On the Picard numbers of abelian varieties in positive characteristic

TL;DR

This paper investigates the possible Picard numbers of abelian varieties over algebraically closed fields of positive characteristic, revealing parallels and unique features compared to complex cases, including structural results and characteristic-dependent phenomena.

Contribution

It extends known results about Picard numbers from complex to positive characteristic, highlighting new characteristic p features and pathologies.

Findings

Non-completeness of Picard number range in dimension ≥ 2

Asymptotic completeness as dimension increases

Structure results for large Picard numbers

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.