This is a probably vague question from an outsider:
It is well known that it is possible to embed "the" moduli space of curves $M_g$ with fixed genus g into the moduli space of principally polarized abelian varieties $A_g$ by sending a curve to its Jacobian.
The image of this embedding is called Torelli locus, denoted by $T_g$. Now identifying the jacobian of curves among all pp abelian varieties is the famous Schottky problem which is reasonably well understood thanks to many people using in a beautiful way the theory of integrable systems. In particular it is well known that not all ppav are the Jacobian of a curve.
I'm wondering now if it's nevertheless possible to express (in a reasonable way) the moduli space $A_g$ in terms of Torelli loci? It's probably a stupid idea to hope that one $T_l$, for some $l$, is enough, but maybe it's possible to use "the" tower of Torelli loci? (I haven't checked details here, the $M_g$'s have a tower structure, so I guess one can expect a tower structure for the $T_g$'s as well?!)
Another, related, question would be: Is it possible to embed (in a reasonable way respecting all the involved structures) any ppav into the jacobian of a curve? (I'm not so sure if this is really a reasonable question).
Thank you very much in advance for any help.