Divided powers in Chow rings and integral Fourier transforms

Abstract

We prove that for any monoid scheme M over a field with proper multiplication maps M × M → M , we have a natural PD-structure on the ideal CH>0(M ) ⊂ CH * (M ) with regard to the Pontryagin ring structure. Further we investigate to what extent it is possible to define a Fourier transform on the motive with integral coefficients of the Jacobian of a curve. For a hyperelliptic curve of genus g with sufficiently many k-rational Weierstrass points, we construct such an integral Fourier transform with all the usual properties up to 2 N -torsion, where N = 1 + ⌊log 2 (3g)⌋. As a consequence we obtain, over k =k, a PD-structure (for the intersection product) on 2 N · a, where a ⊂ CH(J) is the augmentation ideal. We show that a factor 2 in the properties of an integral Fourier transform cannot be eliminated even for elliptic curves over an algebraically closed field.the Pontryagin product and the usual product on CH(X) Q are interchanged by the Fourier transform CH(X) Q → CH(X t ) Q , we are led to ask whether one can define an integral version of the Fourier transform. An integral Fourier transform having all the usual properties would allow us to transport the PD-structure on CH * (X) for the Pontryagin product to a PDstructure on CH * (X t ) for the intersection product. Our second main result is that when X is the Jacobian J = J(C) of a hyperelliptic curve C of genus g, this plan works up to 2 N -torsion, with N = 1 + ⌊log 2 (3g)⌋. This relies on two elementary results in intersection theory of J and of powers of C that we establish in section 2. Namely, complementing the results of Collino in [4] we prove that up to 2-torsion the intersections of classes of the Brill-Noether loci W i in J are given by the same formula as in cohomology. Also, we check that a modified diagonal class on C × C × C introduced by Gross and Schoen in [11] is 2-torsion.The idea for the construction of an integral Fourier transform F for hyperelliptic Jacobians is as follows. Working with Q-coefficients one can express F entirely in terms of *products. To be precise, one has F(a) = (−1) g ·j * 2 E(τ ) * j 1, * (a) , where j 1 , j 2 : J → J 2 are the inclusions of the coordinate axes, τ ∈ CH(J 2 ) Q is the class (−1) g · ∆ * F(θ)−j 1, * F(θ)−j 2, * (θ) , and where E(τ ) is the * -exponential of τ . (See (3.7.3) in the text.) Our theorem on divided powers for the Pontryagin product allows us to define the * -exponential integrally. The next ingredient we need is the class F(θ), which, in general, we do not know how to define integrally. However, if C is hyperelliptic and ι : C → J is the embedding associated to a Weierstrass point p 0 ∈ C then we have F(θ) = (−1) g+1 · ι(C) . This puts us in a position to define, for hyperelliptic C, an integral endomorphism F : h(J) → h(J) of the ungraded motive h(J). We prove in Theorem 3.9 that it has all the expected properties up to 2 N -torsion. As a corollary we obtain a natural PD-structure on a large ideal in CH(J) for the intersection product; see Cor. 3.10.Our final result is Theorem 3.11, in which ...

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II

“…We now apply this to C [•] and J. By functoriality, see [15], Thm. 1.6, the homomorphism σ * : CH * (C [•] ) → CH * (J) is a PD-morphism.…”

“…First we recall the main construction of [15], Section 1. We consider a commutative graded monoid scheme M = ⊕ n 0 M n over k such that each M n is a quasi-projective k-scheme and such that the addition maps µ : …”

“…This PDstructure is functorial with respect to push-forward via homomorphisms. We refer to [15], Section 1 for further details. We now apply this to C [•] and J.…”

“…In Section 1 of [15] we have defined natural PD-structures on ideals of classes of positive dimension in CH * (C [•] ) and CH * (J). We are going to prove that our isomorphisms CH * (…”

“…If we work over a field then by the results in [15], Section 1, we have natural PD-structures on the ideal CH >0 (C [•] /S) ⊂ CH * (C [•] /S) and on the ideal in CH * (J/S)[t] u generated by CH >0 (J/S) together with all classes u [m] for m 1. In Thm.…”