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 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 vK 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 vL the associated normalised valuation ew of L and let \scriptstyle{\mathcal{O}} be the valuation ring of L under vL. 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 ψL/K which in turn is the inverse of the function ηL/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 Gt(L/K) = Gs(L/K) where s = ψL/K(t).

These higher ramification groups Gt(L/K) are defined for any real t ≥ −1, but since vL 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 {Gt(L/K) : t ≥ −1} if Gt(L/K) ≠ Gu(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 {Gt(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]

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. ↑ 4.0 4.1 4.2 Serre (1979) IV.3, p.76
  5. ↑ Neukirch (1999) Theorem 8.9, p.68

References