Ramification group

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 , O L , p for the valuation, the ring of integers and its maximal ideal for L . As a consequence of Hensel's lemma, one can write O L = O K [ α ] for some α ∈ L where O K is the ring of integers of K . (This is stronger than the primitive element theorem.) Then, for each integer i ≥ − 1 , we define G i to be the set of all s ∈ G that satisfies the following equivalent conditions.

The group G i is called i -th ramification group. They form a decreasing filtration,

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,

The study of ramification groups reduces to the totally ramified case since one has G i = ( G 0 ) i for i ≥ 0 .

One also defines the function i G ( s ) = w ( s ( α ) − α ) , s ∈ G . (ii) in the above shows i G is independent of choice of α 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 ∈ G ,

Fix a uniformizer π of L . Then s ↦ s ( π ) / π induces the injection G i / G i + 1 → U L , i / U L , i + 1 , i ≥ 0 where U L , 0 = O L × , U L , i = 1 + p i . (The map actually does not depend on the choice of the uniformizer.) It follows from this

In particular, G 1 is a p-group and G is solvable.

The ramification groups can be used to compute the different D L / K of the extension L / K and that of subextensions:

If H is a normal subgroup of G , then, for σ ∈ G , i G / H ( σ ) = 1 e L / K ∑ s ↦ σ i G ( s ) .

Combining this with the above one obtains: for a subextension F / K corresponding to H ,

If s ∈ G i , t ∈ G j , i , j ≥ 1 , then s t s − 1 t − 1 ∈ G i + j + 1 . In the terminology of Lazard, this can be understood to mean the Lie algebra gr ⁡ ( G 1 ) = ∑ i ≥ 1 G i / G i + 1 is abelian.

Example: the cyclotomic extension

The ramification groups for a cyclotomic extension K n := Q p ( ζ ) / Q p , where ζ is a p n -th primitive root of unity, can be described explicitly:

where e is chosen such that p e − 1 ≤ s < p e .

Example: a quartic extension

Let K be the extension of Q₂ generated by x₁= 2 + 2   . The conjugates of x₁ are x₂= 2 − 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. 2 generates π²; (2)=π⁴.

Now x₁-x₃=2x₁, which is in π⁵.

and x₁-x₂= 4 − 2 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 ( D K / Q 2 ) = 3+3+3+1+1 = 11. so that the different 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 ≥ − 1 , let G u denote G i where i the least integer ≥ u . In other words, s ∈ G u ⇔ i G ( s ) ≥ u + 1. Define ϕ by

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 ≤ 0 . Then ϕ ( u ) = u for − 1 ≤ u ≤ 0 . It is immediate that ϕ is continuous and strictly increasing, and thus has the continuous inverse function ψ defined on [ − 1 , ∞ ) . Define G v = G ψ ( v ) . G v is then called the v-th ramification group in upper numbering. In other words, G ϕ ( 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

(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 = ϕ 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 ϕ ( 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

is just