CHAPTER 1 Algebraic theory of complex multiplication The theory of complex multiplication. . . is not only the most beautiful part of mathematics but also of all science. — David Hilbert 1.1. Introduction 1.1.1. Lifting questions. A natural question early in the theory of abelian vari- eties is whether every abelian variety in positive characteristic admits a lift to char- acteristic 0. That is, given an abelian variety A0 over a field κ with char(κ) 0, does there exist a local domain R of characteristic zero with residue field κ and an abelian scheme A over R whose special fiber Aκ is isomorphic to A0? We may also wish to demand that a specified polarization of A0 or subring of the endomor- phism algebra of A0 (or both) also lifts to A. (The functor A Aκ from abelian R-schemes to abelian varieties over κ is faithful, by consideration of finite ´etale torsion levels see the beginning of 1.4.4.) Suppose there is an aﬃrmative solution A to such a lifting problem over some local domain R as above. Let’s see that we can arrange for a solution to be found over a local noetherian domain (that is even complete). This rests on a direct limit technique (that is very useful throughout algebraic geometry), as follows. Observe that for the directed system of noetherian local subrings Ri with local inclusions Ri → R, we have R = lim − Ri. In [34, IV3, §8–§12 IV4, §17] there is an exhaustive development of the technique of descent through direct limits. The principle is that if {Di} is a directed system of rings with limit D, and if we are given a “finitely presented” algebro-geometric situation over D (a diagram of finitely many D-schemes of finite presentation, equipped with with finitely many D-morphisms among them and perhaps some finitely presented quasi-coherent sheaves on them, some of which may be D-flat, etc.) then the entire structure descends to Di for suﬃciently large i. Moreover, if we increase i enough then we can also descend “reasonable” properties (such as flatness for morphisms or sheaves, and properness, surjectivity, smoothness, and having geometrically connected fibers for morphisms), any two descents become isomorphic after increasing i some more, and so on. The results of this direct limit formalism are intuitively plausible, but their proofs can be rather non-obvious to the uninitiated (e.g., descending the properties of flatness and surjectivity). We will often use this limit formalism without much explanation, and we hope that the plausibility of such results is suﬃcient for a non- expert reader to follow the ideas. Everything we need is completed proved in the cited sections of [34]. As a basic example, since the condition of being an abelian 13 http://dx.doi.org/10.1090/surv/195/02

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