Integral aspects of Fourier duality for abelian varieties

• Published in: arXiv.org 2024-07
• Main Authors: Junaid Hasan, Hassan, Hazem, Lin, Milton, Manivel, Marcella, McBeath, Lily, Moonen, Ben
• Format: Article
• Language: English
• ISSN: 2331-8422

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 \(i\in \{1,\ldots,g\}\).