### Comments by Anweshi

The essential point is what Emerton mentioned, ie the analogy with Minkowski's theorem on number fields with ramification. The basic principle is that "arithmetic is geometry". Number rings are in a sense zero dimensional objects, elliptic curves one dimensional objects and abelian varieties correspond to higher dimensions. So we have Minkowski's theorem. And we ask, can we extend it to higher dimensions? Tate, after setting up the theory correctly as in his famous survey article on the arithmetic of elliptic curves, proved it rather trivially for elliptic curves(as Emerton mentions). Now the task is for abelian varieties.

Fontaine comes along, and proves that it is indeed the case. But the proof turns out to be much more complicated than expected. He built a whole lot of "Fontaine theory" around this. It goes into $p$-adic Hodge theory, $p$-adic Galois representations etc. He worked on it for some 15 years in isolation, it is said. The first major success of his theory was this theorem, and later it gained popularity. Now it is a major stream of research in arithmetic geometry. 

References:

* Neukirch, Algebraic number theory, for the general philosophy that "arithmetic is geometry"/
* Notes of Robert Coleman's course on *[Fontaine's theory of the mysterious functor](http://math.berkeley.edu/~coleman/fontaine.html)*
*  The Bourbaki expose of Bearnadette Perrin-Riou. Fonctions L p-adiques des représentations p-adiques, Astérisque 229, (1995).
* Tate, The Arithmetic of Elliptic Curves, Survey Article, Inventiones.


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. This is all intimately connected with them, as Emerton mentions. In fact, you can find a particular viewpoint by Fontaine on Finite Flat group Schemes.

There could be also be a **motivic explanation** of this. The reason I think so is the following.  I have heard the answer that there is no elliptic curve over $F_1$ because from the zeta functions the motives turn out to be mixed Tate. But, on the other hand, my own "proof" of this fact was that if there were an elliptic curve or abelian variety over $F_1$, it would be extensible to $Spec\ Z$ and there by Fontaine's theorem the only abelian scheme is the trivial one. Ever since I have wondered, whether it is possible to substitute Fontaine's theory arguments with motivic ones.

Emerton clarifies in this connection: From a number theorist's point of view, p-adic Hodge theory is one of the key ingredients in the theory of motives, so these arguments are motivic, in a certain sense. (Perhaps one can say that p-adic Hodge theory encodes arithmetic properties of motives in a way analogously to the way that Hodge theory encodes geometric and analytic properties.)

Thus, by Emerton's answer, Fontaine theory seems to be thus a deeper part of motives. However, this "no abelian variety over Z" theorem of Fontaine was the first major application of Fontaine's theory. I imagined, if any results of Fontaine's theory were to be replaced by usual motivic arguments, then this ought to be the first candidate

Before stopping, I must mention the intimate connection all this has with Iwasawa theory. Fontaine's theory is very much tangled with it, as could be seen in the expose of Perrin-Riou. However the more knowledgeable people should clarify on this.

This might be an apt place to mention the [conference][1] in honor of Fontaine. He is about to retire, after his great achievements.

### 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.


  [1]: http://www.ihp.jussieu.fr/ceb/Trimestres/T10-1/C1/