Hasse–Arf theorem

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] Contents 1 Statement 1.1 Higher ramification groups 1.2 Statement of the theorem 2 Example 3 Non-abelian extensions 4 Notes 5 References 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 {displaystyle 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 {displaystyle G_{s}(L/K)={sigma in G,:,v_{L}(sigma a-a)geq s+1{text{ for all }}ain {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 {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 {displaystyle p^{n}} , {displaystyle p} residue characteristic and {displaystyle G(i)} be the subgroup of {displaystyle G} of order {displaystyle p^{n-i}} . The theorem says that there exist positive integers {displaystyle i_{0},i_{1},...,i_{n-1}} such that {displaystyle G_{0}=cdots =G_{i_{0}}=G=G^{0}=cdots =G^{i_{0}}} {displaystyle G_{i_{0}+1}=cdots =G_{i_{0}+pi_{1}}=G(1)=G^{i_{0}+1}=cdots =G^{i_{0}+i_{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}} ... {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 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

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