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)
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.
scientific article; zbMATH DE number 4152511
scientific article