Hasse–Arf theorem

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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

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Higher ramification groups

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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

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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

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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

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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

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  1. ^ 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.
  2. ^ H. Hasse, Normenresttheorie galoisscher Zahlkörper mit Anwendungen auf Führer und Diskriminante abelscher Zahlkörper, J. Fac. Sci. Tokyo 2 (1934), pp.477–498.
  3. ^ 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.
  4. ^ a b c Serre (1979) IV.3, p.76
  5. ^ Neukirch (1999) Theorem 8.9, p.68

References

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