Faltings heights of abelian varieties with complex multiplication

On Faltings heights of abelian varieties with complex multiplication

This expository article introduces some conjectures and theorems related to the Faltings heights of abelian varieties with complex multiplication. The topics include the Colmez conjecture, the averaged Colmez conjecture, and the André–Oort conjecture.

Abelian Varieties with Complex Multiplication (for Pedestrians)

(June 7, 1998.) This is the text of an article that I wrote and disseminated in September 1981, except that I’ve updated the references, corrected a few misprints, and added a table of contents, some footnotes, and an addendum. The original article gave a simplified exposition of Deligne’s extension of the Main Theorem of Complex Multiplication to all automorphisms of the complex numbers. The a...

Real abelian varieties with complex multiplication

Division fields of Abelian varieties with complex multiplication

L’accès aux archives de la revue « Mémoires de la S. M. F. » (http://smf. emath.fr/Publications/Memoires/Presentation.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de...

Complex Multiplication, Rationality and Mirror Symmetry for Abelian Varieties

We show that complex multiplication on abelian varieties is equivalent to the existence of a constant rational Kähler metric. We give a sufficient condition for a mirror of an abelian variety of CM-type to be of CM-type. We also study the relationship between complex multiplication and rationality of a toroidal lattice vertex algebra.