Hasse–Arf theorem#Higher ramification groups

{{main|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 $v\_K$ 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 $\backslash scriptstyle\{\backslash mathcal\{O\}\}$ 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

:$G\_s(L/K)=\backslash \{\backslash sigma\backslash in\; G\backslash ,:\backslash ,v\_L(\backslash sigma\; a-a)\backslash geq\; s+1\; \backslash text\{\; for\; all\; \}a\backslash in\backslash mathcal\{O\}\backslash \}.$

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

:$\backslash eta\_\{L/K\}(s)=\backslash int\_0^s\; \backslash frac\{dx\}$

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 {G^t(L/K) : t ≥ −1} 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.

With the above set up of an abelian extension L/K, the theorem states that the jumps of the filtration {G^t(L/K) : t ≥ −1} are all rational integers.Neukirch (1999) Theorem 8.9, p.68

Suppose G is cyclic of order $p^n$, $p$ residue characteristic and $G(i)$ be the subgroup of $G$ of order $p^\{n-i\}$. The theorem says that there exist positive integers $i\_0,\; i\_1,\; ...,\; i\_\{n-1\}$ such that

:$G\_0\; =\; \backslash cdots\; =\; G\_\{i\_0\}\; =\; G\; =\; G^0\; =\; \backslash cdots\; =\; G^\{i\_0\}$

:$G\_\{i\_0\; +\; 1\}\; =\; \backslash cdots\; =\; G\_\{i\_0\; +\; p\; i\_1\}\; =\; G(1)\; =\; G^\{i\_0\; +\; 1\}\; =\; \backslash cdots\; =\; G^\{i\_0\; +\; i\_1\}$

:$G\_\{i\_0\; +\; p\; i\_1\; +\; 1\}\; =\; \backslash cdots\; =\; G\_\{i\_0\; +\; p\; i\_1\; +\; p^2\; i\_2\}\; =\; G(2)\; =\; G^\{i\_0\; +\; i\_1\; +\; 1\}$

:...

:$G\_\{i\_0\; +\; p\; i\_1\; +\; \backslash cdots\; +\; p^\{n-1\}i\_\{n-1\}\; +\; 1\}\; =\; 1\; =\; G^\{i\_0\; +\; \backslash cdots\; +\; i\_\{n-1\}\; +\; 1\}.$Serre (1979) IV.3, p.76

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

• $G\_0\; =\; Q\_8$
• $G\_1\; =\; Q\_8$
• $G\_2\; =\; \backslash Z/2\backslash Z$
• $G\_3\; =\; \backslash Z/2\backslash Z$
• $G\_4\; =\; 1$

The upper numbering then satisfies

• $G^n\; =\; Q\_8$   for $n\; \backslash leq\; 1$
• $G^n\; =\; \backslash Z/2\backslash Z$   for
• $G^n\; =\; 1$   for

so has a jump at the non-integral value $n=3/2$.

{{Reflist}}

• {{Neukirch ANT}}
• {{citation | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local Fields |translator-last=Greenberg|translator-first= Marvin Jay|translator-link1= Marvin Greenberg | series=Graduate Texts in Mathematics | volume=67 | publisher=Springer-Verlag | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 | mr=554237}}

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