Complex multiplication cycles on elliptic modular threefolds

Schoen, Chad

doi:10.1215/s0012-7094-86-05343-3

Cited by 33 publications

(27 citation statements)

References 9 publications

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“…The first examples of smooth projective threefolds with the property that Griff 2 (W Q ) ⊗ Q has infinite dimension were given by Schoen in [9,Theorem 4.7]. It is implicit in the proof of this result that there are infinitely many primes p such that an infinite number of the cycles considered by Schoen are defined over a p-adic field K p .…”

Section: Remarkmentioning

confidence: 99%

Algebraic cycles on products of elliptic curves over p-adic fields

Rosenschon

Srinivas

2007

Math. Ann.

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Section: Remarkmentioning

confidence: 99%

Algebraic cycles on products of elliptic curves over p-adic fields

Rosenschon

Srinivas

2007

Math. Ann.

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“…Then the universal generalized elliptic curve with level-N structure φ : E → M is the canonical minimal smooth completion of φ : E → M [Shioda, 1972], [Deligne and Rapoport, 1973]. Let Then 2 E is not smooth: Using the local coordinates of [Deligne, 1969, Lemme 5.5] or [Scholl, 1990, §2], compare also [Schoen, 1986], one can check that the points over c ∈ M ∞ that are a product of two double points of E c are rational double points in 2 E . If we let 2 E ∞ 0 = 2 E sing ⊂ 2 E ∞ denote the reduced subscheme of 2 E consisting of all these points, for all c ∈ M ∞ , then applying [Deligne, 1969, Lemmes 5.4, 5.5] or [Scholl, 1990, Prop.2.1.1, Thm.3.1.0(i)] gives us the following description of the desingularization 2 E of 2 E .…”

Section: Elliptic Modular Surfaces and Threefoldsmentioning

confidence: 99%

“…Then the smooth completion φ : E → M of E over the compactification M of M obtained by adjoining the cusps is an elliptic modular surface [Shioda, 1972]. The fibre product 2 E := E × M E over M has only rational double points for singularities, and by blowing these up we get the nonsingular elliptic modular threefold 2 E that is the main focus of our attention; such threefolds have also been studied in [Schoen, 1986]. For the fibre products E × M · · · × M E (k ≥ 1 times) there is a natural desingularization k E due to [Deligne, 1969], but see also [Scholl, 1990]; the first-named author of the present paper has looked at the cohomology and the Hodge structure of these k E , and verified the generalized Hodge conjecture for them [Gordon, 1993].…”

Section: Introductionmentioning

confidence: 99%

Chow motives of elliptic modular threefolds

Gordon¹,

Murre²

1999

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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The main result of this paper is the proof for elliptic modular threefolds of some conjectures formulated by the second-named author and shown by Jannsen to be equivalent to a conjecture of Beilinson on the filtration on the Chow groups of smooth projective varieties. These conjectures are known to be true for surfaces in general, but for elliptic modular surfaces we obtain more precise results which are then used in the proof of the conjectures for elliptic modular threefolds.Let φ : E → M be the universal elliptic curve with level-N structure, whose smooth completion is an elliptic modular surface E. An elliptic modular threefold is a desingularization 2 E of the fibre product E × M E. The first main result is that there exists a decomposition of the diagonal ∆( 2 E) modulo rational equivalence as a sum of mutually orthogonal idempotent correspondences π i which lift the Künneth components of the diagonal modulo homological equivalence. These correspondences act on the Chow groups of 2 E, and secondly we show that π i · CH j ( 2 E) = 0 for i < j or i > 2j; the implication of this is that there is a filtration on CH j ( 2 E) that has j steps, as predicted by the general conjectures. The third main result is that the first step of this filtration, the kernel of π 2j acting on CH j ( 2 E), coincides with the kernel of the cycle class map from CH j ( 2 E) into the cohomology H 2j ( 2 E); which is to say that there is a natural, geometric description for this step of the filtration. We also identify F 2 CH 3 ( 2 E) as the Albanese kernel. As a by-product of our methods we also obtain some information about the Chow groups of the Chow motives for modular forms k W defined by Scholl, for k = 1 and 2, for example that CH 2 ( 1 W) = CH 2 Alb (E), and that CH 3 ( 2 W) = F 3 CH 3 ( 2 E) lives at the deepest level of the filtration, within the Albanese kernel.Typeset by A M S-T E X 1 1.1.6. The functor V(S) → M(S). There is a natural contravariant functor from V(S) to M(S), given by associating to a morphism f : X → Y of smooth projective S -schemes the class of the transpose of its graph, [ t Γ f ] ∈ CH d S (Y ) (Y × S X, Q), and associating to X in V(S) the object (X, [∆(X)]), where ∆(X) denotes the diagonal in X × S X . When S = Spec k we write h(X) := (X, [∆(X)]).

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“…The group of cycles on X of codimension d homologous to zero modulo algebraic equivalence is called the d-th Griffiths group of X and is denoted by G d (X); we will mainly consider the Q-vector space G d Q (X) = G d (X) ⊗ Q, or equivalently, the Griffiths group modulo torsion. Clemens [5] then showed that G d Q (X) can have infinite dimension over Q. Clemens' example is given by a general quintic hypersurface in P 4 , and this opened the way to the construction of other examples (see [1], [2], [13], [17]).…”

Section: Introductionmentioning

confidence: 99%