Divided powers in Chow rings and integral Fourier transforms

We prove that for any monoid scheme M over a field with proper multiplication maps M x M --> M, we have a natural PD-structure on the ideal CH>0(M) subset of 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 + left perpendicularlog(2)(3g)right perpendicular. As a consequence we obtain, over k = (k) over tilde, a PD-structure (for the intersection product) on 2(N) . a, where a subset of 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.