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 $��: A\to S$ 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 $��_A$ the determinant of the sheaf of differentials of $��$ and by $d$ the rank of the locally free sheaf $��_*L$. In this paper, we shall prove the following results: (i). there is an isomorphism {\rm det}(��_*L)^{\otimes 24}\cong (e^*��_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 $��_A$ such that there exists an isometry {\rm det}(��_*\bar{L})^{\otimes 2m}\cong (e^*\bar��_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 $L$.

30 pages