Hasse–Arf theorem
Describes arithmetic nature of jumps in ramification filtration.
In local class field theory, the Hasse–Arf theorem describes the jumps in the upper numbering filtration of a finite Galois extension's Galois group. Helmut Hasse proved the special case where the residue fields are finite, and Cahit Arf established the general result.
Quick Facts
- Field
- Mathematics, local class field theory
- Proved by
- Helmut Hasse (special case), Cahit Arf (general result)
- Subject
- Jumps of upper numbered higher ramification groups
- Extension type
- Finite abelian extension
Facts from the source article.
Statement
The theorem concerns the upper numbered higher ramification groups of a finite abelian extension L/K. Given a discrete normalised valuation v of K with residue field characteristic p > 0, which extends uniquely to L as w, let v_L be the associated normalised valuation of L and let O_L be its valuation ring. The Galois group G has s-th ramification groups G_s(L/K) for real s ≥ −1. The upper numbering is obtained via the function ψ_{L/K}, the inverse of η_{L/K}. The groups G^t(L/K) = G_s(L/K) with s = ψ_{L/K}(t) change in discrete jumps. A jump occurs at t if G^t(L/K) ≠ G^u(L/K) for any u > t. The Hasse–Arf theorem states that for an abelian extension, these jumps are all rational integers.
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