On the Picard numbers of abelian varieties

We study the possible Picard numbers of abelian varieties of given dimension $g$. If $R_g$ denotes the set of realizable Picard numbers, then $R_g$ is bounded by $g^2$. We show that, for $g$ at least $3$, the set $R_g$ always has gaps and we analyze the nature of these gaps. We further prove that the Picard numbers are asymptotically complete in $[1,g^2]$ as $g$ goes to infinity. Finally we show that every Picard number which can be realized over the complex numbers can already be realized by an abelian variety defined over a number field.