Hasse–Arf theorem
In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite was originally proved by Helmut Hasse,[1][2] and the general result was proved by Cahit Arf.[3][4]
Statement
Higher ramification groups
The theorem deals with the upper numbered higher ramification groups of a finite abelian extension L/K. So assume L/K is a finite Galois extension, and that vK is a discrete normalised valuation of K, whose residue field has characteristic p > 0, and which admits a unique extension to L, say w. Denote by vL the associated normalised valuation ew of L and let be the valuation ring of L under vL. Let L/K have Galois group G and define the s-th ramification group of L/K for any real s ≥ −1 by
So, for example, G−1 is the Galois group G. To pass to the upper numbering one has to define the function ψL/K which in turn is the inverse of the function ηL/K defined by
The upper numbering of the ramification groups is then defined by Gt(L/K) = Gs(L/K) where s = ψL/K(t).
These higher ramification groups Gt(L/K) are defined for any real t ≥ −1, but since vL is a discrete valuation, the groups will change in discrete jumps and not continuously. Thus we say that t is a jump of the filtration {Gt(L/K) : t ≥ −1} if Gt(L/K) ≠ Gu(L/K) for any u > t. The Hasse–Arf theorem tells us the arithmetic nature of these jumps.
Statement of the theorem
With the above set up, the theorem states that the jumps of the filtration {Gt(L/K) : t ≥ −1} are all rational integers.[4][5]
Example
Suppose G is cyclic of order , residue characteristic and be the subgroup of of order . The theorem says that there exist positive integers such that
- ...
- [4]
Non-abelian extensions
For non-abelian extensions the jumps in the upper filtration need not be at integers. Serre gave an example of a totally ramified extension with Galois group the quaternion group Q8 of order 8 with
- G0 = Q8
- G1 = Q8
- G2 = Z/2Z
- G3 = Z/2Z
- G4 = 1
The upper numbering then satisfies
- Gn = Q8 for n≤1
- Gn = Z/2Z for 1<n≤3/2
- Gn = 1 for 3/2<n
so has a jump at the non-integral value n=3/2.
Notes
- H. Hasse, Führer, Diskriminante und Verzweigunsgskörper relativ Abelscher Zahlkörper, J. Reine Angew. Math. 162 (1930), pp.169–184.
- H. Hasse, Normenresttheorie galoisscher Zahlkörper mit Anwendungen auf Führer und Diskriminante abelscher Zahlkörper, J. Fac. Sci. Tokyo 2 (1934), pp.477–498.
- Arf, C. (1939). "Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper". J. Reine Angew. Math. (in German). 181: 1–44. Zbl 0021.20201.
- Serre (1979) IV.3, p.76
- Neukirch (1999) Theorem 8.9, p.68
References
- 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 (1979), Local Fields, Graduate Texts in Mathematics, 67, translated by Greenberg, Marvin Jay, Springer-Verlag, ISBN 0-387-90424-7, MR 0554237, Zbl 0423.12016