Algebraic cycles and values of L-functions. II
[For part I see J. Reine Angew. Math. 350, 94-108 (1984; Zbl 0527.14008).] In this article, the author refines and illustrates conjectures made by \textit{A. A. Beilinson} [J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Math. 24, 181-238 (1984; Zbl 0588.14013)] and himself [see part I cited above]. For X a smooth projective variety over a number field k, the author considers the ``coniveau filtration'' F on \(H^.(X_{\bar k},{\mathbb{Q}}_{\ell})\) defined by \[ F^ iH^.(X_{\bar k},{\mathbb{Q}}_{\ell})= \lim Ker(H^.(X_{\bar k},{\mathbb{Q}}_{\ell})\to H^.(X_{\bar k}-y,{\mathbb{Q}}_{\ell})), \] where Y runs over the closed subschemes of \(X_{\bar k}\) of codimension i. He conjectures that the L-functions associated to the \(\ell\)-adic representations \(gr^ i_ F H^ j\) of Gal\((\bar k/k)\) are independent of \(\ell\). He also considers a certain filtration F on the Chow groups of X and conjectures that the rank of \(gr^ i_ F CH^ r(X)\) is equal to the order of vanishing of \(L(gr^ i_ F H^{2r-1},s)\) at \(s=r\). A connection is suggested between \(gr^ i_ F H^{2r-1}(X_{\bar k},{\mathbb{Q}}_{\ell}(r))\) and the Tate module of the torsion part of \(gr^ iCH^ r(X)\) and a generalization of a formula of Birch and Swinnerton-Dyer becomes meaningful. Moreover there are cycle maps from \(gr^ i_ F CH^ r(X)_{{\mathbb{Q}}}\) to \(H^ 1(\bar k/k, gr^ i_ F H^{2r-1}(X_{\bar k},{\mathbb{Q}}_{\ell}(r))\). - A result of \textit{R. Y. Yager} [Ann. Math., II. Ser. 115, 411-449 (1982; Zbl 0496.12010)] fits into this framework. The author finally discusses an example studied by \textit{B. Schoen} in his thesis [``Algebraic cycles on desingularized nodal hypersurfaces'' (Univ. Chicago 1982)] a quintic hypersurface in \(P^ 4\) with 125 double points. The conjectures imply that the Griffiths group of codimension two cycles on this variety should be of rank 0 over \({\mathbb{Q}}\) but of rank 1 over \({\mathbb{Q}}(\sqrt{5})\). An explicit cycle is exhibited which should be of infinite order.
scientific article; zbMATH DE number 4021313
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