The Mordell-Lang conjecture is a generalization of the well-known Faltings’s theorem on finiteness of rational points on curves of genus at least 2. The statement is roughlt stated as follows:
Let be a number field and an abelian variety over . The intersection of an algebraic subvariety with a subgroup of finite rank is contained in a finite union of cosets (of abelian subvarieties of ) contained in .
Thus the points of are captured by finitely many translates of abelian subvarieties. It implies Faltings’s theorem using Jacobian and the Abel-Jacobi map. The purpose of this post is try to explain Anand Pillay’s proof Mordell-Lang for function fields in characteristic zero. The primary reference is his 2004 paper Mordell–Lang conjecture for function fields in characteristic zero, revisited.
The formal statement is as follows:
Let be algebraically closed fields of characteristic zero and an abelian variety over . Let be an irreducible subvariety of over and a finite rank subgroup of (i.e. is a finite-dimensional -vector space). Suppose is Zariski dense in and has trivial stablizer in , then up to a translate is contained in an abelian subvariety of that descends to an abelian variety defined over .
This is firstly proved by Buium in 1992. The positive characteristic case is first done by Hrushovski using model-theoretic method (Zilber’s trichotomy for Zariski geometry). Note that the conclusion here is stronger than the one in Hrushovski’s paper, since we assume in addition that has trivial stablizer.
A result from complex geometry using deformation-theoretic method
To motivate Pillay’s proof we mention the following amazing result due to Ueno:
If is a complex torus and an analytic subvariety of with trivial stabilizer, then is an algebraic variety.
This feels like a Chow’s theorem for complex tori: Even though complex tori are not algebraic in general, their ‘generic’ analytic subvarieties are. The proof of these results illustrate the power of taking derivative in diophantine questions over function fields. (E.g. Fermat’s last theorem for function fields is a triviality simply by differentiating the equation .) In fact, I believe the Gauss-Manin connection (a way to differentiate cohomology classes) plays a crucial role in all approaches to Mordell-type conjectures, in one way or another. As an example, a recent approach to Faltings’s theorem by Lawrence-Venkatesh uses the -adic period mapping, whose construction boils down to the Gauss-Manin connection.
Going back to Ueno’s result, it is an easy corollary of the following infinitesimal version of Chow’s theorem by Campana.
Let be an analytic space, an analytic family of analytic compact cycles in parametrized by a compact analytic space and denote the graph of . There is a bimeromorphic (birational?) embedding of into for some that is compatible with projection to . Here denotes the sheaf of differential operators of order and is the total space of the Grassmanian sheaf of -dimensional subspaces of .
This thoerem implies that a stablizer-free analytic subvariety is Moishezon (having enough meromorphic functions), and being a subvariety of the complex torus it also Kähler, hence projective by Moishezon’s theorem.
This suggests deformation-theoretic method (combined with some finiteness assumption such as compactness) can be useful in showing algebracity result, once we are in a setting that allows us to take derivative. One natural setting for doing this is differential algebraic groups.
Preliminaries on differential algebra
We first define what an algebraic -group is and this requires the language of differential algebra. A good reference is this paper by Pillay. Let be a field (characteristic zero) with a derivation and the field of constant (elements killed by ). Note that WLOG, we can extend scalar to make differentially closed (this means that any finite system of differential polynomial equations and inequations over that has a solution in some extension of already has a solution over , analogous to being algebraically closed). An algebraic -variety (defined by Buium) is an irreducible variety over equipped with a derivation on the structure sheaf of extending (so it allows us to differentiate sections in $X(K)). If is an algebraic group over and is compatible with the multiplication and inversion of , then is an algebraic -group.
We define the -twisted tangent bundle as follows: If is locally cut out by over , then is locally cut out by these equations together with for where are new variables and denotes the polynomial obtained by differentiating the coefficients of with . Note that if is defined over , the coefficients of are in and thus annhilated by , in which case is the tangent bundle of .
Note that if is an algebraic -group, there is a natural algebraic -group structure on given by sending We also have a natural algebraic -group homomorphism given by projection.
Pillay actually define algebraic -group in a more explicit way, by replacing the derivation with the equivalent datum of a section which is also a homomorphism over ; it gives a -rational splitting of as a semidirect product of and , the Lie algebra of thought of as a -variety. This also amounts to the datum of a -rational that is a left inverse to and a crossed homomorphism, i.e. we have for . We call a -subvariety if maps to .
Arguably the most important definition is the following: giving an algebraic -variety , we define (or if is understood) to be If is the zero section and is defined over , this is precisely , which is a much smaller object than . (Pillay says it is a finite-dimensional differential algebraic group, I haven’t looked into the definition of it but it presumably has something to do with some model-theoretic dimension?) On the other hand, is Zariski-dense in (this reminds me of the fact that the -points in an algebraic variety defined over is Zariski dense, for solution see this mathoverflow post).
Main theorem
Now we can state and prove an analogous algebracity result in this differential algebraic setting:
Suppose that is an algebraic -group, is a -subvariety of with trivial stablizer. Assume also that and generates . Then comes from , i.e. there exists an algebraic group defined over such that is isomorphic to where is the zero section.
The conclusion is commonly called (strongly?) isotrivial in deformation theory literature.
Proof sketch: The -jet of at is the dual space of where is the maximal ideal of the local ring of at . This is a finite dimensional -vector space. Similarly, we define for any subvariety of containing . The general principle is again
If is a member of an algebraic family of subvarieties, all passing through ,then is determined (in this family) by for sufficiently large .
In this case this is almost immediate since the family is algebraic to begin with. Since has trivial stablizer, for , we have implies . If , contains . Thus by the general prinicple, for sufficiently large , the map taking gives a birational embedding of into . The dual space is equipped with a connection over , which in turn induces one on . Since is differentially closed, we have a fundamental system of solutions of the equation , that is, there exists a tuple of elements of which is simultaneously an -basis of and a -basis of the solution space . (This certainly reminds me of some constructions in Riemann-Hilbert correspondence and -adic Hodge theory.)
Now suppose , then is a -variety of , and essentially the same argument as above applies and shows that , a -submodule of , admits a finite -basis that is also a -basis for the solution space . Thus is a -rational point of and we have obtained a birational isomorphism of with a subvariety of such that for generic , is rational over . Note that, as is Zariski-dense in , is defined over and it is not hard to take it from here.
To apply this theorem to Mordell-Lang it remains to produce a finite-dimensional differential algebraic subgroup of containing . This is done by Buium in the paper mentioned above. More precisely, his construction gives the following:
There is a connected commutative algebraic -group and a surjective homomorphism (of algebraic groups) such that:
is unipotent (and is thus the unipotent radical of G,as is abelian);
is injective;
.
Using this it is not hard to finish the proof. Maybe I will come back and revisit his construction, as it seems rather magical!