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 $\pi \colon X\to S$ 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 $n=0,\ldots,g$. 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 $i\in \{1,\ldots,g\}$.