Hasse–Arf theorem

In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of a filtration of the Galois group of a finite Galois extension. A special case of it was originally proved by Helmut Hasse, and the general result was proved by Cahit Arf.

Statement

Higher ramification groups

Main article: Ramification group

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 the theorem says that there exist strictly integers $i_{0},i_{1},...,i_{n-1}$ such that

$G_{0}=\cdots =G_{i_{0}}=G=G^{0}=\cdots =G^{i_{0}}$

$G_{i_{0}+1}=\cdots =G_{i_{0}+pi_{1}}=G(1)=G^{i_{0}+1}=\cdots =G^{i_{0}+i_{1}}$

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

...

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

References

• Jürgen Neukirch, Algebraic Number Theory, Springer (1999).
• Serre, Jean-Pierre (1980), Local Fields, Berlin, New York: Springer-Verlag, ISBN 9780387904245, MR0554237 {{citation}}: Check |author1-link= value (help)