On the Picard numbers of abelian varieties in positive characteristic

2018/03/23 by Laface, Roberto
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.08704

Abstract

In this paper, we study the set R_g^(p) of possible Picard numbers of abelian varieties of dimension g over algebraically closed fields of characteristic p>0. We show that many of the results for complex abelian varieties have analogues in positive characteristic: non-completeness in dimension g ≥ 2, asymptotic completeness as g → +∞, structure results for abelian varieties of large Picard number. On the way, we highlight and discuss new characteristic p>0 features and pathologies: non-additivity of the range of Picard numbers, supersingularity index of an abelian variety, dependence of R_g^(p) on p, relation to the p-rank and the Newton polygon.

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