Families of abelian varieties with many isogenous fibres
Let be a subvariety of the moduli space of principally polarised abelian varieties of dimension over the complex numbers. Suppose that contains a Zariski dense set of points which correspond to abelian varieties from a single isogeny class. A generalisation of a conjecture of André and Pink predicts that is a weakly special subvariety. We prove this when using the Pila–Zannier method and the Masser–Wüstholz isogeny theorem. This generalises results of Edixhoven and Yafaev when the Hecke orbit consists of CM points and of Pink when it consists of Galois generic points.
2010 Mathematics Subject Classification:11G18, 14K02
Let denote the Siegel moduli space of principally polarised abelian varieties of dimension . We consider the following conjecture. In particular we prove the conjecture when is a curve, and make some progress on higher–dimensional cases.
Let be the isogeny class of a point . Let be an irreducible closed subvariety of such that is Zariski dense in . Then is a weakly special subvariety of .
Conjecture 1.1 holds when is a curve.
Let be the isogeny class of a point . Let be an irreducible closed subvariety of such that is Zariski dense in .
Then there is a special subvariety which is isomorphic to a product of Shimura varieties with , and such that
for some irreducible closed subvariety .
Theorem 1.3, but not Theorem 1.2, depends on results concerning the hyperbolic Ax–Lindemann conjecture from recent preprints of Pila and Tsimerman [pila-tsimerman:ax-lindemann] and of Ullmo [ullmo:hyp-ax-lind].
Conjecture 1.1 is a consequence of the Zilber–Pink conjecture on subvarieties of Shimura varieties [pink:zp]. For a statement of the Zilber–Pink conjecture and proof that it implies Conjecture 1.1, see section 2.
Conjecture 1.1 is slightly more general than the case of the following conjecture of André and Pink, because the isogeny class of is sometimes bigger than the Hecke orbit: by isogeny class we mean the set of points such that the corresponding abelian variety is isogenous to , with no condition of compatibility between isogeny and polarisations. On the other hand the Hecke orbit consists of those points for which there is a polarised isogeny between the principally polarised abelian varieties – that is, an isogeny satisfying , where and are the polarisations. In the case of there is no difference between Hecke orbits and Pink’s generalised Hecke orbits.
[[andre:g-functions] Chapter X Problem 3, [pink:conj] Conjecture 1.6] Let be a mixed Shimura variety over and the generalised Hecke orbit of a point . Let be an irreducible closed algebraic subvariety such that is Zariski dense in . Then is a weakly special subvariety of .
Some cases of Conjecture 1.1 are already known: If the point is Galois generic, then the isogeny class and the Hecke orbit coincide, and Conjecture 1.1 follows from equidistribution results of Clozel, Oh and Ullmo, as was shown by Pink [pink:conj]. When is a special point, Theorem 1.2 was proved by Edixhoven and Yafaev [edixhoven-yafaev] by exploiting the fact that Galois orbits of special points are contained in for suitable Hecke operators and these Galois orbits tend to be large compared to the degree of . When corresponds to a product of elliptic curves, Habegger and Pila [habegger-pila:unlikely] proved the theorem using the method we extend here.
The terminology “weakly special subvariety” was introduced by Pink [pink:conj], although the concept was first studied by Moonen [moonen:linearity-I]. Moonen showed that a subvariety of a Shimura variety is totally geodesic (in the sense of differential geometry) if and only if it satisfies the following definition. An algebraic subvariety of is called weakly special if there exist a sub-Shimura datum of , a decomposition
and a point such that is the image in of . In other words, to say that is weakly special means that we can choose , , in the conclusion of Theorem 1.3 such that is a single point in . For more details, see section LABEL:subsec:special-subvars.
In this article we will use a characterisation of weakly special subvarieties due to Ullmo and Yafaev [uy:characterisation]: is weakly special if and only if an irreducible component of is algebraic, where is the quotient map and is the Siegel upper half space. Here we call a subvariety of algebraic if it is a connected component of for some algebraic variety . In order to prove Theorem 1.3 we require a strengthening of this characterisation called the hyperbolic Ax–Lindemann conjecture: if is a maximal algebraic subvariety of , then is algebraic. A proof of the Ax–Lindemann conjecture for was recently announced by Pila and Tsimerman [pila-tsimerman:ax-lindemann].
Our proof of Theorems 1.3 and 1.2 follows the method proposed by Pila and Zannier for proving the Manin–Mumford and André–Oort conjectures [pila-zannier]. This is based upon counting rational points of bounded height in certain analytic subsets of , and applying the Pila–Wilkie counting theorem on sets definable in o-minimal structures.
The central part of the proof of Theorems 1.3 and 1.2 is in section LABEL:sec:main-thm. This uses a strong version of the Pila–Wilkie counting theorem involving definable blocks. The other ingredients are an upper bound for the heights of matrices in relating isogenous points, proved in section LABEL:sec:matrix-heights, and a lower bound for the Galois degrees of principally polarised abelian varieties in an isogeny class, derived from the Masser–Wüstholz isogeny theorem [mw:isogeny-avs].
In section LABEL:sec:isog-fg-fields we use a specialisation argument to prove a version of the Masser–Wüstholz isogeny theorem for finitely generated fields of characteristic , generalising the original theorem which was valid only over number fields. This is necessary in order to prove Theorems 1.3 and 1.2 for points and subvarieties defined over and not only over .
Now we consider some generalisations of Theorem 1.2. Theorem 1.2 immediately implies Conjecture 1.4 for curves in Shimura varieties of Hodge type, if we restrict to usual Hecke orbits. This is because, by the definition of a Shimura variety of Hodge type, there is a finite morphism for some such that the image of each Hecke orbit in is contained in a Hecke orbit of , and is weakly special if and only if is weakly special. However this does not imply Conjecture 1.4 for generalised Hecke orbits in Shimura varieties of Hodge type, as a generalised Hecke orbit in may map into infinitely many isogeny classes in .
Because of the use of the Masser–Wüstholz theorem, our method applies only to Shimura varieties parameterising abelian varieties i.e. those of Hodge type. In particular, let us compare with Theorem 1.2 of [edixhoven-yafaev]. Take any Shimura datum . Edixhoven and Yafaev generalise Hecke orbits by choosing a representation of and considering a set of points where the induced -Hodge structures are isomorphic. The Masser–Wüstholz theorem can be used only when these Hodge structures have type . Hence our method lacks a key advantage of Edixhoven and Yafaev’s formulation, namely that they can replace by a subgroup so that is Hodge generic, or by its adjoint group.
This restriction to isogeny classes of abelian varieties rather than generalised Hecke orbits is related to our inability to prove the full Conjecture 1.1. In the case of the André–Oort conjecture, a conclusion as in Theorem 1.3 implies the full conjecture by induction on (see [ullmo:hyp-ax-lind]). This is because, when is of the form , special points in project to special points in .
This does not work for Conjecture 1.1 because the hypothesis that contains a dense set of points from a single isogeny class does not imply the same thing for , where we fix a point in order to realise as a subvariety of . The problem is that the decomposition need not have an interpretation in terms of moduli of abelian varieties. For example this happens in André and Borovoi’s example of a subvariety which decomposes as a product of Shimura varieties but where the generic abelian variety in the family parameterised by is simple.
I am grateful to Emmanuel Ullmo for suggesting to me the problem treated in this paper and for regular conversations during its preparation. I would also like to thank Barinder Banwait for his comments on an earlier version of the manuscript, and Gaël Rémond for remarks on LABEL:isogeny-bound. I am grateful to the referee for their detailed comments.
This paper was published in Journal für die reine und angewandte Mathematik (Crelles Journal), 2015, Issue 705, p. 211–231 (DOI: 10.1515/crelle-2013-0058). The published version is available at www.degruyter.com.
There is a gap in the proof of LABEL:matrix-height-bound-fg in the published version of this paper. This was discovered by Gabriel Dill during his ongoing PhD studies (2016). I am very grateful to Gabriel both for pointing out this gap and for finding a method of fixing it, which has been added to the arXiv version of the paper as LABEL:correction:rat-rep-height-bound-plus.
2. Shimura varieties and the Zilber–Pink conjecture
In this article we consider only the moduli space of abelian varieties and its subvarieties, however to place Conjecture 1.1 in its proper context we need to consider Shimura varieties. For the convenience of the reader, we briefly summarise the theory of Shimura varieties and the Zilber–Pink conjecture in this section. This contains no original material: the primary sources are [deligne:shimura-vars] for Shimura varieties, [moonen:linearity-I] for special and weakly special subvarieties (special subvarieties are what Moonen calls subvarieties of Hodge type) and [pink:zp] for the Zilber–Pink conjecture. We also prove that the Zilber–Pink conjecture implies Conjecture 1.1 by an argument which is essentially due to Pink.
2.1. Shimura varieties
A Shimura datum is defined to be a pair where is a reductive algebraic group over and is a -conjugacy class of homomorphisms
satisfying the following conditions:
the Hodge structure on the adjoint representation of induced by has type contained in ;
induces a Cartan involution of the adjoint group of ;
has no factor defined over on which the projection of is trivial.
Under these conditions, each connected component of is a Hermitian symmetric domain on which the identity component of acts holomorphically.
Let be a compact open subgroup of , where is the ring of finite adeles. We define
where acts diagonally on on the left and acts on only on the right. Deligne [deligne:shimura-vars] showed that can be given the structure of an algebraic variety over a number field, called a Shimura variety.
Choose a connected component of . Let be the preimage of the identity component (in the analytic topology) of in ; this is the stabiliser of . Let .
The image of in is called the neutral component of . As a complex manifold it is canonically isomorphic to
where is a congruence subgroup of .
Let be a Shimura datum and let be the adjoint group of , where is the centre of . We get a new Shimura datum by letting be the -conjugacy class of morphisms containing for , where is the quotient map . The map is an injection whose image is a union of connected components of ([moonen:linearity-I] section 2.1).
2.2. Moduli of abelian varieties
The fundamental example of a Shimura variety is the moduli space of principally polarised abelian varieties of dimension . We recall briefly that a polarisation of an abelian variety of is an isogeny to the dual variety satisfying a certain positivity condition. Any polarisation induces a symplectic form . A polarisation is principal if it has degree – that is if it is an isomorphism .
The moduli space of principally polarised abelian varieties of dimension is associated with the Shimura datum where is the group of symplectic similitudes and the conjugacy class of Hodge parameters can be identified with the union of the Siegel upper and lower half spaces
The action of on is given by
Taking , we get the Shimura variety whose -points are in bijection with isomorphism classes of principally polarised abelian varieties of dimension over .
2.3. Hecke correspondences
Let be a Shimura datum and a compact open subgroup. Choose and let . The inclusion induces a finite morphism
Because is normalised by , we also have an automorphism which sends the double coset to . We define a finite correspondence on by