We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas’s results on an integral version of Grothendieck–Riemann–Roch. If S is smooth quasi-projective of dimension d over a field and \(\pi :X\rightarrow S\) 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 \(\textrm{CH}(X;\Lambda )\) with coefficients in the ring \(\Lambda = \mathbb {Z}[1/(2g+d+1)!]\) . If X admits a polarization \(\theta \) of degree \(\nu (\theta )^2\) we further construct an \(\mathfrak {sl}_2\) -action on \(\textrm{CH}(X;\Lambda _\theta )\) with \(\Lambda _\theta = \Lambda [1/\nu (\theta )]\) , and we show that \(\textrm{CH}(X;\Lambda _\theta )\) is a sum of copies of the symmetric powers \(\textrm{Sym}^n(\textrm{St})\) of the 2-dimensional standard representation, for \(n=0,\ldots ,g\) . For an abelian variety over an algebraically closed field, we use our results to produce torsion classes in \(\textrm{CH}^i(X;\Lambda _\theta )\) for every \(i\in \{1,\ldots ,g\}\) .
