As defined by Karen Smith here, beginning of section 3? 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.
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asked Jun 6, 2014 at 16:37
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1$\begingroup$ Recalling the definition would help... $\endgroup$– ACLCommented Jun 6, 2014 at 21:52
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1$\begingroup$ Apparently the question is in the post title. As such, it has a stupid answer: yes, insofar as a product of split varieties is split. $\endgroup$– Allen KnutsonCommented Jun 7, 2014 at 1:13
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1$\begingroup$ An abelian variety $A$ is $F$-split iff it is ordinary. I don't recall a reference right now, but if one defines ordinary to mean that the map induced by Frobenius on $H^1(A, O_A)$ is injective, then it is easy to prove. (Most abelian varieties, meaning a dense Zariski open subset of the moduli space, are ordinary.) $\endgroup$– nafCommented Jun 7, 2014 at 4:19
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$\begingroup$ Thanks for all the comments. Yes, the title could be improved, I am a little MO-rusty right now, sorry. Ulrich's comment is exactly the kind of answer I am looking for, would be great if someone knows a reference though. $\endgroup$– Hailong DaoCommented Jun 7, 2014 at 8:05
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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:).
