# Picard number of principally polarized abelian varieties

Let $A$ be an abelian variety of dimension $n$. Over $\mathbb{C}$, at least, it is known that the Picard number (that is, the rank of the Neron-Severi group of $A$) is less than or equal to $n^2$, with equality if and only if $A$ is isogenous to the self product of an elliptic curve with complex multiplication.

Is there a bound on the Picard number if $A$ is simple? In other words, can a simple abelian variety have Picard number $n^2-1$?

• Can someone point me to a proof of this: "Let $A$ be an abelian variety of dimension $n$. Over $\mathbb{C}$, at least, it is known that the Picard number (that is, the rank of the Néron-Severi group of $A$) is less than or equal to $n^2$, with equality if and only if $A$ is isogenous to the self product of an elliptic curve with complex multiplication." Commented Feb 21, 2022 at 21:27

A tight bound for simple $A/\mathbb{C}$ is $\rho(A) \leq 3n/2$. This follows from Proposition 5.5.7 in Birkenhake-Lange. If $A$ does not have indefinite quaternionic multiplication, the stronger bound $\rho(A) \leq n$ holds.