This is the transcription to blog format of a talk I gave at the UC Berkeley Student Arithmetic Geometry Seminar about several topics related to Fontaine’s famous result that there are no abelian schemes over .
Statement of the result and some consequences
Two statements of the main result
The goal of this post is to discuss the consequences of, and nearby results, to the following famous theorem of Fontaine:
Theorem 1(Fontaine, ’81): There does not exist a non-trivial abelian variety with everywhere good reduction.
Recall that we say that an abelian variety has good reduction at a prime if there exists an abelian scheme (equiv. an abelian scheme ) such that (resp. . Recall that an abelian scheme is a smooth proper group scheme with connected fibers (one usually requires constant fiber dimension, but this is automatic in our case—when is integral).
Intuitively, this means that an abelian variety has good reduction at when there is a ‘good notion’ of what the ‘reduction modulo ‘ of means. In particular, not to state the obvious, the reduction being .
This result is usually not stated in this format in fact, it’s stated in the following equivalent form:
Theorem 2: There does not exist a non-trivial abelian scheme .
One might think that these are exactly the same, but there is something to be said. Namely, just because we can find a model why does this mean that we can find a global model ?
Proof(Theorem 1=Theorem 2): Clearly if is an abelian scheme then has everywhere good reduction in particular with model for all . Thus, we see that Theorem 1 implies Theorem 2.
The converse is the non-obvious step. The key observation is that there one needn’t try to glue the models together to get some group scheme over . Instead, there is always a canonically associated smooth group scheme associated to an which is a candidate for our desired abelian scheme . This allows us to bypass any nasty gluings and, instead, just compare with these various local factors .
This cryptically describe group scheme is none other than the Néron model of over . Thus, we aim to show that is proper which will then imply that its automatically an abelian scheme—the connectedness of the fibers will then follow from standard theory around the theorem on formal functions (e.g. see corollary 2.23 in Illusie’s Topics in Algebraic Geometry).
To see that is proper we proceed as follows. Note that it suffices to show that for all primes there is a neighborhood of such that is proper. That said, note that the abelian scheme must spread out to an abelian scheme for some open . Note then that and are both Néron models for over and thus, by unicity of Néron models, . Since is proper the result follows.
Both of these equivalent variations of the theorem of Fontaine are useful in their own right—each shedding a different light on a different aspect of why Fontaine’s theorem is stupendous. Let us explain these two perspectives (which, again, by the above equivalence are purely cosmetic).
To me the fact that there is no abelian variety with everywhere good reduction is a statement about Galois representations and their rigidity. Namely, it follows from the Néron-Ogg-Shafarevich that has good reduction at if and only if the -adic representation
is unramified at for all (equivalently some )–an extension of this to the case when is due to Coleman-Iovita in which case unramified is replaced by crystalline (a condition from -adic Hodge theory). Thus, we can interpret the non-existence of such an abelian variety as a sort of non-existence of an extremely nice geometric representation. This is the perspective that the actual proof of Fontaine takes.
The other perspective, the fact that there are no abelian schemes over is more a rigidity statement about varieties over . Namely, any abelian scheme over would provide, in some sense, a ‘universally definable’ abelian scheme. Namely, such an abelian scheme would give rise to an abelian scheme over any scheme : namely . Considering how different abelian varieties act over different characteristics, and how they act in sometimes incommensurate ways, it’d be surprising that we could find one that placates the ‘s for all . The statement of a Fontaine is a confirmation of this suspicion.
A result on curves
Now, while the theorem of Fontaine is astounding in its own right, it has some stupendous consequences that follow with very little work
First and foremost amongst these consequences is the following:
Theorem 3: The only smooth proper curve is .
Here a curve means that every fiber is a geometrically connected variety of dimension .
Proof: The clever idea is to realize that if is a curve, then , the Jacobian of , should be an abelian variety with everywhere good reduction. That said, this idea has one tiny hiccup. Namely, what if is trivial? This happens, for example, if and, more generally, if has genus . Thus, we must do something different in that case.
So, as we saw above, the proof breaks into two cases and . Note, here, that means the genus of any fiber of . This is well-defined (in the sense of being independent of fiber) by smooth proper base change or, much more simply, the fact that the Euler characteristic of the structure sheaf is locally constant function on the base (for flat proper families).
Let us first deal with the case when . Then, we claim that the Jacobian has everywhere good reduction, and since this contradicts Fontaine’s theorem. There are two ways to see this. Namely, one can use the fact that the Jacobian scheme (defined as the connected component of the relative Picard scheme) is defined for much more general objects than projective varieties over fields. In fact, the curve has its own Jacobian which serves as a model of showing that it has good reduction.
A simpler way of seeing that has everywhere good reduction is to recall that the Néron-Ogg-Shafarevich criterion implies that has good reduction at if and only if is unramified (for some/any ). But,
as Galois representations and so, consequently, it suffices to see that the -adic cohomology of is unramified. But, as is well-known, and implicit in the computation of the cohomology of a curve
as Galois representations. So, since admits a model over some the representation , and thus is unramified from where the good reduction of at follows.
Remark: The Albanese variety of an , which for curves is the dual of the Jacobian, is the ‘-motive’ associated to . In other words, and share the same first cohomologies.
Regardless, we see that if then has everywhere good reduction and so contradicts Fontaine’s theorem.
So, suppose now that is a smooth proper curve of genus . Note then that every geometric fiber of must be the projective line (since, trivially, all genus curves over an algebraically closed field are such). Thus, is a so-called Brauer-Severi scheme—a scheme which is étale locally on the base isomorphic to projective space (see Grothendieck’s article Le Groupe de Brauer in Dix Exposes for more details). Thus, defines a class in the Brauer group which is trivial if and only if . But, (see this post) from where the claim follows.
There are two natural questions that one might ask following Theorem 3.
First, can we give an analogue of Theorem 2? Namely, is it true that if has everywhere good reduction (again that for every there exists a smooth proper curve such that ) then ? Again, this may seem obvious, but we have a major snag: Néron models may not exist for arbitrary curves.
That said, I believe a proof can be easily adapted with a little extra work for the genus case:
Theorem 4: Let be a smooth proper geometrically connected curve with everywhere good reduction. Then, .
Proof: It’s known (see, for example, the ninth chapter of Qing Liu’s text) that if then has a unique minimal regular model over . I claim that is smooth. To see this, again, it suffices to work Zariski locally on the base in which case one can imitate the proof as in Theorem 2.
Remark: It’s known, but with considerably more work, that in the above setup has a Néron model. For example, see this paper.
We are thus reduced to the case when . Note that since has genus that it defines a class which is trivial if and only if . Thus, it suffices to show that . Now, from global class field theory the group fits into the following short exact sequence
where the first map is the obvious one (pullback of the cohomology class/base change of the central simple algebra) and the second one, the so-called invariant map , is the ‘sum coordinates map’ once the canonical identifications
In particular, we see that if is such that for all prime, then . This doesn’t quite follow from the injectivity of the above map, but the injectivity followed by the fact that (where ranges over all places of ). In particular, if for all prime, but , then which is impossible. Thus, and so by the injectivity of the map, .
Now, note that for all prime the image of in is precisely . Thus, by the previous paragraph, it suffices to show that for all . But, by assumption there exists a genus curve such that . So, in particular, is the image of under the map
That said, since is a Henselian local ring with residue field , we know from a theorem of Grothendieck (see Corollary 2.13 of Chapter 4 of Milne’s text on étale cohomology) that
and it’s a classic fact (implied, for example, by Wedderburn’s theorem) that . Thus, we see that as desired. Since was arbitrary, we conclude that as desired.
The second question is whether one can deduce Theorem 1 from Theorem 3 (or, equivalently, Theorem 2 from Theorem 4). Namely, can one deduce that if all curves with good reduction everywhere are that all abelian varieties with everywhere good reduction are trivial?
There are two impediments to such a claim
- Not all abelian varieties are Jacobians.
- Whether the proof that good reduction at of implies good reduction of at is reversible.
Both of these are, as far as I can now tell, difficult to fix. Namely, it’s a classic fact that every abelian variety over a field is a quotient of a Jacobian (cf. Milne’s text on on abelian varieties, specifically III.10). One may be able to finagle this in to something meaningful, but I haven’t thought about it too deeply.
Remark: I must have been sleep deprived when I wrote this initially for I had written the obviously false statement “Every abelian variety is isogenous to a Jacobian”. Thanks to Arithmetica for pointing this out in the comments!
But, 2. is definitively not fixable. Namely, there are curves for which their Jacobian has good reduction at a prime but the curve itself does not (see the nice answer of Noam Elkies here).
Remark: From the last paragraph we know that the -adic cohomology of a curve is not enough to detect its reduction type since, after all, a curve and its Jacobian have the same first cohomology groups (and the other cohomology groups of a curve are non-interesting). Thus, one may wonder if there is anything at all analogous to Néron-Ogg-Shafarevich for curves.
Somewhat surprisingly the answer is yes. In fact, instead of caring about the action of the inertia group on the cohomology of the curve (Néron-Ogg-Shafarevich saying that for an abelian variety good reduction is equivalent to triviality of this action) one looks at the ‘action’ of the inertia group on the étale fundamental group of the curve. More specifically, if (here where is a DVR) is a curve, then a -point gives a splitting of
and thus a conjugation action on . The induced map is independent of the choice of -point and a Theorem of Oda says that this homomorphism can determine reduction type—in fact, one only needs to check the action on the third term of the derived sequence. See section 10.2 of this for details.
Where this fails
Before we continue on, it’s worth mentioning in what ways Fontaine’s theorem can fail. Specifically, Fontaine’s theorem is stated specifically for and this manifests itself clearly in the proof which relies heavily on the Odlyzko bounds . So, what happens if you’re not dealing with ? We’ll mention later that Fontaine’s results do extend a little beyond , but let’s mention some cases where it does fail.
Perhaps the best way of producing abelian varieties , for a number field, with everywhere good reduction is to deal with abelian varieties with CM (complex multiplication). Recall that is said to have CM if contains a number field of dimension . Note that some authors might call this rational CM and reserve the phrase ‘CM’ for what we’d call geometric CM (i.e. that contains such a field).
The reason why abelian varieties with CM are good places to look for abelian varieties with everywhere good reduction is the following:
Theorem 5: Let be CM. Then, there exists a finite extension such that has everywhere good reduction.
Proof(Sketch): The abelian variety will have good reduction almost everywhere trivially. So, we only need to take care of finitely many primes, say the set of which is . Let be a rational prime such that for all . Then, it suffices to show that for each the image is finite where
since we can then kill these non-trivial inertial actions by a finite base extension. But, since has a CM we know that has image lying in . In particular, has abelian image and thus factors through .
In particular, factors through for all . But, by local class field theory we know that is isomorphic to which is a product of a finite group and a free -module. Also, is a product of a finite group and a free module over
for similar reasons.
Since there are no continuous morphisms to (since is topologically nilpotent in the latter and not in the former) we see that has image in a finite group.
An explicit example of a non-CM elliptic curve with everywhere good reduction was given by Tate as follows. Let
and . Then, Tate showed that
has everywhere good reduction over . For more explicit examples see this paper of Dembélé and Kumar.
Ideas of proof
Not surprisingly the proof of Theorem 1 is exceedingly difficult. But, before we give the vaguest of ideas of how this is attacked, let us mention the following. The elliptic curve case of Theorem 1 was known well before the general proof of Fontaine. Namely, Tate proved that there are no elliptic curves with everywhere good reduction by very elementary means. We outline this now.
It suffices to prove Theorem 2. So, let be an elliptic scheme. It’s known that a sufficient (and perhaps necessary, but I’m not positive) condition for an elliptic scheme with affine to be isomorphic to a Weierstrass form (i.e. a cubic in ) is that is free. That said, it’s a vector bundle and since all vector bundles on are free we conclude that must be a Weierstrass form.
Remark: The above can be stated in a slightly less obnoxious way for our particular needs. Namely, since we know that has a global minimal Weierstrass form—we can do it at all primes individually and the fact that we’re class number implies that we can globally work one prime at a time. Then, one can easily see that this global minimal Weierstrass form is our .
Thus, we know that is isomorphic to a cubic in with coefficients , and . Since has everywhere good reduction this implies that . One can then explicitly show that the Diophantine equation
has no integral solutions. Of course, this does not, at all, extend to higher dimensions.
So, about Fontaine’s proofs. Fontaine has given two proofs of this result. The first is found in l n’y a pas de variété abélienne sur which, essentially, comes down to super, super fine analysis of ramification of finite flat group schemes over . It’s this paper that most readily implies that there are no abelian varieties with everywhere good reduction over other fields (e.g. ). For a nice overview, with ample background, to this proof see these notes of Schoof.
He later gave another more conceptual proof of the result using -adic Hodge theory. Namely, in Schémas lisse et propres sur he proves the following astounding result which greatly generalizes Theorem 1:
Theorem 6(Fontaine, ’93): Let be a smooth proper scheme. Then, for all and the Hodge number where .
Here, the Hodge number is as usual:
This, to me, is absolutely astounding. Namely, knowing that a scheme is proper and smooth over gives one an astounding control on the cohomology of the generic fiber. This is entirely unexpected, and somewhat mysterious—why ?
That said, we can immediately see that Theorem 6 implies Theorem 1 and Theorem 2. Namely, if is an abelian scheme of positive dimension then by Theorem 6
which is impossible (here we have used the fact that abelian varieties (more generally group schemes) are parallelizable—they have trivial cotangent sheaf). Similarly, if is a curve then
thus proving that the genus of (in which case the rest is done as in the proof of Theorem 3).
This proof uses the fact that if is such a scheme, then for any the representation is unramified at every and crystalline at and deduces strong restrictions on the cohomology of as a result (the Hodge numbers coming into play via the -conjecture—which is a theorem).
In the direction of what happens if one allows just very mild bad reduction we have the following theorem of Abrashkin:
Theorem 7(Abrashkin): Let be a smooth projective variety such that has good reduction away from and bad semi-stable reduction at . Then, .
Here, denotes the second Betti number of whose definition can be given, for example, as .
As an example of this, let be an abelian variety of dimension at least . Then, note that
that said . Thus, it can’t be the case that has good reduction outside of and semi-stable reduction at .
Analogies and intuitions
One thing that one might be interested in is why one would expect Fontaine’s theorem to hold. In particular, one might try and analogize the statement to other situations and prove the result holds there. This is what we try and do in this section.
In particular, we are interested in studying abelian schemes over , , and . The first of these is a reasonable quest due to the classic analogy between and (or, if you prefer, the number field/function field analogy). The case is a simpler first step to solving the case over , and is a toy case where things should be easier.
But, before we start, we need to decide what it is that we’re trying to show exactly. Namely, what we’d like to suss out is whether in all three of these cases all curves/abelian schemes have to be ‘isotrivial’. By this, we mean a ‘constant family’ of such objects. Rigorously, a family of curves/abelian schemes is isotrivial if there exists an object such that —it’s a constant family with fiber .
This seems like the right generalization in the sense that such a result, and Fontaine’s theorem, say that the only curves/abelian schemes are the obvious ones. For there are none, and over there are just the isotrivial ones. Of course, one could say ‘something something something something’, but I am not that man. And, as the man himself says…
The case over the complex line
Let us start with what is the most intuitive case. What are the abelian schemes over ? In particular, do there exist any which are not isotrivial?
One might start with the case of elliptic schemes where, again, one might proceed as in Tate’s proof over . Namely, since has only trivial vector bundles we can write such an elliptic scheme as a Weierstrass form. Moreover, since we’re over characteristic we can even write it in the usual edulcoration
where . Then, one can try and show that
has no solution where . That said, again, this lacks the conceptual air to be satisfying and it also lacks any obvious recourse to higher-dimensions.
That said, one might have an intuition that it would be hard to make a non-isotrivial family of curves/abelian varieties because there can be no ‘twisting’. Namely, one might have the following topological intuition. Let
be an abelian scheme and consider its analytification
Then, since is a smooth and proper map we know, by Ehresmann’s lemma, that is a locally trivial fiber bundle. But, locally trivial fiber bundles on the complex line are, well, trivial. Thus, topologically the family is trivial. But, abelian varieties are essentially composed of two parts: topology+Hodge filtration (see the end of this post for details). We’ve now shown that the topological portion of the family is trivial, and so one might then imagine that this can be leveraged to show isotriviality in any case where we have a simply connected base.
Of course, there is a snag. Just because the topological aspect of the family (the local system of singular homology groups) is trivial there is no, a priori, reason to believe that the filtration aspect must also be trivial. The precise issue will be made clear in the following proof:
Theorem 8: Let be a smooth connected algebraic variety such that is simply connected. Then, every family of abelian varieties is isotrivial.
Proof: Let be such an abelian variety. By 4.4.3 of Deligne’s Theorie de Hodge II there is an equivalence of categories between abelian schemes and polarizable -variations of Hodge structure. The map is given by
In particular, since is simply connected then the -local system is a constant local system but, a priori, one might be worried that the Hodge filtration is not constant.
That said, here is a way one might remedy this. Since is compactifiable by Hironaka’s theorem and by the discussion following Theorem 11 of this article we may conclude that is constant and thus, by Deligne’s theorem, that is isogenous to a constant family . But, the kernel of some isogeny must be a finite group constant group scheme . Thus, as desired.
Remark: Using the results mentioned in this mathoverflow post, namely the cited rigidity result of Strauch, one can use the above cited result of Peters and Steenbrink to show that not only must a -VHS on a compactifiable complex manifold be constant (as a VHS) but also that the same is true for integral Hodge structures. Thus, one can bypass the above argument dealing with the isogeny category.
In the special case of the affine line, the above cited mathoverflow post also contains a ‘more elementary’ proof.
In particular, we derive the following two corollaries:
Corollary 9: Let be a smooth proper curve. Then, is isotrivial.
Proof: The proof is as in the proof of Theorem 3 or Theorem 4 for the positive genus case, and for the genus case it follows, again, from the fact that .
Remark: It should be noted that the above argument doesn’t totally emphasize the key property that is algebraic. It was used in two places. First, Deligne’s result about classifying abelian schemes in terms of -VHS (of the right type) required algebraicity of . That said, the veracity of the statement where is an arbitrary smooth connected complex manifold is not obvious to me. The second place we used it was in the context of the Peters and Steenbrink paper to know that was compactifiable.
Now, these seem like silly remarks, but as soon as one leaves the algebraic category Theorem 8 becomes wildly false. For example consider the following family of elliptic curves over , the upper half-plane:
where and are the usual modular forms of level and weights and . This is, of course, not an isotrivial family even though is a simply connected complex manifold. That said, this not a contradiction since is not algebraic!
The case of the line over a finite field
We’d now like to deal with the case of the line . Namely, we’d like to suss out whether or not there are non-isotrivial curves/abelian varieties. To spoil the drama, let us remark that we will not be able to prove anything substantive for genus greater than or abelian varieties of dimension greater than .
So, let us begin with the obvious case: genus curves. We then have the following obvious extension of the above ideas:
Theorem 10: Let be a smooth proper curve of genus where or . Then, .
The proof, again, is just the fact that
and, of course, this extends to show that any curve of genus 0 over is just as long as . This follows from the fact that
as is fairly easily computed.
Let us now move on to the slightly more interesting case of genus curves. We start, not shockingly, with the case of elliptic schemes. Namely, is every elliptic scheme , where , isotrivial?
Here is where we want to use the case of the complex line as spiritual guidance. Namely, there we saw that the operative thing about was that it was simply connected and that it forced the constancy of the local system (and the Hodge filtration). Of course, is not simply connected, we can’t access something like integral cohomology, and we have no such thing as a Hodge filtration.
So, at first glance, it seems like it’s literally impossible to adapt the proof in any meaningful way. That said, we do have ways of working around each of these difficulties. Namely, the line over is not (étale) simply connected but it is ‘prime-to-p’ simply connected—its fundamental group has no prime-to-p quotients. We cannot access the integral cohomology but we can access the -cohomology as the sheaf . And, what we lack in our ability to deal with Hodge theory we’ll make up for in an understanding of the moduli space of elliptic curves with a fixed cohomology class.
NB: Let’s assume that is not a power of or for the sake of convenience.
In particular, let us state the following rigorously:
Theorem 11: Let be an elliptic scheme. Then, is isotrivial.
Proof: Let us begin by noting that for all coprime to we have that is a finite étale group scheme over with (geometric fiber) isomorphic to . We claim that for arbitrarily large it is, in fact, constant.
To see this, note that it is constant if and only if the monodromy action is trivial on the fiber for some/any geometric point . Since the actual geometric point is irrelevant we suppress it from the notation. But, note that the action of on the fiber of corresponds to an element of the group
but for any we have that
has prime-to- order and thus, since has no prime-to-p quotients, this map must be trivial.
Thus, we see that for arbitrarily large the group scheme is constant. Thus, we can choose, for arbitrarily large , trivializations
The pair then defines a map where, here, denotes the open modular curve of level . This then extends to a map (where is the canonical compactification of ). But, for the genus is positive, and so any map from the projective line is constant.
This says precisely that the map factors through a point which says precisely that the family and thus, in particular , is constant.
We would now like to extend this to the case over which will largely just be a calculation of twists:
Theorem 12: Let be an abelian scheme. Then, is isotrivial.
Proof: By Theorem 11 we know that is isotrivial, let’s say isomorphic to for an elliptic curve. We claim that is actually defined over . To see this, we merely note that
and thus this common -invariant lies in and thus in . So, we may assume without loss of generality that for .
So, we know now that is a -twist of . So, it suffices to show that all such twists are isotrivial.
That said, note that such twists are classified by
where these automorphisms are as elliptic schemes over . But, it’s easy to see that all such automorphisms are just automorphisms of and thus all the twists classes are represented by twists of , and thus are isotrivial.
One should be able to extend Theorem 12 to all genus curves, by analyzing the Jacobian, but I have not checked this entirely.
One is very hopeful after all of this. We might be able to extend the above to prove that every curve/abelian scheme over , or at least , is isotrivial. Unfortunately, our world comes crashing down—it’s false. See this mathoverflow post. Thus, this is an example where the number field/fucntion field analogy breaks down a little bit.
Remark: It’s somewhat interesting to note the following. One might be able to summarize the above proof’s idea as: leverage the moduli space of elliptic curves (or, rather, some decorated elliptic curves) to show that all elliptic schemes are isotrivial. Thus, the question of isotriviality becomes one of pure geometry: is the geometry of the moduli space such that it can support interesting functions from the affine line/
This is interesting because in this (already mentioned) mathoverflow post Donu Arapura gives a proof that has no abelian schemes using, again, the geometry of some moduli space (or, rather, a cover of it). That said, his approach is to use the complex differential geometry of a moduli space to show that the affine line cannot support non-isotrivial abelian schemes (a hyperbolicity result). This cannot be extended to some sort of interesting algebraic condition on any of the locally symmetric varieties that are quotients of this moduli space else, in theory, this should translate to the positive characteristic world where, as mentioned above, the result is false for higher-dimensional abelian varieties!