Periods of abelian varieties with complex multiplication
type: Item
instance of: scholarly article
title: Periods of abelian varieties with complex multiplication
zbMATH Open document ID: 0826.14028
DOI: 10.2307/2946559
author: Pierre Colmez
published in: Annals of Mathematics. Second Series
publication date: 1995-11-14T00:00:00Z
review text: 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 \textit{G. W. Anderson} [Compos. Math. 57, 153-217 (1986; Zbl 0591.14001)]. The author deduces this last statement from a study by \textit{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 \(Q)\).
reviewed by: Daniel Bertrand
Mathematics Subject Classification ID: 14K22
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 510251
zbMATH Keywords: \(p\)-adic periods of an abelian variety
zbMATH Keywords: product formula for periods
zbMATH Keywords: complex multipictions
zbMATH Keywords: Lubin-Tate formal groups
zbMATH Keywords: Dieudonné modules
zbMATH Keywords: Faltings' height
zbMATH Keywords: Chowla-Selberg formula
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946559
OpenAlex ID: W2322428437
description: scientific article; zbMATH DE number 510251