TITLE: Picard number and torsion of Neron-Severi group of abelian varieties over a number field
QUESTION [8 upvotes]: Let $A$ be an abelian variety over a number field $k$ and let $NS_A$ denote its Neron-Severi scheme. Then the group of $k$-rational points of $NS_A$ is a finitely generated abelian group, i.e. $NS_A(k)=H^0(G_k,NS_A(\bar k)) = \mathbb{Z}^\rho \oplus Torsion$. I'd like to know:
What does a torsion element look like? Can it be ample? Is there an easy example, maybe of an elliptic curve, where the torsion of $NS_A(k)$ is non-trivial?
Is the rank $\rho$ of $NS_A(k)$, i.e. the Picard number, bounded by 1 and the square of the dimension of $A$? 
Is the group of $\bar k$-rational points of $NS_A$ also a finitely generated abelian group?

REPLY [6 votes]: Regarding the bounds for the Picard number $\rho$, the lower bound $\rho \geq 1$ comes from the fact that there exist divisors which are not algebraically equivalent to 0 (alternatively, one can argue that $\operatorname{id}_A$ is a symmetric endomorphism of $A$). The upper bound $\rho \leq (\dim A)^2$ (which holds only in characteristic 0) is Exercise 2.6(5) in Birkenhake-Lange's book Complex abelian varieties.
It is possible to show further that if $A$ is an abelian variety over $\mathbf{C}$ of dimension $\geq 2$, then $\rho = (\dim A)^2$ if and only if $A$ is isogenous to a power of some CM elliptic curve (see Exercise 5.6(10) in the same book).