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] The theorem deals with the upper numbered higher ramification groups of a finite abelian extension
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
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.
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] Suppose G is cyclic of order
The theorem says that there exist positive integers
such that 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 so has a jump at the non-integral value