Abelian varieties  Mumford D.
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Название: 
Abelian varieties 
Автор: 
Mumford D. 
Категория: 
Математика

Тип: 
Книга 
Дата: 
30.12.2008 15:25:32 
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41 
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I don't believe the word "Jacobian" is cvor used in this dook. Rather stubbornly I wanted to prove that tho theory of abolian varieties could be developed without tho crutch of " reduction to Jacobians ". One of the main reasons this is possiblo is that I have used systematically tho highor cohomology groups: I am especially fond of the proof of the main theorem of §8, which replaces Theorem 4, p. 99 of Lang [L]. But I have to admit that some people might feel Lang's argument is more geometric. For a treatment of Jacobians, the reader should look at Weil's and Lang's books: especially the very important Theorem 31, p. 117 of Weil, which Lang strangely omits. For abelian schemes, some of the basic facts can bo found in my book, Geometric Invariant Theory, Ch. G, [Ml]. This area lias been greatly clarified by recent work of Raynaud which should appear soon. The connection of modular forms with moduli spaces of abelian varieties can be found in Baily [B] and Shimura [Sh], as well as in their talks at tho Boulder Summer Institute [BAI]. A purely algebrogeometric treatment of the "thetanull werte", which are special modular forms, is in my paper [M2]. It is interesting to ask whether further ties between the analytic and algebraic theories oxist: e.g. an algebraic definition of the Eisenstein series as a section of a line bundle on the moduli врасе. For the Dieudonne theory, see Manin [Ma], Oort [0], the Seminaire HeidelbergStrasbourg [DG], Tate [Tl], and the papers of Barsotti [Bt]. Among the vast literature on the arithmetic theory, let me only mention the Neron model [N] and the stable reduction theorem for this [Gl], Kodaira [K], the MordellWeil theorem [LN], the report of Cassels' [C], and Tate [T2].
Some of the material in this book is new and has not been published elsewhere. This includes the results of §16 on the index of a nondegenerate line bundle, and the results of §23 on the thetagroups 
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