TITLE: Schottky locus in genus 2
QUESTION [10 upvotes]: Let $\phi_g : \mathcal{M}_g \rightarrow \mathcal{A}_g$ be the period mapping from the open moduli space of genus $g$ Riemann surfaces to the moduli space of $g$-dimensional principally polarized abelian varieties over $\mathbb{C}$.  Thus for a Riemann surface $S$ the image $\phi_g(S)$ is the Jacobian of $S$. The Schottky problem consists in determining the image of $\phi_g$.
It is classical that $\text{Im}(\phi_2)$ is exactly the set of abelian varieties that are not isomorphic to a product of elliptic curves.  This is asserted in many places, but I have not been able to find a nice discussion of it in the literature.  Does anyone know one?  The more down-to-earth, the better.

REPLY [2 votes]: According to MR0364265 (51 #520) Oort, Frans; Ueno, Kenji "Principally polarized abelian varieties of dimension two or three are Jacobian varieties" (J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 377–381) covers this (I don't have access to the paper and am going solely by the review).