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Motivation

I learned about this question from a wonderful article Rational points on curves by Henri Darmon. He gives a list of statements (some are theorems, some conjectures) of the form

  • the set $\{$ objects $\dots$ over field $K$ with good reduction everywhere except set $S$ $\}$ is finite/empty

One interesting thing he mentions is about abelian schemes in the most natural case $K = \mathbb Q$, $S$ empty. I think according to the definition we have a trivial example of relative dimension 0.

Question

Why is the set of non-trivial abelian schemes over $\mathop{\text{Spec}}\mathbb Z$ empty?

Reference

This is proven in Il n'y a pas de variété abélienne sur Z by Fontaine, but I'm asking because: (1) Springer requires subscription, (2) there could be new ideas after 25 years, (3) the text is French and could be hard to read (4) this knowledge is worth disseminating.

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It was Shafarevich who at the 1962 ICM asked whether this set was empty, so maybe he should get the mathoverflow "reputation". :) According to Parshin, Abrashkin proved the result independently of Fontaine at about the same time: see the addendum at the end of Fontaine's paper, and see Abrashkin, V. A. Galois modules of group schemes of period $p$ over the ring of Witt vectors. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 4, 691--736, 910; translation in Math. USSR-Izv. 31 (1988), no. 1, 1--46. –  Bjorn Poonen Jan 6 '10 at 4:42
    
I found a way to "donate reputation to Shafarevich": I can post a question with bounty "on his behalf". Would you like to try giving a question? –  Ilya Nikokoshev Jan 6 '10 at 12:33
    
@Bjorn. Then how did it so happen that this theorem is known in the name of Fontaine alone? Perhaps, in the context of the "big picture" of Fontaine theory? –  Anweshi Jan 10 '10 at 15:27
    
@Anweshi: I'm not so sure that this is true. Careful people often cite it as "proved independently by Fontaine and Abrashkin". Here's a conjecture as to why Fontaine gets more hits and Abrashkin though: when Faltings proved Mordell there was a book written containing a lot of the ideas of the proof and accessible to grad students called "Arithmetic Geometry" by Cornell-Silverman, and in one of the articles there, much read by grad students presumably, they say it's "a very recent theorem of Fontaine". –  Kevin Buzzard Feb 10 '10 at 7:20
    
On the other hand, Russian translation of Lang's Diophantine Geometry contains a very good survey of Zarkhin and Parshin about finiteness problems in diophantine geometry, including a sketch of Faltings' proof, and they give attribution to Abrashkin and Fontaine (independently). That's where I read it first. So it depends on the country. –  Victor Protsak Jun 9 '10 at 0:49
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5 Answers

up vote 27 down vote accepted

It's a result related in spirit to Minkowski's theorem that $\mathbb Q$ admits no non-trivial unramified extensions. If $A$ is an abelian variety over $\mathbb Q$ with everywhere good reduction, then for any integer $n$ the $n$-torsion scheme $A[n]$ is a finite flat group scheme over $\mathbb Z$. Although this group scheme will be ramified at primes $p$ dividing $n$, Fontaine's theory shows that the ramification is of a rather mild type: so mild, that a non-trivial such family of $A[n]$ can't exist.

In the last 25 years, there has been much research on related questions, including by Brumer--Kramer, Schoof, and F. Calegari, among others. (One particularly interesting recent variation is a joint paper of F. Calegari and Dunfield in which they use related ideas to construct a tower of closed hyperbolic 3-manifolds that are rational homology spheres, but whose injectivity radii grow without bound.)

EDIT: I should add that the case of elliptic curves is older, due to Tate I believe, and uses a different argument: he considers the equation computing the discriminant of a cubic polynomial $f(x)$ (corresponding to the elliptic curve $y^2 = f(x)$) and shows that this solution equation has no integral solutions giving a discriminant of $\pm 1$.

This direction of argument generalizes in different ways, but is related to a result of Shafarevic (I think) proving that there are only finitely many elliptic curves with good reduction outside a finite set of primes. (A result which was generalized by Faltings to abelian varieties as part of his proof of Mordell's conjecture.)

Finally, one could add that in Faltings's argument, he also relied crucially on ramification results for $p$-divisible groups, due also to Tate, I think, results which Fontaine's theory generalizes. So one sees that the study of ramification of finite flat groups schemes and $p$-divisible groups (and more generally Fontaine's $p$-adic Hodge theory) plays a crucial role in these sorts of Diophantine questions. A colleague describes it as the ``black magic'' that makes all Diophantine arguments (including Wiles' proof of FLT as well) work.

P.S. It might be useful to give a toy illustrative example of how finite flat group schemes give rise to mildly ramified extensions: consider all the quadratic extensions of $\mathbb Q$ ramified only at $2$: they are ${\mathbb Q}(\sqrt{-1}),$ ${\mathbb Q}(\sqrt{2})$, and ${\mathbb Q}(\sqrt{-2})$, with discriminants $-4$, $8$, and $-8$ respectively. Thus ${\mathbb Q}(\sqrt{-1})$ is the least ramified, and not coincidentally, it is the splitting field of the finite flat group scheme $\mu_4$ of 4th roots of unity.

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Re: generalized by Faltings to abelian varieties. That's one more reason I ask: Darmon seems to call this a Shafarevich Conjecture (page 16) rather then a result. –  Ilya Nikokoshev Jan 5 '10 at 23:33
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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.) –  Emerton Jan 6 '10 at 0:23
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One might say that the elliptic curve case follows from nonexistence of weight 2 level 1 cusp forms + modularity, but there may be some hidden circular reasoning (and it seems like a backwards way to look at things). –  S. Carnahan Jan 6 '10 at 5:13
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I don't think there is any hidden circular reasoning there, and in fact is a perfectly good way to think about it. Actually, Tate's result served as an early consistency check for the modularity conjecture. (Just as his result on the non-existence of continuous representations $\rho:G_{\mathbb Q} \to GL_2(\bar{F}_2)$ unramified outside 2 served as an early consistency check on Serre's conjecture.) –  Emerton Jan 6 '10 at 5:17
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I should also add, regarding my paranthetical remark, that Tate's non-existence result for mod 2 2-dim'l Galois reps. unram. outside 2 is a crucial ingredient in the proof of Serre's conjecture by Khare and Wintenberger, so logically that situation is quite different from the elliptic curve case, where Tate's non-existence result plays no role in the proof of modularity. –  Emerton Jan 6 '10 at 6:02
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Here's another motivation for believing that there is no abelian variety over Z: the existence of such a beast would imply that the "motivic" expectations concerning L-functions are false. More precisely, it is not very difficult to show (Mestre does this in a paper in Compositio in 1986 using explicit formulas) that if an abelian variety over the rationals of dimension g at least 1 has the property that its L-function is entire with the expected functional equation, then its conductor is at least 10^{g}, and in particular is >1.

(There's a very short proof of a weaker fact, sufficient to "imply" the theorem of Fontaine and Abrashkin, in Th. 5.51 of my book with Iwaniec, though the printed version has an unfortunate mistake, and -- I must apologize here for not checking the history at the time -- we did not mention Abrashkin...)

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Fontaine's initial result was proven using his results on Hodge-Tate decomposition which refine Tate's theorem. But later on (around 1990, Schémas propres et lisses sur $\mathbf Z$), he revisited his theorem using Faltings's theorem (that the $p$-adic cohomology of a proper variety with good reduction mod $p$ is crystalline) and showed that proper smooth schemes over $\mathbf Z$ have a fairly special cohomology in degree $\leq 3$ (the Hodge numbers $h^{i,j}$ are zero for $i\neq j$ and $i+j\leq 3$).

As for Abelian varieties, this theorem has also been proved independently by Abrashkin.

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Here is a nice script of Schoof: http://www.cems.uvm.edu/~voight/notes/274-Schoof.pdf

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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. (update: I think others provided some good 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,... ?

  • ilya's comment: I think it would be very useful if somebody posted a question along the lines of what Thomas suggests, especially filling in some background from Illusie's paper (I would do it, but I don't have the paper itself).

** Anweshi's comment:** Fontaine's theory uses a great deal of crystalline cohomology. For instance please see Robert Coleman's notes referred above.

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