On the Picard numbers of abelian varieties in positive characteristic

Roberto Laface

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arXiv · 1803.08704

On the Picard numbers of abelian varieties in positive characteristic

Roberto Laface

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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 $g \geq 2$, asymptotic completeness as $g \rightarrow +\infty$, 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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Roberto Laface. 2018-03-23. On the Picard numbers of abelian varieties in positive characteristic. https://arxiv.org/abs/1803.08704

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