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.[1] (This is stronger than the primitive element theorem.) Then, for each integer i \ge -1, we define G_i to be the set of all s \in G that satisfies the following equivalent conditions.

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,

The study of ramification groups reduces to the totally ramified case since one has G_i = (G_0)_i for i \ge 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,

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 \ge 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]

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 \ne 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 \ge 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 \ge 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 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}) = 3+3+3+1+1 = 11. so that the different \mathfrak{D}_{K/Q}=π11.

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

Ramification groups in upper numbering

If u is a real number \ge -1, let G_u denote G_i where i the least integer \ge u. In other words, s \in G_u \Leftrightarrow i_G(s) \ge 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 \le 0.[10] Then \phi(u) = u for -1 \le u \le 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^n_K / (U^n_K \cap N_{L/K}(L^*)) \ .

Notes

  1. Neukirch (1999) p.178
  2. since G/G_0 is canonically isomorphic to the decomposition group.
  3. Serre (1979) p.62
  4. Conrad
  5. Use U_{L, 0}/U_{L, 1} \simeq l^\times and U_{L, i}/U_{L, i+1} \approx l^+
  6. Serre (1979) 4.1 Prop.4, p.64
  7. Serre (1979) 4.1. Prop.3, p.63
  8. Serre (1979) 4.2. Proposition 10.
  9. Serre (1967) p.156
  10. Neukirch (1999) p.179
  11. Serre (1967) p.155
  12. Neukirch (1999) p.180
  13. Serre (1979) p.75
  14. Neukirch (1999) p.355
  15. Snaith (1994) pp.30-31

See also

References

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