Integral aspects of Fourier duality for abelian varieties

We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas's work on integral Grothendieck-Riemann-Roch. If S is smooth quasi-projective of dimension d over a field and X S is a g-dimensional abelian scheme, we prove, under very mild assumptions on X/S, that all classical results about Fourier duality, including the existence of a Beauville decomposition, are valid for the Chow ring CH (X;) with coefficients in the ring = Z1/ (2g+d+1) !. If X admits a polarization of degree () ² we further construct an sl₂-action on CH (X;_) with _ = 1/ (), and we show that CH (X;_) is a sum of copies of the symmetric powers Symⁿ (St) of the 2-dimensional standard representation, for n=0, , g. For an abelian variety over an algebraically closed field, we use our results to produce torsion classes in CHⁱ (X;_) for every i \1, , g\.