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 of a finite Galois extension of local fields. We shall write for the valuation, the ring of integers and its maximal ideal for . As a consequence of Hensel's lemma, one can write for some where is the ring of integers of .[1] (This is stronger than the primitive element theorem.) Then, for each integer , we define to be the set of all that satisfies the following equivalent conditions.

  • (i) operates trivially on
  • (ii) for all
  • (iii)

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

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

The Galois group and its subgroups are studied by employing the above filtration or, more specifically, the corresponding quotients. In particular,

  • where are the (finite) residue fields of .[2]
  • is unramified.
  • 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 for .

One also defines the function . (ii) in the above shows is independent of choice of and, moreover, the study of the filtration is essentially equivalent to that of .[3] satisfies the following: for ,

Fix a uniformizer of . Then induces the injection where . (The map actually does not depend on the choice of the uniformizer.[4]) It follows from this[5]

  • is cyclic of order prime to
  • is a product of cyclic groups of order .

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

The ramification groups can be used to compute the different of the extension and that of subextensions:[6]

If is a normal subgroup of , then, for , .[7]

Combining this with the above one obtains: for a subextension corresponding to ,

If , then .[8] In the terminology of Lazard, this can be understood to mean the Lie algebra is abelian.

Example: the cyclotomic extension

The ramification groups for a cyclotomic extension , where is a -th primitive root of unity, can be described explicitly:[9]

where e is chosen such that .

Example: a quartic extension

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

Now x1x3 = 2x1, which is in π5.

and which is in π3.

Various methods show that the Galois group of K is , cyclic of order 4. Also:

and

so that the different

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

Ramification groups in upper numbering

If is a real number , let denote where i the least integer . In other words, Define by[10]

where, by convention, is equal to if and is equal to for .[11] Then for . It is immediate that is continuous and strictly increasing, and thus has the continuous inverse function defined on . Define . is then called the v-th ramification group in upper numbering. In other words, . Note . The upper numbering is defined so as to be compatible with passage to quotients:[12] if is normal in , then

for all

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

Herbrand's theorem states that the ramification groups in the lower numbering satisfy (for where is the subextension corresponding to ), and that the ramification groups in the upper numbering satisfy .[13][14] 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 is abelian, then the jumps in the filtration are integers; i.e., whenever is not an integer.[15]

The upper numbering is compatible with the filtration of the norm residue group by the unit groups under the Artin isomorphism. The image of under the isomorphism

is just[16]

See also

Notes

  1. Neukirch (1999) p.178
  2. since is canonically isomorphic to the decomposition group.
  3. Serre (1979) p.62
  4. Conrad
  5. Use and
  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, Corps locaux. Ch. IV, §4, Proposition 18
  10. Serre (1967) p.156
  11. Neukirch (1999) p.179
  12. Serre (1967) p.155
  13. Neukirch (1999) p.180
  14. Serre (1979) p.75
  15. Neukirch (1999) p.355
  16. Snaith (1994) pp.30-31

References

  • B. Conrad, Math 248A. Higher ramification groups
  • Fröhlich, A.; Taylor, M.J. (1991). Algebraic number theory. Cambridge studies in advanced mathematics. 27. Cambridge University Press. ISBN 0-521-36664-X. Zbl 0744.11001.
  • Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 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. 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.
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