  We prove that for any monoid scheme M over a field with proper multiplication
maps from M x M to M, we have a natural PD-structure on the ideal CH_{>0}(M) 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 is the integral part of 1 + log_2(3g). As a consequence we obtain, over
an algebraically closed field, a PD-structure (for the intersection product) on
2^N A, where A is the augmentation ideal of CH(J). 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.
