Hasse–Arf theorem
# 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, and the general result was proved by Cahit Arf.

## 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 $v_K$ 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 $v_L$ the associated normalised valuation ew of L and let $\scriptstyle {\mathcal {O}}$ be the valuation ring of L under $v_L.$ Let L/K have Galois group G and define the s-th ramification group of L/K for any real s ≥ −1 by

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

So, for example, $G_{−1}$ is the Galois group G. To pass to the upper numbering one has to define the function $ψ_{L/K}$ which in turn is the inverse of the function $η_{L/K}$ defined by

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

The upper numbering of the ramification groups is then defined by $G^t(L/K)$ = $G_s(L/K)$ where s = $ψ_{L/K}(t).$

These higher ramification groups $G^t(L/K)$ are defined for any real t ≥ −1, but since $v_L$ 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 ${G^t(L/K)$ : t ≥ −1} if $G^t(L/K)$ ≠ $G^u(L/K)$ 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 ${G^t(L/K)$ : t ≥ −1} are all rational integers.

## Example

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

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

## 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 $Q_8$ of order 8 with

* $G_0$ = $Q_8$
* $G_1$ = $Q_8$
* $G_2$ = Z/2Z
* $G_3$ = Z/2Z
* $G_4$ = 1

The upper numbering then satisfies

* $G^n$ = $Q_8$ for n≤1
* $G^n$ = Z/2Z for 1<n≤3/2
* $G^n$ = 1 for 3/2<n

so has a jump at the non-integral value n=3/2.

