# On the Picard numbers of abelian varieties

## TL;DR

This paper investigates the range and distribution of Picard numbers for abelian varieties of fixed dimension, revealing gaps, asymptotic completeness, and field of definition properties.

## Contribution

It establishes bounds, identifies gaps in realizable Picard numbers, and proves all realizable numbers over complex numbers are also realizable over number fields.

## Key findings

- Picard numbers are bounded by the square of the dimension.
- Gaps exist in the set of realizable Picard numbers for dimensions at least 3.
- Asymptotically, all numbers in [1, g^2] are realizable as the dimension grows.

## Abstract

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.