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Le théorème de Tate-Poitou pour les corps de fonctions des courbes définies sur les corps locaux de dimension N. (The Tate-Poitou theorem for function fields of curves defined over local fields of dimension N). (French) Zbl 0703.11064

Let X be a projective, geometrically irreducible, smooth curve over a field k. Assume that k is a local field of dimension N in the sense of Parshin-Kato; i.e., there exists a chain of fields \(k_ 0,k_ 1,...,k_ N=k\), such that \(k_ 0\) is a finite field of characteristic p and, for each i, the field \(k_ i\) is complete with respect to a discrete valuation with residue field equal to \(k_{i-1}\). Assume, moreover, that the field \(k_ 0\) contains the q-th-roots of unity, where q is a given integer prime to p. The author shows in the paper the existence of a duality between the finite groups \(H^{N+2}(X,\mu_ q)\) and \(H^ 1(X,\mu_ q)\), if \(0\leq N\leq 4\). For \(N=0\), the theorem yields results which are already well-known.
Reviewer: P.Bayer

MSC:

11R58 Arithmetic theory of algebraic function fields
14H05 Algebraic functions and function fields in algebraic geometry
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References:

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