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[edit]

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.

  • (i) s operates trivially on \mathcal O_L / \mathfrak p^{i+1}.
  • (ii) w(s(x) - x) \ge i+1 for all x \in \mathcal O_L
  • (iii) w(s(\alpha) - \alpha) \ge 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.[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 \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,

  • i_G(s) \ge i + 1 \Leftrightarrow s \in G_i.
  • i_G(t s t^{-1}) = i_G(s).
  • i_G(st) \ge \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 \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]

  • 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 \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[edit]

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[edit]

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[edit]

  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[edit]

References[edit]