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Complex multiplication cycles on elliptic modular threefolds. (English) Zbl 0623.14018
The author studies one-cycles, modulo algebraic equivalence, on certain elliptic modular threefolds. To be more precise, he considers a desingularization \(\tilde W(N)\) of the fibre product \(Y(N)\times_{\pi}Y(N)\), where \(\pi: Y(N)\to X(N)\) is Shioda’s elliptic modular surface for level N structure, which is viewed as a generalized elliptic curve over the modular curve X(N). The main result is that the \({\mathbb{Q}}\)-vector space \(B_ 1(\tilde W(N))\otimes {\mathbb{Q}}\) of one- cycles, modulo algebraic equivalence, of \(\tilde W(N)\) is infinite dimensional when \(N\geq 3\). In order to establish this he studies certain cycles in the complex multiplication fibres which are homologous to zero, but not algebraically equivalent to zero. Using an idea of Bloch the author is able to show that the images of the cycles are of infinite order. Finally, by choosing an appropriate infinite collection of these cycles, which behave sufficiently well with respect to the action of Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}})\), he shows that the classes of these cycles are linearly independent in \(B_ 1(\tilde W(N))\otimes {\mathbb{Q}}\).
Reviewer: S.Kamienny

MSC:
14J30 \(3\)-folds
14C15 (Equivariant) Chow groups and rings; motives
14K22 Complex multiplication and abelian varieties
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
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