Comments by Anweshi
There could be a motivic explanation of this. In any case, there is a whole lot "Fontaine theory" built around this. It goes into $p$-adic Hodge theory, $p$-adic Galois representations etc. References:
- the Bourbaki expose of Perrin-Riou
- notes of Robert Coleman's course on Fontaine's theory of the mysterious functor
It could be also worthwhile to have a look at the articles on finite flat group schemes in the volume Arithmetic Geometry of Cornell and Silverman, and in the volume Modular forms and Fermat's Last Theorem by Cornell, Silverman and Stevens.
Comment by Ilya
I think this should be indeed related to motives; quite a lot is known about the motives of abelian varieties. I'll look up the references.