On the determinant bundles of abelian schemes

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Let π: S be an abelian scheme over a scheme S which is quasi-projective over an affine noetherian scheme and let be a symmetric, rigidified, relatively ample line bundle on. We show that there is an isomorphism (π*)øtimes 24≃(π*ω)øtimes 12d of line bundles on S, where d is the rank of the (locally free) sheaf π*. 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 (π*)øtimes (24/N)≃(π*ω)øtimes (12d/N).