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

Main article: Ramification group

The theorem deals with the upper numbered higher ramification groups of a finite abelian extension ${\displaystyle L/K}$. So assume ${\displaystyle L/K}$ is a finite Galois extension, and that ${\displaystyle v_{K}}$ 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 ${\displaystyle v_{L}}$ the associated normalised valuation ew of L and let ${\displaystyle \scriptstyle {\mathcal {O}}}$ be the valuation ring of L under ${\displaystyle v_{L}}$. Let ${\displaystyle L/K}$ have Galois group G and define the s-th ramification group of ${\displaystyle L/K}$ 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 ψ_L/K which in turn is the inverse of the function η_L/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 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.^[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 ${\displaystyle Q_{8}}$ of order 8 with

• ${\displaystyle G_{0}=Q_{8}}$
• ${\displaystyle G_{1}=Q_{8}}$
• ${\displaystyle G_{2}=\mathbb {Z} /2\mathbb {Z} }$
• ${\displaystyle G_{3}=\mathbb {Z} /2\mathbb {Z} }$
• ${\displaystyle G_{4}=1}$

The upper numbering then satisfies

• ${\displaystyle G^{n}=Q_{8}}$   for ${\displaystyle n\leq 1}$
• ${\displaystyle G^{n}=\mathbb {Z} /2\mathbb {Z} }$   for ${\displaystyle 1<n\leq 3/2}$
• ${\displaystyle G^{n}=1}$   for ${\displaystyle 3/2<n}$

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