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 work on integral Grothendieck-Riemann-Roch. If $S$ is smooth quasi-projective of dimension $d$ over a field and $\pi :X\to S$ 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 $\mathrm{CH}(X;\Lambda )$ 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 $\mathrm{CH}(X;{\Lambda }_{\theta })$ with ${\Lambda }_{\theta }=\Lambda [1/\nu (\theta )]$, and we show that $\mathrm{CH}(X;{\Lambda }_{\theta })$ is a sum of copies of the symmetric powers ${\mathrm{Sym}}^n(\mathrm{St})$ of the $2$-dimensional standard representation, for $n=0,\dots ,g$. For an abelian variety over an algebraically closed field, we use our results to produce torsion classes in ${\mathrm{CH}}^i(X;{\Lambda }_{\theta })$ for every $i\in \{1,\dots ,g\}$.

22 pages