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

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

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

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

$G_{i_0 + p i_1 + 1} = \cdots = G_{i_0 + p i_1 + p^2 i_2} = G(2) = G^{i_0 + i_1 + 1}$

...

$G_{i_0 + p i_1 + \cdots + p^{n-1}i_{n-1} + 1} = 1 = G^{i_0 + \cdots + i_{n-1} + 1}.$^[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

1. H. Hasse, Führer, Diskriminante und Verzweigunsgskörper relativ Abelscher Zahlkörper, J. Reine Angew. Math. 162 (1930), pp.169–184.
2. H. Hasse, Normenresttheorie galoisscher Zahlkörper mit Anwendungen auf Führer und Diskriminante abelscher Zahlkörper, J. Fac. Sci. Tokyo 2 (1934), pp.477–498.
3. Arf, C. (1939). "Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper". J. Reine Angew. Math. (in German) 181: 1–44. Zbl 0021.20201.
4. Serre (1979) IV.3, p.76
5. 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