### 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](http://math.berkeley.edu/~coleman/fontaine.html)*

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.