On the periods of CM abelian varieties. An introduction to P. Colmez's article "Périodes des variétés abéliennes à multiplication complexe"

Prof. Damian Rössler (Université Paul Sabatier, Toulouse)

April 07, 15:30 - 16:30, G 19.2
April 08, 14:00 - 15:00, G 19.1
April 09, 10:00 - 11:00, G 19.1
April 09, 15:30 - 16:30, G 19.1
April 10, 10:00 - 11:00, G 19.2

This is a preparation course for the summer school workshop "Periods and heights of CM abelian varieties", taking place from July 6 to 11, 2014 in Alpbach, Austria.

Abstract

The aim of this mini-course is to provide an overview of the article “Périodes des variétés abéliennes à multiplication complexe” of P. Colmez (Annals of Math. 138, no. 3, 625–683). In this article, the author generalises (in part conjecturally) the formula of Chowla and Selberg to any abelian variety with complex multiplication. We shall consider the following topics:
(1) the conjectural “product formula” of P. Colmez
(2) formal groups with complex multiplication; Cartier's generalisation of Lubin-Tate formal groups
(3) the computation of the valuations of the p-adic periods of formal groups with CM;
the p-adic periods of abelian varieties with CM
(4) the distribution relations of p-adic and classical periods of CM abelian varieties
(5) the periods of Fermat curves; proof of the product formula in the case of abelian CM extensions of \(\bf Q\)
(6) [if there is enough time] the link with Arakelov theory; the arithmetic fixed point formula (this is not part of Colmez’s article)