The Torelli locus - the image of the moduli space of curves (M_g) in the moduli space of abelian varieties (A_g) - is much-studied but still mysterious. In characteristic p, A_g has a beautiful stratification by the isomorphism type of A[p], and examples show that Mg is far from transverse to this stratification. In an ongoing project, we develop tools to understand (and perhaps make principled conjectures about) which strata of A_g meet M_g. In this talk, we explain some of the structures involved and give new results about them. Parts of this are joint work with Bryden Cais and Rachel Pries.