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Moduli of abelian varieties and Newton polygons. (English. Abridged French version) Zbl 0734.14016
Summary: Let \({\mathcal A}\) be the moduli space \({\mathcal A}={\mathcal A}_{g,1,n}\otimes {\mathbb{F}}_ p\), where g, p and n (prime to p) are fixed, of principally polarized abelian varieties with level-n-structure in characteristic p. Consider Newton polygons. Grothendieck showed that under specialization Newton polygons go up, and Koblitz conjectured the converse: any pair of Newton polygons one lying above the other can be realized by a specialization of abelian varieties. In this note we give a survey of a proof of this conjecture. We study closed sets obtained by the stratification of \({\mathcal A}\) by Newton polygons, and in fact we derive a stronger statement. The dimensions of these strata and other properties are obtained. Also we indicate a new proof of a conjecture by Manin (every symmetric Newton polygon can be realized by an abelian variety in characteristic p).

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14K10 Algebraic moduli of abelian varieties, classification
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