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The ramification is tame when the ramification indices are all relatively prime to the residue characteristic p of , otherwise wild. This condition is important in Galois module theory. A finite generically étale extension B / A {\displaystyle B/A} of Dedekind domains is tame if and only if the trace Tr : B → A {\displaystyle \operatorname ...
Let S v be the set of equivalence classes of extensions of v to L and let G be the Galois group of L over K. Then G acts on S v by σ[w] = [w ∘ σ] (i.e. w is a representative of the equivalence class [w] ∈ S v and [w] is sent to the equivalence class of the composition of w with the automorphism σ : L → L; this is independent of the ...
More precisely, the conductor computes the non-triviality of higher ramification groups: if s is the largest integer for which the "lower numbering" higher ramification group G s is non-trivial, then (/) = / +, where η L/K is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups.
The tame ramification part ε is defined in terms of the reduction type: ε=0 for good reduction, ε=1 for multiplicative reduction and ε=2 for additive reduction. The wild ramification term δ is zero unless p divides 2 or 3, and in the latter cases it is defined in terms of the wild ramification of the extensions of K by the division points ...
The following procedure (Neukirch, p. 47) solves this problem in many cases. The strategy is to select an integer θ in O L so that L is generated over K by θ (such a θ is guaranteed to exist by the primitive element theorem), and then to examine the minimal polynomial H(X) of θ over K; it is a monic polynomial with coefficients in O K.
with the summation taken over four ramification points. The formula may also be used to calculate the genus of hyperelliptic curves. As another example, the Riemann sphere maps to itself by the function z n, which has ramification index n at 0, for any integer n > 1. There can only be other ramification at the point at infinity.
In mathematics, the Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field of K then A has good reduction if and only if the ℓ-adic Tate module T ℓ of A is unramified.
The conjugacy classes , …, are rational if for any element and any integer relatively prime to the order of , the element belongs to . Assume G {\displaystyle G} is a centerless group, and fix a rigid list of rational conjugacy classes c = ( c 1 , … , c n ) {\displaystyle \mathbf {c} =(c_{1},\ldots ,c_{n})} .