Integral aspects of Fourier duality for abelian varieties

Abstract 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 $$\pi :X\rightarrow S$$ π : 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 $$\textrm{CH}(X;\Lambda )$$ CH ( X ; Λ ) with coefficients in the ring $$\Lambda = \mathbb {Z}[1/(2g+d+1)!]$$ Λ = Z [ 1 / ( 2 g + d + 1 ) ! ] . If X admits a polarization $$\theta $$ θ of degree $$\nu (\theta )^2$$ ν ( θ ) 2 we further construct an $$\mathfrak {sl}_2$$ sl 2 -action on $$\textrm{CH}(X;\Lambda _\theta )$$ CH ( X ; Λ θ ) with $$\Lambda _\theta = \Lambda [1/\nu (\theta )]$$ Λ θ = Λ [ 1 / ν ( θ ) ] , and we show that $$\textrm{CH}(X;\Lambda _\theta )$$ CH ( X ; Λ θ ) is a sum of copies of the symmetric powers $$\textrm{Sym}^n(\textrm{St})$$ Sym n ( St ) of the 2-dimensional standard representation, for $$n=0,\ldots ,g$$ n

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