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
- 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 simpler motivic explanation of this, without getting into the intricacies of Fontaine theory. 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 clarified to me 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 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.
Comments by Emerton
(1) There were earlier applications of Fontaine's results on finite flat group schemes; e.g. they played a role in Mazur's proof of boundedness of torsion of elliptic curves over $\mathbb Q$. I say this just to emphasize that Fontaine's theory didn't really develop in isolation. His theory is deep and technical, and it took people time to absorb it. But the theory of finite flat group schemes and $p$-divisible groups has a long history intertwined with arithmetic: there are results going back to Oda, Raynaud, and Tate; Fontaine generalized these; they were used by Mazur in his work, and by Faltings; Fontaine generalized further to $p$-adic Hodge theory (a theory whose existence was in part conjectured earlier by Grothendieck, motivated by, among other things, the work of Tate); ... . One shouldn't think of these ideas as being esoteric (despite the ``black magic'' label); they are and always have been at the forefront of the interaction between geometry and arithmetic, in one guise or another. (As another illustration, Fontaine's theory also closely ties in with earlier themes in the work of Dwork.)
(2) I'm not sure that there is any particular kind of usual motivic argument. The phrase motive conjures up a lot of different images in different peoples minds, but one way to think of what motivic means is that it is the study of geometry via structures on cohomology. From this point of view, $p$-adic Hodge theory is certainly a natural and important tool.
Here are some papers that give illustrations of $p$-adic Hodge theoretic reasonsing in what might be regarded as a motivic context:
Grothendieck, Un theoreme sur les homomorphismes de schemas abeliens, a wonderful paper. Although the results are essentially recovered and generalized by Delignes work in his Hodge II paper, it gives a fantastic illustration of how $p$-adic Hodge theoretic methods can be used to deduce geometric theorems.
Kisin and Wortmann, A note on Artin motives
Kisin and Lehrer, Eigenvalues of Frobenius and Hodge numbers
James Borger, Lambda-rings and the field with one element
These three are chosen to illustrate how $p$-adic Hodge theory arguments can be used to make geometric/motivic deductions. The paper of Borger is also an attempt in part to provide foundations for the theory of schemes over the field of one element, and illustrates how $p$-adic Hodge theory plays a serious role in their study.
Maulik, Poonen, Voisin, Neron-Severi groups under specialization, a terrific paper, which illustrates the possibility of using either $p$-adic Hodge-theoretic arguments or classical Hodge-theoretic arguments to make geometric deductions. (This is the same kind of complementarity as in Grothendieck's paper above compared to Deligne's Hodge II.)
Comments by Anweshi.
@Emerton, or anybody else: If there is something which does not make sense in my foray into "motivic" pictures, or something else which does not make sense, please feel free to erase and edit in whatever way you wish.
a further question by Thomas:
The great references given above let me ask about the current status of the many conjectures and open questions in Illusie's survey, e.g. finiteness theorems, crystalline coefficients, geometric semistability,... ?
