This paper establishes a sequence of relations for the valuations of the \(p\)-adic periods of an abelian variety \(A\) with complex multiplictions, which the author announced in two previous papers [cf. C. R. Acad. Sci., Paris, Sér. I 309, No. 3, 139-142 (1989; Zbl 0701.14040) and 313, No. 13, 899-904 (1991; Zbl 0760.14017)]. The proofs rely on the study of Lubin-Tate formal groups and on Fontaine’s theory of Dieudonné modules. The main output is an exact expression for these valuations in terms of the \(p\)-part of a linear combination \(Z(\Phi)\) (which depends on the CM type \(\Phi\) of \(A\)) of special values of derivatives of Artin \(L\)- functions. This in turn enables the author to give a meaning to the sum over all \(p\)’s of these expressions, and to relate it to the stable Faltings’ height \(h(A)\) of \(A\). He then conjectures a direct relation between \(h(A)\) and \(Z(\Phi)\), which amounts to predicting that the (renormalized) product of the absolute values of all \((p\)-adic and archimedean) periods of \(A\) is equal to 1.
This conjectural “product formula” is finally proved in the case of an abelian CM field, where it sharpens a rationality statement of G. W. Anderson [Compos. Math. 57, 153-217 (1986; Zbl 0591.14001)]. The author deduces this last statement from a study by R. F. Coleman [in: \(p\)- adic Analysis, Proc. Int. Conf., Trento 1989, Lect. Notes Math. 1454, 173-193 (1989; Zbl 0757.14013)] of Fermat curves, and deduces from it a new proof of the Chowla-Selberg formula for the periods of CM elliptic curves. The “product formula” is in fact the leading thread of the paper. It is discussed in the introduction in a delightful and convincing (albeit heuristic) way.
(Typographical remarks: on p. 633, line 6, the sign of \(\log D\) should be changed; on p. 663, the \(h_k\) in formula II. 2.3.1 denotes the degree of \(K\) over \(Q)\).