Inmathematics, specifically inlocal class field theory, theHasse–Arf theorem is a result concerning jumps of theupper numbering filtration of theGalois group of a finiteGalois extension. A special case of it when the residue fields are finite was originally proved byHelmut Hasse,^[1][2] and the general result was proved byCahit Arf.^[3][4]

The theorem deals with the upper numbered higher ramification groups of a finiteabelian extension${\displaystyle L/K}$. So assume${\displaystyle L/K}$ is a finite Galois extension, and that${\displaystyle v_K}$ is adiscrete normalised valuation ofK, whoseresidue field has characteristicp > 0, and which admits a unique extension toL, sayw. Denote by${\displaystyle v_L}$ the associated normalised valuationew ofL and let${\displaystyle \mathcal{O}}$ be thevaluation ring ofL under${\displaystyle v_L}$. Let${\displaystyle L/K}$ haveGalois groupG and define thes-th ramification group of${\displaystyle L/K}$ for any reals ≥ −1 by

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

So, for example,G₋₁ is the Galois groupG. To pass to the upper numbering one has to define the functionψ_L/K which in turn is the inverse of the functionη_L/K defined by

${\displaystyle {\eta }_{L/K}(s)={\int }_0^s\frac{dx}{|G_0:G_x|}.}$

The upper numbering of theramification groups is then defined byG^t(L/K) = G_s(L/K) wheres = ψ_L/K(t).

These higher ramification groupsG^t(L/K) are defined for any realt ≥ −1, but sincev_L is a discrete valuation, the groups will change in discrete jumps and not continuously. Thus we say thatt is a jump of the filtration {G^t(L/K) : t ≥ −1} ifG^t(L/K) ≠ G^u(L/K) for anyu > t. The Hasse–Arf theorem tells us the arithmetic nature of these jumps.

With the above set up of an abelian extensionL/K, the theorem states that the jumps of the filtration {G^t(L/K) : t ≥ −1} are allrational integers.^[4][5]

SupposeG 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

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

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

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

...

${\displaystyle G_{i_0+p{i}_1+\cdots +p^{n-1}{i}_{n-1}+1}=1=G^{i_0+\cdots +i_{n-1}+1}.}$^[4]

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 thequaternion 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${\displaystyle n\le 1}$
• ${\displaystyle G^n=\mathbb{Z}/2\mathbb{Z}}$   for
• ${\displaystyle G^n=1}$   for

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

• Neukirch, Jürgen (1999).Algebraische Zahlentheorie.Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin:Springer-Verlag.ISBN 978-3-540-65399-8.MR 1697859.Zbl 0956.11021.
• Serre, Jean-Pierre (1979),Local Fields, Graduate Texts in Mathematics, vol. 67, translated byGreenberg, Marvin Jay,Springer-Verlag,ISBN 0-387-90424-7,MR 0554237,Zbl 0423.12016

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