Hasse–Arf theorem: Difference between revisions
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Latest revision as of 10:29, 26 April 2024
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
[edit]Higher ramification groups
[edit]The theorem deals with the upper numbered higher ramification groups of a finite abelian extension . So assume is a finite Galois extension, and that 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 the associated normalised valuation ew of L and let be the valuation ring of L under . Let have Galois group G and define the s-th ramification group of 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
[edit]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
[edit]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
[edit]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 of order 8 with
The upper numbering then satisfies
- for
- for
- for
so has a jump at the non-integral value .
Notes
[edit]- ^ Hasse, Helmut (1930). "Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper". J. Reine Angew. Math. (in German). 162: 169–184. doi:10.1515/crll.1930.162.169. MR 1581221.
- ^ H. Hasse, Normenresttheorie galoisscher Zahlkörper mit Anwendungen auf Führer und Diskriminante abelscher Zahlkörper, J. Fac. Sci. Tokyo 2 (1934), pp.477–498.
- ^ Arf, Cahit (1939). "Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper". J. Reine Angew. Math. (in German). 181: 1–44. doi:10.1515/crll.1940.181.1. MR 0000018. Zbl 0021.20201.
- ^ a b c Serre (1979) IV.3, p.76
- ^ Neukirch (1999) Theorem 8.9, p.68
References
[edit]- Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 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, vol. 67, translated by Greenberg, Marvin Jay, Springer-Verlag, ISBN 0-387-90424-7, MR 0554237, Zbl 0423.12016