a refinement of the Galois group ${\displaystyle G}$ of a finite ${\displaystyle L/K}$ Galois extension of local fields. We shall write ${\displaystyle w,{\mathcal {O}}_{L},{\mathfrak {p}}}$ for the valuation, the ring of integers and its maximal ideal for ${\displaystyle L}$. As a consequence of Hensel's lemma, one can write ${\displaystyle {\mathcal {O}}_{L}={\mathcal {O}}_{K}[\alpha ]}$ for some ${\displaystyle \alpha \in L}$ where ${\displaystyle {\mathcal {O}}_{K}}$ is the ring of integers of ${\displaystyle K}$.^[3] (This is stronger than the primitive element theorem.) Then, for each integer ${\displaystyle i\geq -1}$, we define ${\displaystyle G_{i}}$ to be the set of all ${\displaystyle s\in G}$ that satisfies the following equivalent conditions.

• (i) ${\displaystyle s}$ operates trivially on ${\displaystyle {\mathcal {O}}_{L}/{\mathfrak {p}}^{i+1}.}$
• (ii) ${\displaystyle w(s(x)-x)\geq i+1}$ for all ${\displaystyle x\in {\mathcal {O}}_{L}}$
• (iii) ${\displaystyle w(s(\alpha )-\alpha )\geq i+1.}$

The group ${\displaystyle G_{i}}$ is called ${\displaystyle i}$-th ramification group. They form a decreasing filtration,

${\displaystyle G_{-1}=G\supset G_{0}\supset G_{1}\supset \dots \{*\}.}$

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

The Galois group ${\displaystyle G}$ and its subgroups ${\displaystyle G_{i}}$ are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular,

• ${\displaystyle G/G_{0}=\operatorname {Gal} (l/k),}$ where ${\displaystyle l,k}$ are the (finite) residue fields of ${\displaystyle L,K}$.^[4]
• ${\displaystyle G_{0}=1\Leftrightarrow L/K}$ is unramified.
• ${\displaystyle G_{1}=1\Leftrightarrow L/K}$ 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 ${\displaystyle G_{i}=(G_{0})_{i}}$ for ${\displaystyle i\geq 0}$.

One also defines the function ${\displaystyle i_{G}(s)=w(s(\alpha )-\alpha ),s\in G}$. (ii) in the above shows ${\displaystyle i_{G}}$ is independent of choice of ${\displaystyle \alpha }$ and, moreover, the study of the filtration ${\displaystyle G_{i}}$ is essentially equivalent to that of ${\displaystyle i_{G}}$.^[5] ${\displaystyle i_{G}}$ satisfies the following: for ${\displaystyle s,t\in G}$,

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

Fix a uniformizer ${\displaystyle \pi }$ of ${\displaystyle L}$. Then ${\displaystyle s\mapsto s(\pi )/\pi }$ induces the injection ${\displaystyle G_{i}/G_{i+1}\to U_{L,i}/U_{L,i+1},i\geq 0}$ where ${\displaystyle U_{L,0}={\mathcal {O}}_{L}^{\times },U_{L,i}=1+{\mathfrak {p}}^{i}}$. (The map actually does not depend on the choice of the uniformizer.^[6]) It follows from this^[7]

• ${\displaystyle G_{0}/G_{1}}$ is cyclic of order prime to ${\displaystyle p}$
• ${\displaystyle G_{i}/G_{i+1}}$ is a product of cyclic groups of order ${\displaystyle p}$.

In particular, ${\displaystyle G_{1}}$ is a p-group and ${\displaystyle G_{0}}$ is solvable.

The ramification groups can be used to compute the different ${\displaystyle {\mathfrak {D}}_{L/K}}$ of the extension ${\displaystyle L/K}$ and that of subextensions:^[8]

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

If ${\displaystyle H}$ is a normal subgroup of ${\displaystyle G}$, then, for ${\displaystyle \sigma \in G}$, ${\displaystyle i_{G/H}(\sigma )={1 \over e_{L/K}}\sum _{s\mapsto \sigma }i_{G}(s)}$.^[9]

Combining this with the above one obtains: for a subextension ${\displaystyle F/K}$ corresponding to ${\displaystyle H}$,

${\displaystyle v_{F}({\mathfrak {D}}_{F/K})={1 \over e_{L/F}}\sum _{s\not \in H}i_{G}(s).}$

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

The ramification groups for a cyclotomic extension ${\displaystyle K_{n}:=\mathbf {Q} _{p}(\zeta )/\mathbf {Q} _{p}}$, where ${\displaystyle \zeta }$ is a ${\displaystyle p^{n}}$-th primitive root of unity, can be described explicitly:^[11]

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

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

Let K be the extension of Q₂ generated by ${\displaystyle x_{1}={\sqrt {2+{\sqrt {2}}}}}$. The conjugates of ${\displaystyle x_{1}}$ are ${\displaystyle x_{2}={\sqrt {2-{\sqrt {2}}}}}$, ${\displaystyle x_{3}=-x_{1}}$, ${\displaystyle x_{4}=-x_{2}}$.

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. ${\displaystyle {\sqrt {2}}}$ generates π²; (2)=π⁴.

Now ${\displaystyle x_{1}-x_{3}=2x_{1}}$, which is in π⁵.

and ${\displaystyle x_{1}-x_{2}={\sqrt {4-2{\sqrt {2}}}},}$ which is in π³.

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

${\displaystyle G_{0}=G_{1}=G_{2}=C_{4}.}$

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

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

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

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

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

where, by convention, ${\displaystyle (G_{0}:G_{t})}$ is equal to ${\displaystyle (G_{-1}:G_{0})^{-1}}$ if ${\displaystyle t=-1}$ and is equal to ${\displaystyle 1}$ for ${\displaystyle -1<t\leq 0}$.^[13] Then ${\displaystyle \phi (u)=u}$ for ${\displaystyle -1\leq u\leq 0}$. It is immediate that ${\displaystyle \phi }$ is continuous and strictly increasing, and thus has the continuous inverse function ${\displaystyle \psi }$ defined on ${\displaystyle [-1,\infty )}$. Define ${\displaystyle G^{v}=G_{\psi (v)}}$. ${\displaystyle G^{v}}$ is then called the v-th ramification group in upper numbering. In other words, ${\displaystyle G^{\phi (u)}=G_{u}}$. Note ${\displaystyle G^{-1}=G,G^{0}=G_{0}}$. The upper numbering is defined so as to be compatible with passage to quotients:^[14] if ${\displaystyle H}$ is normal in ${\displaystyle G}$, then

${\displaystyle (G/H)^{v}=G^{v}H/H}$ for all ${\displaystyle v}$

(whereas lower numbering is compatible with passage to subgroups.)

Herbrand's theorem states that the ramification groups in the lower numbering satisfy ${\displaystyle G_{u}H/H=(G/H)_{v}}$ (for ${\displaystyle v=\phi _{L/F}(u)}$ where ${\displaystyle L/F}$ is the subextension corresponding to ${\displaystyle H}$), and that the ramification groups in the upper numbering satisfy ${\displaystyle G^{u}H/H=(G/H)^{u}}$.^[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 ${\displaystyle G}$ is abelian, then the jumps in the filtration ${\displaystyle G^{v}}$ are integers; i.e., ${\displaystyle G_{i}=G_{i+1}}$ whenever ${\displaystyle \phi (i)}$ 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 ${\displaystyle G^{n}(L/K)}$ under the isomorphism

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

is just^[18]

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

• Finite extensions of local fields

• 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.