Canonical key formula for projective abelian schemes

In this paper we prove a refined version of the canonical key formula for projective abelian schemes in the sense of Moret-Bailly, we also extend this discussion to the context of Arakelov geometry. Precisely, let $\pi: A\to S$ be a projective abelian scheme over a locally noetherian scheme $S$ with unit section $e: S\to A$ and let $L$ be a symmetric, rigidified, relatively ample line bundle on $A$. Denote by $\omega_A$ the determinant of the sheaf of differentials of $\pi$ and by $d$ the rank of the locally free sheaf $\pi_*L$. In this paper, we shall prove the following results: (i). there is an isomorphism {\rm det}(\pi_*L)^{\otimes 24}\cong (e^*\omega_A^\vee)^{\otimes 12d} which is canonical in the sense that it is compatible with arbitrary base-change; (ii). if the generic fibre of $S$ is separated and smooth, then there exist positive integer $m$, canonical metrics on $L$ and on $\omega_A$ such that there exists an isometry {\rm det}(\pi_*\bar{L})^{\otimes 2m}\cong (e^*\bar{\omega}_A^\vee)^{\otimes md} which is canonical in the sense of (i). Here the constant $m$ only depends on $g,d$ and is independent of $L$.