Diophantine problems and $p$-adic period mappings
We give an alternative proof of Faltings’s theorem (Mordell’s conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings’s original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of -adic Galois representations in a family of algebraic varieties. The key inputs into this analysis are the comparison theorems of -adic Hodge theory, and explicit topological computations of monodromy.
By the same methods we show that, in sufficiently large dimension and degree, the set of hypersurfaces in projective space, with good reduction away from a fixed set of primes, is contained in a proper Zariski-closed subset of the moduli space of all hypersurfaces. This uses in an essential way the Ax–Schanuel property for period mappings, recently established by Bakker and Tsimerman.
Let be a number field. This paper has two main goals.
Firstly, we will give a new proof of the finiteness of -rational points on a smooth projective -curve of genus . The proof is closely related to Faltings’s proof [Faltings], but is based on a closer study of the variation of -adic Galois representations in a family; it makes no usage of techniques specific to abelian varieties.
Secondly, we give an application of the same methods to a higher-dimensional situation. Consider the family of degree- hypersurfaces in and let be the complement of the discriminant divisor in this family; we regard as a smooth -scheme. For a finite set of primes, points of correspond to proper smooth hypersurfaces of degree in . It is very reasonable to suppose that is finite modulo the action of for and all . We shall show at least that, if and , then is contained in a proper Zariski closed subset of . For this higher-dimensional result, we use a very recent theorem of Bakker and Tsimerman, the Ax–Schanuel theorem for period mappings.
We can obtain a still stronger theorem along a subvariety of if one has control over monodromy. Namely, if is the Zariski closure of integral points, our result actually implies that the monodromy of the universal family of hypersurfaces must drop over each component of . It is possible that this imposes a stronger codimension condition on than simply “proper” but we do not know for sure.
Note that, without the result of Bakker and Tsimerman, one can still prove that lies in a proper -analytic subvariety of but one cannot prove the second statement about .
A simple toy case to illustrate the methods is given by the -unit equation, which we analyze in §LABEL:Sunit.
1.2. Outline of the proof
Consider a smooth projective family over , where is itself a smooth -variety; we suppose this extends to a family over the ring of -integers of , for some finite set of places of (containing all the archimedean places).
For call the fiber over . We want to bound , making use of the fact that, if extends to , then admits a smooth proper model over . That one can thus reduce Mordell’s conjecture to finiteness results for varieties with good reduction was observed by Parshin [Parshin] and then used by Faltings in his proof of the Mordell conjecture [Faltings].
Choosing a rational prime that is unramified in and not below any prime of , write for the Galois representation of on the -adic geometric étale cohomology of , i.e. . As observed by Faltings, one deduces from Hermite–Minkowski finiteness that there are only finitely many possibilities for the semisimplification of (denoted by ). In the contexts of interest, we will complete the proof by establishing the following result.
(*) For the families that we study, there exists a place of above such that the map
has finite fibers. (Here is the absolute Galois group of .)
Faltings proves a much stronger statement when is an abelian scheme over , using a remarkable argument with heights: every is semisimple and determines up to isogeny. While our approach gives less information, it yields results even when is not abelian.
Our analysis uses -adic Hodge theory. However we make no use of -adic Hodge theory in families: we need only the statements over a local field. Under the correspondence of -adic Hodge theory, the restricted representation from () corresponds to a filtered -module, namely the de Rham cohomology of over equipped with its Hodge filtration and a semilinear Frobenius map. The variation of this filtration is described by a period mapping; in this setting, this is a -analytic mapping
Therefore, the variation of the -adic representation with is controlled by (1.2). The basic, and very naive, “hope” of the proof is that injectivity of the period map (1.2) should force (1.1) to be injective.
However, (*) does not follow directly from injectivity of the period map, that is to say, from Torelli-type theorems.
Firstly, different filtrations on the underlying -module can give filtered -modules which are abstractly isomorphic, the isomorphism being given by a linear endomorphism commuting with . Hence, one needs to know not only that the period mapping (1.2) is injective, but that its image has finite intersection with an orbit of the action of the centralizer of on the period domain. In other words, we must analyze a question of “exceptional intersections” between the image of a period map and an algebraic subvariety. Even if this is addressed, we obtain only that has finite fibers; but (1.1) addresses the semisimplification of the global Galois representation , rather than itself.
Thus one must overcome the following three problems, and in a sense most of the paper is showing that they can be effectively overcome in at least the two situations we consider:
Showing that the centralizer of is not too large, and
Showing that the image of the period mapping has finite intersection with an orbit of this centralizer, and
Controlling in some a priori way the extent to which can fail to be semisimple.
We will discuss in turn how we deal with (a), (b) and (c). For (a) and (b), we use different techniques in the curve case and in the hypersurface case. For (c), we just note for now that we certainly do not show that is always semisimple; but nor does one need such a strong statement.
1.3. Problem (a): controlling the centralizer of
As we have explained, we need a method to ensure the centralizer of the crystalline Frobenius acting on the cohomology of a fiber is not too large. For example, if so that is simply a -linear map, we must certainly rule out the possibility that is a scalar!
This issue, that might have too large a centralizer and thus (*) might fail, already occurs in the simplest possible example. When analyzing the -unit equation, it is natural to take and to be the Legendre family, so that is the curve . Unfortunately (*) fails: for , if we write for the representation of the Galois group on the (rational) Tate module of , then belongs to only finitely many isomorphism classes so long as the reduction is not equal to or .
Again we proceed in two different ways:
In general, Frobenius is a semilinear operator on a vector space over an unramified extension of ; semilinearity alone gives rise to a nontrivial bound (Lemma 2.1) on the size of its centralizer, which, in effect, becomes stronger as gets larger.
In the application to Mordell, it turns out that we can always put ourself in a situation where is rather large. This forces the Frobenius centralizer to be small. We explain this at more length below.
In the case of hypersurfaces, we do not have a way to enlarge the base field as in (i). Our procedure is less satisfactory than in case (i), in that it gives much weaker results:
We are of course able to choose the prime , and we choose it (via Chebotarev) so that the crystalline Frobenius at has centralizer that is as small as possible. To do this, we fix an auxiliary prime , and first use the fact (from counting points over extensions of ) that crystalline Frobenius at has the same eigenvalues as Frobenius on acting on -adic cohomology; thus it is enough to choose such that the latter operator has small centralizer. One can do this via Chebotarev, given a lower bound on the image of the global Galois representation, and for this we again use some -adic Hodge theory (cf. [Sen]). Another approach, by point-counting, is outlined in Lemma LABEL:end_lemma.
Let us explain point (i) above by example. In our analysis of the -unit equation in §LABEL:Sunit, we replace the Legendre family instead by the family with fiber
for a suitable large integer . In our situation, the corresponding map will now only have finite fibers, at least on residue disks where is not a square – an example of the importance of enlarging .
Said differently, we have replaced the Legendre family with a family with the following composite structure:
where the second map is given by , and is simply the restriction of the Legendre family over . The composite defines a family over with geometrically disconnected fibres, and this disconnectedness is, as we have just explained, to our advantage.
It turns out that the families introduced by Parshin (see [Parshin, Proposition 9]), in his reduction of Mordell’s conjecture to Shafarevich’s conjecture, automatically have a similar structure. That is to say, if is a smooth projective curve, Parshin’s families factorize as
where is finite étale and is a relative curve.
There is in fact a lot of flexibility in this construction; in Parshin’s original construction the covering is obtained by pulling back multiplication by on the Jacobian, and as such each geometric fiber is a torsor under . We want to ensure that the Galois action on each fiber of has large image – with reference to the discussion above, this is what allows us to ensure that the auxiliary field is of large degree. We use a variant where each fiber admits a -equivariant map to (for a suitable auxiliary prime ). The Weil pairing alone implies that the Galois action on this is nontrivial, and this (although very weak) is enough to run our argument.
1.4. Problem (b): controlling the image of the period map
We must show that the image of the -adic period map
cannot intersect an algebraic subvariety of the flag variety in an unexpectedly large set. (In fact, the algebraic subvariety is an orbit of a subgroup, but we will not make use of this.)
First of all, one can transfer this question to the same question about the complex period map: the -adic and complex-analytic period maps are given by evaluation of the same -rational power series. This is a straightforward but crucial argument – see Lemma LABEL:vCpowerseries.
Once this is done, we use two methods:
For Mordell’s conjecture, where we have , it is enough to compute the monodromy of the family . This computation was not simple (at least for us). It is related to computations of Looijenga [Looijenga]. Our strategy is roughly that the monodromy representation of extends to a certain mapping class group, and we deduce large monodromy from the same assertion for the mapping class group. In the latter setting, we make careful use of Dehn twists.
For hypersurfaces, we have , and the monodromy argument would give only that the exceptional set is a proper -analytic subvariety of . One wants to get a proper Zariski-closed subvariety (for example, this permits one, in principle at least, to make an inductive argument on the dimension, although we do not try to do so here.) We obtain this only by appealing to a remarkable recent result of Bakker and Tsimerman, the Ax–Schanuel theorem for period mappings: this is a very powerful and general statement about the transcendence of period mappings.
1.5. Problem (c): How to handle the failure of semisimplicity
Let . The local Galois representation can certainly be very far from semisimple, and thus we cannot hope to use -adic Hodge theory alone to constrain semisimplicity.
However, the Hodge weights of a global representation are highly constrained by purity (Lemma LABEL:globalsimple). This means, for example, that any global subrepresentation of corresponds, under -adic Hodge theory, to a Frobenius-stable subspace whose Hodge filtration is numerically constrained. Now (assuming we have arranged that the Frobenius has small centralizer) there are not too many choices for a Frobenius-stable subspace; on the other hand, the Hodge filtration varies as varies -adically. Thus one can at least hope to show that such a “bad” exists only for finitely many . In this way we can hope to show that is simple for all but finitely many . (In practice, we prove a much weaker result.)
The purity argument is also reminiscent of an argument at the torsion level in Faltings’s proof (the use of Raynaud’s results on [Faltings, p. 364]).
We use this argument both for Mordell’s conjecture and for hypersurfaces. The linear algebra involved is fairly straightforward for curves (see Claim 1 and its proof in Section LABEL:highergenus) but becomes very unwieldy in the higher-dimensional case. To handle it in a reasonably compact way we use some combinatorics related to reductive groups (§LABEL:GG_combinatorics). However this argument is not very efficient and presumably gives results that are far from optimal.
1.6. Effectivity; comparison with Chabauty–Kim and Faltings
It is of interest to compare our method with that of Chabauty, and the nonabelian generalizations thereof due to Kim.
Let be a projective smooth curve over with Jacobian . Fix a finite place . The classical method of Chabauty proceeds by considering as the intersection of global points on the Jacobian and local points on the curve, inside . If the rank of is less than the -dimension of (i.e. the genus of the curve) it is easy to see this intersection is finite.
We can reinterpret this cohomologically. Let be the -adic Tate module of , where is a prime below . There is a Kummer map and we obtain a mapping
which, explicitly speaking, sends to the extension between the trivial representation and realized by cohomology of the punctured curve for a suitable basepoint . By this discussion, and its local analogue, we get a diagram
(Here the global and local Galois representations are extensions of by the trivial representation.) Kim generalizes this picture, replacing by deeper quotients of . The key difficulty to be overcome is to obtain control over the size of the space of global Galois representations (e.g. the rank of ).
Our picture is very much the same: we have a map from to global Galois representations. In the story just described arises from the cohomology of an open variety – the curve punctured at and an auxiliary point. In the situation of our paper, will arise from the cohomology of a smooth projective variety – a covering of branched only at .
What does this gain? Our global Galois representations are now pure and (presumably) semisimple. Therefore our space of global Galois representations should be extremely small. On the other hand, what we lose is that the map is now no longer obviously injective.
Kim has remarked to one of us (A.V.) that it would be of interest to consider combining these methods in some way. For example, one might consider more carefully the situation of a non-projective family .
We expect that our method of proof can be made algorithmic in the same sense as the method of Chabauty. For example, given a curve as above, one would be able to “compute” a finite subset which contains ; “compute” means that there is an algorithm that will compute all the elements of to a specified -adic precision in a finite time. However, the resulting method is completely impractical, as we now explain.
Firstly, our argument relies on Faltings’s finiteness lemma for Galois representations (Lemma LABEL:finiteness) to give a finite list of possibilities for . We expect that Faltings’s proof can easily be made algorithmic; but there may be very, very many such representations.
Secondly, we would need to explicitly compute the comparisons furnished by -adic Hodge theory. For a given local Galois representation , we need to calculate to some finite precision the filtered -module associated to it by the crystalline comparison isomorphism of -adic Hodge theory. We expect that this should be possible, but we are not aware of any known algorithm to achieve this.
To conclude let us compare our method to Faltings’s original proof. That proof gives much more than ours does: it gives the full Shafarevich and Tate conjectures for abelian varieties, as well as semisimplicity of the associated Galois representation. Our proof gives none of these; it gives nothing about the Tate conjecture, and (at least without further effort) it does not give the Shafarevich conjecture but only its restriction to a one-dimensional subfamily of moduli of abelian varieties. Moreover, our proof is also in some sense more elaborate, since it requires the use of tricks and delicate computations to avoid the various complications that we have described. Its only real advantage in the Mordell case seems to be that it is in principle algorithmic in the sense described above. In our view, the real gain of the method is the ability to apply it to families of higher-dimensional varieties. Our results about hypersurfaces are quite modest, but we regard them as a proof of concept for this idea.
1.7. Structure of the paper
§2 contains notation and preliminaries.
We suggest the reader start with §LABEL:fibers and §LABEL:Sunit to get a sense of the argument.
§LABEL:fibers sets up the general formalism and the structure of the argument. We relate Galois representations to a -adic period map using crystalline cohomology; and we connect the -adic period map to a complex period map and monodromy. The Section ends with Proposition LABEL:finitesetoforbits, a preliminary form of our main result.
§LABEL:Sunit gives a first application: a proof of the -unit theorem, using a variant of the Legendre family. This is much simpler than the proof of Mordell and can be considered a “warm-up.”
§§LABEL:Mordelloutline – LABEL:KP_monodromy give the proof of the Mordell conjecture. §LABEL:Mordelloutline describes the strategy of the proof: we apply a certain refined version of Proposition LABEL:finitesetoforbits, formulated as Proposition LABEL:finitepointsoncurves, to a specific family of varieties that we call the Kodaira–Parshin family. §LABEL:highergenus is the proof of Proposition LABEL:finitepointsoncurves. In particular this is where we take advantage of “geometrically disconnected fibers”; the argument also deals with a technical issue relating to semisimplification. In §LABEL:rationalpoints we introduce the Kodaira-Parshin family and §LABEL:KP_monodromy is purely topological: it computes the monodromy of the Kodaira–Parshin family.
§§LABEL:hypersurface – LABEL:bound_frob study families of varieties of higher dimension. §LABEL:hypersurface introduces a recent transcendence result of Bakker and Tsimerman which is needed to study families over a higher-dimensional base. §LABEL:Hodge_numbers_funny proves the main result, Proposition LABEL:transprop, which shows that fibers of good reduction lie in a Zariski-closed subset of the base. The argument however invokes a “general position” result in linear algebra, Proposition LABEL:linalg, whose proof takes up §LABEL:GG_combinatorics. In §LABEL:bound_frob we suggest an alternative argument, not used in the rest of the paper, to bound the size of the Frobenius centralizer.
This paper owes, of course, a tremendous debt to the work of Faltings – indeed, all the main tools come from his work. Some of the ideas originated in a learning seminar run at Stanford University on Faltings’s proof [Faltings].
The 2017 Stanford PhD thesis [BL] of B.L. contained an earlier version of the arguments of this paper. In particular, that thesis presented a proof of the Mordell conjecture conditional on an assumption about monodromy, and verified that assumption for a certain Kodaira–Parshin family in genus .
We thank Brian Conrad for many helpful conversations and suggestions. A.V. would like to thank Benjamin Bakker, Andrew Snowden and Jacob Tsimerman for interesting discussions. B.L. would like to thank Zeb Brady, Lalit Jain, Daniel Litt, and Johan de Jong.
Dan Abramovich, Raymond Cheng, Brian Conrad, Kiran Kedlaya, and Bjorn Poonen made many helpful comments on drafts of this paper.
We thank Brian Conrad for pointing out the proof of Lemma LABEL:Artss, and for simplifying the proof of Lemma LABEL:padic_BT. We thank Jordan Ellenberg for an interesting discussion about monodromy.
During much of the work on this paper, B.L. was supported by a Hertz fellowship and an NSF fellowship and A.V. was supported by an NSF grant. During the final stages of writing A.V. was an Infosys member at the Institute for Advanced Study. We thank all these organizations for their support of our work.
2. Notation and preparatory results
We gather here some notation and some miscellaneous lemmas that we will use in the text. We suggest that the reader refer to this section only as necessary when reading the main text.
The following notation will be fixed throughout the paper.
a number field
a fixed algebraic closure of
the absolute Galois group
a finite set of finite places of containing all the archimedean places
the ring of -integers
when is understood
a (rational) prime number such that no place of lies above
the completion of at a prime of
a fixed algebraic closure of
the residue field at
the cardinality of
the residue field of , which is an algebraic closure of
the localization of at
By a -set we mean a (discretely topologized) set with a continuous action of .
For a variety over a field of characteristic zero, we denote by the de Rham cohomology of . If is a field extension, we denote by the de Rham cohomology of the base-change , which is identified with .
For any scheme , a family over is an (arbitrary) -scheme . A curve over is a family over for which is smooth and proper of relative dimension and each geometric fiber is connected. (Note that we will also make use of “open” curves, for example in §LABEL:Sunit, but we will avoid using the word “curve” in that context.)
Let be a finite unramified extension of , and the unique automorphism of inducing the -th power map on the residue field. By -module (over ) we will mean a pair , with a finite-dimensional -vector space and a map semilinear over . A filtered -module will be a triple such that is a -module and is a descending filtration on . We demand that each be an -linear subspace of but require no compatibility with . Note that the filtered -modules arising from Galois representations via -adic Hodge theory satisfy a further condition, admissibility, but we will make no use of it in this paper (see [Asterisque, Exposé III, §4.4] and [Asterisque, Exposé III, §5.3.3]).
2.1. Linear algebra
Suppose that is a field automorphism of finite order , with fixed field . Let be an -vector space of dimension , and a -semilinear automorphism. Define the centralizer of in the ring of -linear endomorphisms of via
it is an -vector space. Then
where is now -linear. In particular, .
Let be an algebraic closure of , and let be the set of -embeddings . Then is a -module, and splitting by idempotents of we get a decomposition
where consists of such that for all . (Here the multiplication is for the -module structure, and for the -module structure, on .) Moreover, extends to an -linear endomorphism of ; this endomorphism carries to .
Fix ; then projection to the factor induces an isomorphism
Now is obtained by base extension from the -linear map ; in particular, the dimension of the centralizer on the right is the same as , whence the result. ∎
Let be a finite-index inclusion of groups, and let be a semisimple representation of the group over the characteristic-zero field . Then the induction is also semisimple.