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.

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 ≥ &minus;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&minus;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


 * $$\eta_{L/K}(s)=\int_0^s \frac{dx}{|G_0:G_x|}.$$

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 ≥ &minus;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 ≥ &minus;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 ≥ &minus;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
 * $$G_0 = \cdots = G_{i_0} = G = G^0 = \cdots = G^{i_0}$$
 * $$G_{i_0 + 1} = \cdots = G_{i_0 + p i_1} = G(1) = G^{i_0 + 1} = \cdots = G^{i_0 + i_1}$$
 * $$G_{i_0 + p i_1 + 1} = \cdots = G_{i_0 + p i_1 + p^2 i_2} = G(2) = G^{i_0 + i_1 + 1}$$
 * $$G_{i_0 + p i_1 + \cdots + p^{n-1}i_{n-1} + 1} = 1 = G^{i_0 + \cdots + i_{n-1} + 1}.$$
 * $$G_{i_0 + p i_1 + \cdots + p^{n-1}i_{n-1} + 1} = 1 = G^{i_0 + \cdots + i_{n-1} + 1}.$$

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 The upper numbering then satisfies so has a jump at the non-integral value $$n=3/2$$.
 * $$G_0 = Q_8$$
 * $$G_1 = Q_8$$
 * $$G_2 = \Z/2\Z$$
 * $$G_3 = \Z/2\Z$$
 * $$G_4 = 1$$
 * $$G^n = Q_8 $$  for $$ n \leq 1 $$
 * $$G^n = \Z/2\Z $$  for $$ 1 < n\leq 3/2 $$
 * $$G^n = 1 $$  for $$ 3/2 < n $$