Hasse–Arf theorem

Template:Short description 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

Template:Main 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 $𝒪$ 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

$G_s(L/K)=\{\sigma \in G:v_L(\sigma a-a)\ge s+1\text{ for all }a\in 𝒪\}.$

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

${\eta }_{L/K}(s)=\underset{0}{\overset{s}{\int }}\frac{dx}{|G_0:G_x|}.$

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

$G_0=\cdots =G_{i_0}=G=G^0=\cdots =G^{i_0}$

$G_{i_0+1}=\cdots =G_{i_0+p{i}_1}=G(1)=G^{i_0+1}=\cdots =G^{i_0+i_1}$

$G_{i_0+p{i}_1+1}=\cdots =G_{i_0+p{i}_1+p^2{i}_2}=G(2)=G^{i_0+i_1+1}$

...

$G_{i_0+p{i}_1+\cdots +p^{n-1}{i}_{n-1}+1}=1=G^{i_0+\cdots +i_{n-1}+1}.$

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

• $G_0=Q_8$
• $G_1=Q_8$
• $G_2=\mathbb{Z}/2\mathbb{Z}$
• $G_3=\mathbb{Z}/2\mathbb{Z}$
• $G_4=1$

The upper numbering then satisfies

• $G^n=Q_8$   for $n\le 1$
• $G^n=\mathbb{Z}/2\mathbb{Z}$   for $1<n\le 3/2$
• $G^n=1$   for $3/2<n$

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

References

• Template:Neukirch ANT
• Template:Citation