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There are non-isotrivial families of supersingular abelian varieties of dimension $g$ over $\mathbb P^1_{\overline {\mathbb F_p}}$ if $g\geq 2$; see Goren, E. Z.(3-MGL); Oort, F. Stratifications of Hilbert modular varieties. (English summary) J. Algebraic Geom. 9 (2000), no. 1, 111–154. There are many other papers of Goren and Oort with explicit ...
The same problem already appears in the formulation of the Birch and Swinnerton-Dyer conjecture for say an elliptic curve over a number field. When there is no longer a global minimal Weierstrass model, then we do not have a invariant differential $\omega_E$ that generates the $O_K$-module of differentials on the Néron model of $E$. One way to formulate ...