On the determinant bundles of abelian schemes

Abstract

Let $\pi:\CA\ra S$ be an abelian scheme over a scheme $S$ which is quasi-projective over an affine noetherian scheme and let $\CL$ be a symmetric, rigidified, relatively ample line bundle on $\CA$. We show that there is an isomorphism \det(\pi_*\CL)^{\o times 24}\simeq\big(\pi_*\omega_{\CA}^{\vee}\big)^{\o times 12d} of line bundles on $S$, where $d$ is the rank of the (locally free) sheaf $\pi_*\CL$. We also show that the numbers 24 and $12d$ are sharp in the following sense: if $N>1$ is a common divisor of 12 and 24, then there are data as above such that \det(\pi_*\CL)^{\o times (24/N)}\not\simeq\big(\pi_*\omega_{\CA}^{\vee}\big)^{\o times (12d/N)}.