This is a vague question from an interested outsider:

It is well known that abelian varieties which arise as Jacobian of a curve (or a bit more general as Prym variety) are distinguished by the fact that their associated theta function satisfy the KP equation (Shiota, Mulase, Krichever and others. See also a recent strengthening of this due to Krichever).

In other words those special abelian varieties appear in the Sato Grassmannian as finite dimensional orbits of the KP-flow, i.e. they admit a spectral realization in terms of the KP-flow.

Now, I'm somewhat naively wondering if it's possible to realize general (pp) abelian varieties in a similar manner. Obviously, the KP-flow is not the right thing to consider, but due to recent work of Krichever and Shiota one might wonder if the Toda flow, for example, is the right thing to consider?

In order to enlarge the audience for this question, one can (very naively) reformulate my above question in the following more geometric (but very imprecise, and possibly wrong) way:

1) Is it possible to express an arbitrary (pp) abelian variety in a natural way using the - tower - of moduli spaces of curves $M_g$? (In my mind I imagine rather the tower of Torelli loci $T_g$, where $T_g$ sits inside $A_g$, the moduli space of (pp) abelian varieties). Of course, one can consider moduli spaces with extra structures if needed...

2) Is it possible to express $A_g$, for a fixed $g$, in a natural way using the tower of $M_g$'s resp. $T_g$'s? (In case that makes any sense). For $g \leq 5$ this is possible, in a sense, using only the KP flow and classification results of ppav's in small dimension (they are Prym's in that case).

Comment: The Torelli embedding $M_g \to A_g$ is well known. Roughly speaking, I'm asking for a map in the other direction, but with the whole tower of $M_g$'s as a target (this corresponds roughly speaking to the Toda-flow).

I'm aware of the (technical) imprecisions of my question, but I hope that it's more or less clear what I want.

In any case, thank you very much in advance!

Prym-Tyurin varietydue to Tyurin; this is a principally polarized abelian variety associated to a curve $C$ and a correspondence $\sigma \in \mathrm{End}(JC)$ satisfying certain conditions. Any principally polarized abelian variety is a Prym-Tyurin variety for a curve $C$ of sufficiently high genus. See for instance §12.3 ofComplex abelian varietiesby Birkenhake-Lange. $\endgroup$