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]
L/K
L/K
vK
vL
\scriptstyle{l{O}}
vL
L/K
L/K
Gs(L/K)=\{\sigma\inG:vL(\sigmaa-a)\geqs+1foralla\inl{O}\}.
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
ηL/K
s | |
(s)=\int | |
0 |
dx | |
|G0:Gx| |
.
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 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 are all rational integers.[4]
Suppose G is cyclic of order
pn
p
G(i)
G
pn-i
i0,i1,...,in-1
G0= … =
G | |
i0 |
=G=G0= … =
i0 | |
G |
G | |
i0+1 |
= … =
G | |
i0+pi1 |
=G(1)=
i0+1 | |
G |
= … =
i0+i1 | |
G |
G | |
i0+pi1+1 |
= … =
G | |
i0+pi1+p2i2 |
=G(2)=
i0+i1+1 | |
G |
...
G | |
i0+pi1+ … +pn-1in-1+1 |
=1=
i0+ … +in-1+1 | |
G |
.
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
G0=Q8
G1=Q8
G2=\Z/2\Z
G3=\Z/2\Z
G4=1
Gn=Q8
n\leq1
Gn=\Z/2\Z
1<n\leq3/2
Gn=1
3/2<n
n=3/2