It is a general fact that on *any* simple abelian variety, an effective divisor is ample. The following result underlies the usual algebraic proof of the projectivity of abelian varieties (defined initially only as *complete* group varieties). It is therefore rather standard; the first reference which comes to mind is Lemma 8.5.6 on page 253 in the abelian varieties chapter of the book *Heights in diophantine geometry* by Bombieri and Gubler, from which I quote literally. 

*Let A be an abelian variety and $D$ an effective divisor such that the subgroup*
$$
Z_D : \hspace{3cm}  \{ a \in A \mid a + D = D \}
$$
*is finite. Then $D$ is ample on $A$.*

If the abelian variety $A$ is simple, and $D$ is non-zero, then the $Z_D$ is *a fortiori* finite, since it is a proper algebraic subgroup of $A$; and it follows from the quoted Lemma 8.5.6 that $D$ is ample.

**An application.** *A surjective morphism $f: A \to X$ from a simple abelian variety onto a positive-dimensional projective variety $X$ is finite.*

*Proof.* Choose $H \subset X$ an ample divisor. The divisor $f^*H$ is effective on $A$, hence it is ample. This is equivalent to $f$ being finite.