Integral aspects of Fourier duality for abelian varieties

TL;DR

This paper establishes integral Fourier duality results for abelian schemes, constructs an $rak{sl}_2$-action on their Chow rings, and produces torsion classes, extending classical theories to integral coefficients.

Contribution

It proves Fourier duality and Beauville decomposition for Chow rings of abelian schemes with integral coefficients, and constructs an $rak{sl}_2$-action on these Chow rings.

Findings

Fourier duality holds integrally for Chow rings of abelian schemes.

An $rak{sl}_2$-action on Chow rings is constructed under polarization.

Torsion classes are produced in Chow groups over algebraically closed fields.

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 $i\in \{1,\ldots,g\}$.