Hasse–Arf theorem

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,[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 {\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 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 {\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}.} [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 {\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. ^ a b c Serre (1979) IV.3, p.76
  5. ^ Neukirch (1999) Theorem 8.9, p.68

References