Abstract. In this paper we prove a refined version of the canonical key formula for projective abelian schemes in the sense of Moret-Bailly (cf.[MB]), we also extend this discussion to the context of Arakelov geometry. Precisely, let π : A → S be a projective abelian scheme over a locally noetherian scheme S with unit section e : S → A and let L be a symmetric, rigidified, relatively ample line bundle on A. Denote by ω A the determinant of the sheaf of differentials of π and by d the rank of the locally free sheaf π * L. In this paper, we shall prove the following results: (i). there is an isomorphismwhich 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 ω A such that there exists an isometry⊗md which is canonical in the sense of (i). Here the constant m only depends on g, d and is independent of L.