# 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 ${\displaystyle \scriptstyle {\mathcal {O}}}$ 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

${\displaystyle G_{s}(L/K)=\{\sigma \in G\,:\,v_{L}(\sigma a-a)\geq s+1{\text{ for all }}a\in {\mathcal {O}}\}.}$

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

${\displaystyle \eta _{L/K}(s)=\int _{0}^{s}{\frac {dx}{|G_{0}:G_{x}|}}.}$

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 ${\displaystyle p^{n}}$, ${\displaystyle p}$ residue characteristic and ${\displaystyle G(i)}$ be the subgroup of ${\displaystyle G}$ of order ${\displaystyle p^{n-i}}$. The theorem says that there exist positive integers ${\displaystyle i_{0},i_{1},...,i_{n-1}}$ such that

${\displaystyle G_{0}=\cdots =G_{i_{0}}=G=G^{0}=\cdots =G^{i_{0}}}$
${\displaystyle G_{i_{0}+1}=\cdots =G_{i_{0}+pi_{1}}=G(1)=G^{i_{0}+1}=\cdots =G^{i_{0}+i_{1}}}$
${\displaystyle G_{i_{0}+pi_{1}+1}=\cdots =G_{i_{0}+pi_{1}+p^{2}i_{2}}=G(2)=G^{i_{0}+i_{1}+1}}$
...
${\displaystyle G_{i_{0}+pi_{1}+\cdots +p^{n-1}i_{n-1}+1}=1=G^{i_{0}+\cdots +i_{n-1}+1}.}$[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

1. ^ H. Hasse, Führer, Diskriminante und Verzweigunsgskörper relativ Abelscher Zahlkörper, J. Reine Angew. Math. 162 (1930), pp.169–184.
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, C. (1939). "Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper". J. Reine Angew. Math. (in German). 181: 1–44. Zbl 0021.20201.
4. ^ a b c Serre (1979) IV.3, p.76
5. ^ Neukirch (1999) Theorem 8.9, p.68