As defined by Karen Smith [here, beginning of section 3](http://projecteuclid.org/euclid.mmj/1030132733)? If $E$ is an elliptic curve, then it is when $E$ is ordinary. I wonder about higher dimension cases. Any references would be greatly appreciated. 



Are Abelian varieties (sometimes) globally $F$-split?
The equivalence between ordinarity and Frobenius-splitting for abelian varieties (in fact smooth varieties with trivial tangent bundle) can be found in the paper by Mehta and Srinivas "Varieties in positive characteristic with trivial tangent bundle" (see <https://eudml.org/doc/89874> ). There are also many things about that in Kirti Joshi's work "Frobenius splitting and ordinarity" (the title suggests everything:).


