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}+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 $Q_{8}$ of order 8 with

$G_{0}=Q_{8}$
$G_{1}=Q_{8}$
$G_{2}=\mathbb {Z} /2\mathbb {Z}$
$G_{3}=\mathbb {Z} /2\mathbb {Z}$
$G_{4}=1$

The upper numbering then satisfies

$G^{n}=Q_{8}$ for $n\leq 1$
$G^{n}=\mathbb {Z} /2\mathbb {Z}$ for $1<n\leq 3/2$
$G^{n}=1$ for $3/2<n$

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

Notes

^ Hasse, Helmut (1930). "Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper". J. Reine Angew. Math. (in German). 162: 169–184. doi:10.1515/crll.1930.162.169. MR 1581221.
^ H. Hasse, Normenresttheorie galoisscher Zahlkörper mit Anwendungen auf Führer und Diskriminante abelscher Zahlkörper, J. Fac. Sci. Tokyo 2 (1934), pp.477–498.
^ Arf, Cahit (1939). "Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper". J. Reine Angew. Math. (in German). 181: 1–44. doi:10.1515/crll.1940.181.1. MR 0000018. Zbl 0021.20201.
^ ^a ^b ^c Serre (1979) IV.3, p.76
^ Neukirch (1999) Theorem 8.9, p.68

References

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 by Greenberg, Marvin Jay, Springer-Verlag, ISBN 0-387-90424-7, MR 0554237, Zbl 0423.12016

[1] Hasse, Helmut (1930). "Führer, Diskriminante und Verzweigungskörper relativ-Abelscher Zahlkörper". J. Reine Angew. Math. (in German). 162: 169–184. doi:10.1515/crll.1930.162.169. MR 1581221.

[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, Cahit (1939). "Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper". J. Reine Angew. Math. (in German). 181: 1–44. doi:10.1515/crll.1940.181.1. MR 0000018. Zbl 0021.20201.

[S76-4] Serre (1979) IV.3, p.76

[Neu68-5] Neukirch (1999) Theorem 8.9, p.68

[1]

[2]

[3]

[4]

[5]

Fahrer / Team für die Suche eingeben: