Algebraic cycles and values of L-functions. II. (English) Zbl 0628.14006

Bloch, Spencer

Duke Math. J. 52, 379-397 (1985).

[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 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 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 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.

Reviewer: J.H.M.Steenbrink

MSC:

14C15	(Equivariant) Chow groups and rings; motives
14C05	Parametrization (Chow and Hilbert schemes)
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Keywords:

coniveau filtration; Chow groups; order of vanishing of L; formula of Birch and Swinnerton-Dyer

Citations:

Zbl 0527.14008; Zbl 0588.14013; Zbl 0496.12010

References:

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