Misplaced Pages

Hasse–Arf theorem

Article snapshot taken from[REDACTED] with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
On jumps of upper numbering filtration of the Galois group of a finite Galois extension

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, 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 {\displaystyle L/K} . So assume L / K {\displaystyle L/K} is a finite Galois extension, and that v K {\displaystyle 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 {\displaystyle v_{L}} the associated normalised valuation ew of L and let O {\displaystyle \scriptstyle {\mathcal {O}}} be the valuation ring of L under v L {\displaystyle v_{L}} . Let L / K {\displaystyle L/K} have Galois group G and define the s-th ramification group of L / K {\displaystyle L/K} for any real s ≥ −1 by

G s ( L / K ) = { σ ∈ G : v L ( σ a − a ) ≥ s + 1  for all  a ∈ O } . {\displaystyle 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

η L / K ( s ) = ∫ 0 s d x | G 0 : G x | . {\displaystyle \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(L/K) = Gs(L/K) where s = ψL/K(t).

These higher ramification groups G(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 {G(L/K) : t ≥ −1} if G(L/K) ≠ G(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(L/K) : t ≥ −1} are all rational integers.

Example

Suppose G is cyclic of order p n {\displaystyle p^{n}} , p {\displaystyle p} residue characteristic and G ( i ) {\displaystyle G(i)} be the subgroup of G {\displaystyle G} of order p n − i {\displaystyle p^{n-i}} . The theorem says that there exist positive integers i 0 , i 1 , . . . , i n − 1 {\displaystyle i_{0},i_{1},...,i_{n-1}} such that

G 0 = ⋯ = G i 0 = G = G 0 = ⋯ = G i 0 {\displaystyle G_{0}=\cdots =G_{i_{0}}=G=G^{0}=\cdots =G^{i_{0}}}
G i 0 + 1 = ⋯ = G i 0 + p i 1 = G ( 1 ) = G i 0 + 1 = ⋯ = G i 0 + i 1 {\displaystyle 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 + p i 1 + 1 = ⋯ = G i 0 + p i 1 + p 2 i 2 = G ( 2 ) = G i 0 + i 1 + 1 {\displaystyle 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 + p i 1 + ⋯ + p n − 1 i n − 1 + 1 = 1 = G i 0 + ⋯ + i n − 1 + 1 . {\displaystyle G_{i_{0}+pi_{1}+\cdots +p^{n-1}i_{n-1}+1}=1=G^{i_{0}+\cdots +i_{n-1}+1}.}

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 {\displaystyle Q_{8}} of order 8 with

  • G 0 = Q 8 {\displaystyle G_{0}=Q_{8}}
  • G 1 = Q 8 {\displaystyle G_{1}=Q_{8}}
  • G 2 = Z / 2 Z {\displaystyle G_{2}=\mathbb {Z} /2\mathbb {Z} }
  • G 3 = Z / 2 Z {\displaystyle G_{3}=\mathbb {Z} /2\mathbb {Z} }
  • G 4 = 1 {\displaystyle G_{4}=1}

The upper numbering then satisfies

  • G n = Q 8 {\displaystyle G^{n}=Q_{8}}   for n ≤ 1 {\displaystyle n\leq 1}
  • G n = Z / 2 Z {\displaystyle G^{n}=\mathbb {Z} /2\mathbb {Z} }   for 1 < n ≤ 3 / 2 {\displaystyle 1<n\leq 3/2}
  • G n = 1 {\displaystyle G^{n}=1}   for 3 / 2 < n {\displaystyle 3/2<n}

so has a jump at the non-integral value n = 3 / 2 {\displaystyle n=3/2} .

Notes

  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.
  4. ^ Serre (1979) IV.3, p.76
  5. Neukirch (1999) Theorem 8.9, p.68

References

Categories:
Hasse–Arf theorem Add topic