Preface During the Workshop on Abelian Varieties in Amsterdam in May 2006, the three authors of this book formulated two refined versions of a problem concerning lifting into characteristic 0 for abelian varieties over a finite field. These problems address the phenomenon of CM lifting: the lift into characteristic 0 is required to be a CM abelian variety (in the sense defined in The precise formulations appear at the end of Chapter 1 (see 1.8.5), as problems (I) and (IN). Abelian surface counterexamples to (IN) were found at that time see 2.3.1–2.3.3, and see 4.1.2 for a more thorough analysis. To our surprise, the same counterexam- ples (typical among toy models as defined in 4.1.3) play a crucial role in the general solution to problems (I) and (IN). This book is the story of our adventure guided by CM lifting problems. Ching-Li Chai thanks Hsiao-Ling for her love and support during all these years. He also thanks Utrecht University for hospitality during many visits, including the May 2006 Spring School on Abelian Varieties which concluded with the workshop in Amsterdam. Support by NSF grants DMS 0400482, DMS 0901163, and DMS120027 is gratefully acknowledged. Brian Conrad thanks the many participants in the “CM seminar” at the Univer- sity of Michigan for their enthusiasm on the topic of complex multiplication, as well as Columbia University for its hospitality during a sabbatical visit, and grate- fully acknowledges support by NSF grants DMS 0093542, DMS 0917686, and DMS 1100784. Frans Oort thanks the University of Pennsylvania for hospitality and stimulating environment during several visits. We are also grateful to Burcu Baran, Bas Edixhoven, Ofer Gabber, Johan de Jong, Bill Messing, Ben Moonen, James Parson, Ren´ e Schoof, and Jonathan Wise for insightful and memorable discussions. ix
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