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We prove that for any monoid scheme M over a field with proper multiplicationmaps from M x M to M, we have a natural PD-structure on the ideal CH_{>0}(M) ofCH(M) with regard to the Pontryagin ring structure. Further we investigate towhat extent it is possible to define a Fourier transform on the motive withintegral coefficients of the Jacobian of a curve. For a hyperelliptic curve ofgenus g with sufficiently many k-rational Weierstrass points, we construct suchan integral Fourier transform with all the usual properties up to 2^N-torsion,where N is the integral part of 1 + log_2(3g). As a consequence we obtain, overan algebraically closed field, a PD-structure (for the intersection product) on2^N A, where A is the augmentation ideal of CH(J). We show that a factor 2 inthe properties of an integral Fourier transform cannot be eliminated even forelliptic curves over an algebraically closed field.

Divided powers in Chow rings and integral Fourier transforms