Picard numbers of abelian varieties in positive characteristic

I will introduce and study the set $R_g^{(p)}$ of Picard numbers of abelian varieties of dimension $g$ over algebraically closed fields of characteristic $p>0$. Many of the results for complex abelian varieties have analogs in positive characteristic. However,  there are also new characteristic $p>0$ phenomena , which I will highlight and discuss in detail. In contrast to the case of complex abelian varieties, where the situation is completely clear, there are still some open questions yet to be answered.