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 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.^[4]^[5]

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}+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}}.

^[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 Q₈ of order 8 with

G₀ = Q₈
G₁ = Q₈
G₂ = Z/2Z
G₃ = Z/2Z
G₄ = 1

The upper numbering then satisfies

Gⁿ = Q₈ for n≤1
Gⁿ = Z/2Z for 1<n≤3/2
Gⁿ = 1 for 3/2<n

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

Notes

↑ H. Hasse, Führer, Diskriminante und Verzweigunsgskörper relativ Abelscher Zahlkörper, J. Reine Angew. Math. 162 (1930), pp.169–184.
↑ H. Hasse, Normenresttheorie galoisscher Zahlkörper mit Anwendungen auf Führer und Diskriminante abelscher Zahlkörper, J. Fac. Sci. Tokyo 2 (1934), pp.477–498.
↑ Arf, C. (1939). "Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper". J. Reine Angew. Math. (in German). 181: 1–44. Zbl 0021.20201.
1 2 3 Serre (1979) IV.3, p.76
↑ Neukirch (1999) Theorem 8.9, p.68

References

Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. Zbl 0956.11021. MR 1697859.
Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, 67, Translated from the French by Marvin Jay Greenberg, Springer-Verlag, ISBN 0-387-90424-7, MR 554237, Zbl 0423.12016

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