Algebraic and Geometric Topology: Proceedings of a by A. H. Assadi, P. Vogel (auth.), Andrew Ranicki, Norman

By A. H. Assadi, P. Vogel (auth.), Andrew Ranicki, Norman Levitt, Frank Quinn (eds.)

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Read or Download Algebraic and Geometric Topology: Proceedings of a Conference held at Rutgers University, New Brunswick, USA July 6–13, 1983 PDF

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Additional info for Algebraic and Geometric Topology: Proceedings of a Conference held at Rutgers University, New Brunswick, USA July 6–13, 1983

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For all n ≥ 1, we denote by Dn the degree of the different of the extension Fn /Fn−1 : Dn := deg Diff(Fn /Fn−1 ). 1. Let F = (F0 , F1 , F2 , . ) be a tower of function fields over Fq . Suppose that there exists a constant ∈ R with 0 ≤ < 1 such that the following inequality holds: Dn ≤ · [Fn : Fn−1 ] · Dn−1 , for all n ≥ 2. 1) Then the genus γ(F) of the tower F is finite and the estimate below holds: γ(F) ≤ g(F0 ) − 1 + D1 . 2(1 − ) · [F1 : F0 ] Proof. 1) and from the transitivity of the different that the following inequality   deg Diff(Fn /F0 ) ≤  n−1 j · D1 · [Fn : F1 ] j=0 holds, for all n ≥ 1.

We also define R0 := {x(P ) | P ∈ V0 }. 2) Clearly, this set R0 is a finite subset of Fq ∪ {∞}. 10. Let F = (F0 , F1 , F2 , . 1). 2). b) If β ∈ R and α ∈ Fq ∪ {∞} satisfy the equation ϕ(β) = ψ(α), then α ∈ R. Then the ramification locus of the tower F satisfies V (F) ⊆ {P | P is a place of F0 with x0 (P ) ∈ R}; in particular , V (F) is finite and moreover deg P ≤ #R. 3) P ∈V (F ) Proof. Let P ∈ V (F). There is some n ≥ 0 and a place Q of Fn lying above P such that Q is ramified in the extension Fn+1 /Fn .

The tower T1 is therefore the very particular case = r = 2 of the following tower T2 . 2 The Tower T2 Let be any prime power and let q = r , where r ∈ N and r ≥ 2. Consider the tower T2 over Fq which is given recursively by the equation Y m = (X + 1)m − 1, with m = (q − 1)/( − 1). 2) does define a recursive tower T2 = (F0 , F1 , F2 , . ). 3 are all equal m to ei = 1 (as follows from the equation xm i = (xi−1 + 1) − 1). 2) does define a tower, and the place P0 is totally ramified in all extensions Fn /F0 .

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