Die Tate-Vermutungen für Hilbert-Blumenthal-Flächen. (The Tate conjectures for Hilbert-Blumenthal surfaces)
This paper deals with a special case of the general conjecture of Tate on algebraic cycles. - Consider a smooth projective scheme X defined over a number field K. The Tate conjecture characterizes the algebraic cycles defined over K as those cohomology classes, which are invariant under the action of Gal(\=K/K). Moreover, it relates the rank of the space of algebraic cycles to the order of pole of the Hasse-Weil Zetafunction of X at certain integer points. In the case of Hilbert-Blumenthal surfaces this conjecture was proved for abelian extensions K over \({\mathbb{Q}}\) by \textit{G. Harder, R. P. Langlands} and \textit{M. Rapoport} [J. Reine Angew. Math. 366, 53-120 (1986; Zbl 0575.14004)]. In the present paper the conjecture is proved for non- abelian extensions K over \({\mathbb{Q}}\) using the instability of CM automorphic forms. This phenomenon has been described in the context of L-indistinguishability by \textit{J.-P. Labesse} and \textit{R. P. Langlands} [Can. Math. 31, 726-785 (1979; Zbl 0421.12014)] and in the framework of signature defects by \textit{F. Hirzebruch} and \textit{W. F. Hammond} [Math. Ann. 204, 263-270 (1972; Zbl 0245.10017)]. For an outline of the main steps of the proof see the author's talk in Séminaire de Théorie des Nombres, Paris 1985/86. For the case of a non-abelian extension K over \({\mathbb{Q}}^ a \)different proof of the Tate conjecture for Hilbert- Blumenthal surfaces was given by \textit{V. K. Murty} and \textit{D. Ramakrishnan} using period relations of CM automorphic forms. This paper will be published in the same volume of the Inventiones Math. as the author's.
scientific article; zbMATH DE number 3968747
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