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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of the
upper numbering In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension. Ramificati ...
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 Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...
, and the general result was proved by
Cahit Arf Cahit Arf (; 24 October 1910 – 26 December 1997) was a Turkish mathematician. He is known for the Arf invariant of a quadratic form in characteristic 2 (applied in knot theory and surgery theory) in topology, the Hasse–Arf theorem ...
.


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 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)=\. 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_(s)=\int_0^s \frac. 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 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 are all rational integers.Neukirch (1999) Theorem 8.9, p.68


Example

Suppose ''G'' is cyclic of order p^n, p residue characteristic and G(i) be the subgroup of G of order p^. The theorem says that there exist positive integers i_0, i_1, ..., i_ such that :G_0 = \cdots = G_ = G = G^0 = \cdots = G^ :G_ = \cdots = G_ = G(1) = G^ = \cdots = G^ :G_ = \cdots = G_ = G(2) = G^ :... :G_ = 1 = G^.Serre (1979) IV.3, p.76


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 = Z/2Z *''G''3 = Z/2Z *''G''4 = 1 The upper numbering then satisfies *''G''''n'' = ''Q''8 for ''n''≤1 *''G''''n'' = Z/2Z for 1<''n''≤3/2 *''G''''n'' = 1 for 3/2<''n'' so has a jump at the non-integral value ''n''=3/2.


Notes


References

* * {{DEFAULTSORT:Hasse-Arf theorem Galois theory Theorems in algebraic number theory