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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).

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