Moduli of Supersingular Abelian Varieties

Ke-Zheng Li, Frans Oort (auth.)3540639233, 9783540639237

Abelian varieties can be classified via their moduli. In positive characteristic the structure of the p-torsion-structure is an additional, useful tool. For that structure supersingular abelian varieties can be considered the most special ones. They provide a starting point for the fine description of various structures. For low dimensions the moduli of supersingular abelian varieties is by now well understood. In this book we provide a description of the supersingular locus in all dimensions, in particular we compute the dimension of it: it turns out to be equal to Äg.g/4Ü, and we express the number of components as a class number, thus completing a long historical line where special cases were studied and general results were conjectured (Deuring, Hasse, Igusa, Oda-Oort, Katsura-Oort).

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Table of contents :
Introduction….Pages 1-10
Supersingular abelian varieties….Pages 11-15
Some prerequisites about group schemes….Pages 16-18
Flag type quotients….Pages 19-23
Main results on S g,1 ….Pages 24-27
Prerequisites about Dieudonné modules….Pages 28-34
PFTQs of Dieudonné modules over W ….Pages 35-38
Moduli of rigid PFTQs of Dieudonné modules….Pages 39-50
Some class numbers….Pages 51-54
Examples on S g,1 ….Pages 55-68
Main results on S g,d ….Pages 69-72
Proofs of the propositions on FTQs….Pages 73-83
Examples on S g,d (d>1)….Pages 84-86
A scheme-theoretic definition of supersingularity….Pages 87-95