A bound for the torsion on subvarieties of abelian varieties
We give a uniform bound on the degree of the maximal torsion cosets for subvarieties of an abelian variety. The proof combines algebraic interpolation and a theorem of Serre on homotheties in the Galois representation associated to the torsion subgroup of an abelian variety.
The problem of understanding the distribution of torsion points in subvarieties of abelian varieties was independently raised by Manin and Mumford, who stated the following conjecture.
Conjecture 1.1 (Manin-Mumford).
Let be an algebraic curve of genus , defined over a number field and embedded in its Jacobian . The set of torsion points of which lie in is finite.
It was proved in 1983 by Raynaud [Raynaud83], who soon generalized his theorem to arbitrary subvarieties of an abelian variety.
Theorem 1.2 (Raynaud, [Raynaud83a]).
Let be a subvariety of an abelian variety defined over a number field. Then the Zariski closure of the set of torsion points in is a finite union of translates of abelian subvarieties of by torsion points.
This statement is still true if is replaced by a semi-abelian variety defined over a number field (see [Hindry88]) or even any field of caracteristic (see [McQuillan95]). The aim of this paper is to give a precise bound for the degree of the maximal torsion cosets (translates of abelian subvarieties by torsion points) that appear in Raynaud’s theorem.
1.1. The number of torsion points on curves
In the case of curves, the most spectacular quantitative versions of Raynaud’s theorem are related to a famous conjecture of Coleman. Let be a curve of genus embedded in its Jacobian , all defined over a number field . We denote by the torsion subgroup of and by the finite torsion subset of .
Conjecture 1.3 (Coleman, [Coleman87]).
Let be a prime ideal of above a rational prime such that the following conditions are satisfied:
- is unramified at ,
- has good reduction at .
Then the extension is unramified above .
Using -adic integration theory, Coleman managed to prove his conjecture in several significant cases, for instance if or if has ordinary reduction at . The ramification properties of the field generated by over provide valuable information in view of a quantitative version of the Manin-Mumford conjecture.
Theorem 1.4 (Coleman, [Coleman85]).
Assume that and satisfy the hypotheses of Conjecture 1.3. If has ordinary reduction at and has (potential) complex multiplication, then
The assumptions of the theorem are needed to get a bound of this strength, which is sharp ([Boxall-Grant00, Coleman85]). Using -jets and Coleman’s work, Buium gave an almost unconditional estimate, which can be expressed in the following (slightly weaker) form.
Theorem 1.5 (Buium, [Buium96]).
If and satisfy the hypotheses of Conjecture 1.3:
A suitable choice of can easily be expressed in terms of the discriminant of and the conductor of .
1.2. Uniform bounds in greater dimension
Let now be an abelian variety of dimension , defined over a number field and equipped with an ample line bundle , so that we can define the degree of a subvariety of . The number of maximal torsion cosets associated to can be bounded in terms of and . In fact, Bombieri and Zannier [Bombieri-Zannier96] showed that an uniform estimate can be found, where the dependence on is reduced to its geometric degree.
Later, Hrushovski gave a new proof of the Manin-Mumford conjecture through model theory, which yielded an explicit uniform bound.
Theorem 1.6 (Hrushovski, [Hrushovski01]).
The number of maximal torsion cosets of a subvariety of can be bounded as follows:
Remarks. The symbol means that the stated inequality is true after possibly multiplying the right member by a positive real number depending on . The number is doubly exponential in , and it also depends on a prime of good reduction for .
A breakthrough came with the work of Amoroso and Viada on the effective Bogomolov conjecture for subvarieties of an algebraic torus. They observed that it was more relevant in this setting to describe the geometry of with another parameter. Let be a semi-abelian variety; since it is quasi-projective, we can define the degree of any subvariety of .
If is a subvariety of , let be the smallest such that is the intersection of hypersurfaces of which have degree at most .
Let be a subvariety of and be its torsion subset. It is possible to bound the degree of the -equidimensional part of the Zariski closure of , for an integer .
In the case of tori, Amoroso and Viada’s theorem concerns the Zariski density of points of small height, but it has the following consequence.
Theorem 1.8 (Amoroso-Viada, [Amoroso-Viada09]).
Let and . For any :
Their result was later improved by the second author. Mixing their strategy with ideas of Beukers and Smyth [Beukers-Smyth02] more suited to the study of torsion points, one may find a bound with optimal dependance on .
Theorem 1.9 (Martínez, [Martinez17]).
If and , then
A straightforward consequence of this theorem is an estimate on the number of maximal cosets. A further study provides a variant of this bound which proves conjectures of Aliev and Smyth [Aliev-Smyth:2012], and Ruppert [Ruppert:1993].
1.3. New bounds for the torsion on subvarieties of abelian varieties
Our main theorem is an estimate of the same strength for subvarieties of an abelian variety of dimension defined over a number field (with fixed embedding in projective space).
Combining algebraic interpolation with a theorem of Serre on homotheties in the Galois representation associated to the torsion points of , we prove the following bound.
If is a subvariety of and , then
This immediately translates into a bound for the number of maximal torsion cosets. In the equidimensional case, this only depends on the degree of .
The number of maximal torsion cosets of a subvariety of is bounded as follows:
In the case of curves, we get an improved bound.
Let be a curve in which is not a torsion coset. Then
The dependence on in these three estimates will be explicited below in terms of the constant that appears in Serre’s theorem (which is still rather mysterious).
Remark. Some results on the effective Bogomolov problem ([Galateau10] under a conjecture of Serre on the ordinary primes of , or [Galateau12] for a hypersurface) may be combined with Amoroso and Viada’s method to yield explicit bounds which are polynomial in but weaker than Theorem 1.10 (or even an abelian analogue of Theorem 1.8). It is not suprising since this approach does not fully exploit the properties of torsion points.
The article is organized as follows. In the next section, we discuss and study alternate measures for the degree of subvarieties of , introduce Hilbert functions and recall the classical upper bound (resp. lower bound) proved by Chardin (res. Chardin and Philippon).
In the third section, we state Serre’s theorem, which is a first step towards a famous conjecture of Lang on homotheties in the Galois group of the extensions generated by torsion points of . We use it several times to locate the torsion subset of an irreducible subvariety of that is not a torsion coset. We thus show that , where is an algebraic set that satisfies some important properties and can be described precisely in terms of . We then prove Theorem 1.12, where algebraic interpolation is not needed and a simple application of Bézout’s theorem is sufficient to conclude.
In the last section, our estimates on Hilbert functions allow us to interpolate by a hypersurface of retaining most of the crucial information contained in . We give a proof of Theorem 1.10, and we will finally discuss its optimality in terms of .
Unless stated otherwise, we fix throughout this paper an abelian variety of dimension defined over a number field . We also fix an ample line bundle on . After possibly replacing by , we will assume that is very ample and defines a normal embedding into some projective space . In addition, after possibly tensorizing by , we may assume that is symmetric. By abuse of language, we will say in this article that a real number depends on when it depends on both and .
A projective embedding being fixed, we may now identify every subvariety of (not necessarily irreducible or equidimensional) with its image in . The field of definition of will be denoted by . We will say that is non-torsion if it is not a torsion coset.
We also let be the torsion subgroup of over , and the field generated over by . If is a positive integer, the -torsion subgroup of will be denoted by , and its field of definition by .
2. Geometric preliminaries
Our approach relies strongly on fine interpolation results which follow from estimates on the Hilbert function proved by Chardin and Philippon. Before stating them at the end of this section, we will need to recall some basic geometric properties of abelian varieties, and then introduce various measures of the geometric degree for a subvariety of , that naturally appear in our bounds on Hilbert functions.
2.1. Classical facts on abelian varieties
We gather here classical properties about the geometry of abelian varieties, morphisms and stabilizers, which will be used frequently in the sequel. Let us start with a precious information concerning the translations in .
Lemma 2.1 (Lange-Ruppert, [Lange-Ruppert85]).
The translations in can be defined by homogeneous polynomials in with degree at most .
Notice that a projectively normal embedding in is needed here. Without this assumption, the degree of the homogeneous polynomials can not be so explicitly bounded.
The degree of a subvariety of is invariant under translation. We now describe how it behaves under some isogenies. Let be an irreducible subvariety of , and for a non-zero integer , denote by the isogeny: . By Lemme 6 of [Hindry88], we have
Suppose now that is non-torsion. We will exploit this assumption by looking at the stabilizer of .
The stabilizer of is the algebraic subgroup of defined by
Because we assume that is non-torsion, it follows from Bézout’s theorem ([Hindry88], Lemme 8) that
By Poincaré’s complete reducibility theorem, the abelian variety is isogenous to a product
where is the connected component of which contains , and is an abelian subvariety of ([Hindry88], Lemme 9). After composing with an isogeny whose kernel is , we find a surjective homomorphism
with . Taking large enough so that all the simple factors of are defined over , we may assume that is defined over .
2.2. Degrees of definition and Hilbert functions
A key point in our approach is to use some (classical) refined variants of the projective degree. If is a subvariety of , we define its degree to be the sum of the degrees of its irreducible components. We have already introduced as the minimal degree of hypersurfaces of with intersection . The next definition only retains the projective nature of .
The degree of incomplete (resp. complete) definition of , denoted by (resp. ), is the minimal such that the irreducible components of are irreducible components of an intersection (resp. is an intersection) of hypersurfaces in which all have degree at most .
For a family of subvarieties of , we easily get the following inequality, see for instance ([Martinez17], Lemma 2.6):
We have the following inequalities between our different degrees.
If is an equidimensional variety, we have
The first inequality is straightforward. The image of by a linear map has degree at most , and the variety is the intersection of hypersurfaces of obtained by pull-backs of such linear maps. This shows that
Now, if is a hypersurface of , we have . We choose a set of hypersurfaces of of degree at most and such that . We get:
Assume finally that where the ’s are hypersurfaces of . After possibly removing some of the ’s, we have , where is a hypersurface of . The last inequality is then a direct consequence of Bézout’s theorem. ∎
The degrees of complete and incomplete definition do not necessarily behave as the usual degree with respect to translations in . However, we have the following useful comparison.
Let be a subvariety of . If , then
By Lemma 2.1, if we have a set of complete (resp. incomplete) equations of degree for , we get a set of complete (resp. incomplete) equations of degree for . This proves the announced inequalities. ∎
We now introduce the Hilbert function that can be attached to any projective variety. The incomplete degree of definition naturally arises in a classical lower bound on this function, which explains why we needed to introduce and study this degree.
To a subvariety , there corresponds the homogeneous ideal of polynomials of which vanish on . This defines a graded -module . For a positive integer, we let
the Hilbert function of at . We start with a classical upper bound on the Hilbert function.
Theorem 2.6 (Chardin, [Chardin89]).
If is an equidimensional variety of dimension and , then
On the other hand, a refined version of Chardin and Phillipon’s theorem on Castelnuovo’s regularity yields a lower bound for the Hilbert function when is large enough in a precise sense. The following is Théorème 6.1 of [Amoroso-Viada12] (see [ChardinPhilippon99], Corollaire 3 for the original statement).
Theorem 2.7 (Chardin-Philippon).
Let an union of equidimensional varieties of dimension and . For any integer , we have
Remark. Combining these two bounds provides a powerful interpolation tool, yielding for a well chosen pair of varieties a hypersurface of controlled degree which contains and avoids .
3. Galois properties of torsion points and geometric consequences
In this section, we use a deep theorem of Serre on Galois representations to locate the torsion subset of a subvariety of . Our strategy is primarily based on the classical approach to the Manin-Mumford conjecture initiated by Lang in [Lang65].
3.1. Homotheties in the image of Galois
For every prime number , let be the -adic Tate module of . There is a representation
induced by the action of on the torsion subgroup of . A long-standing conjecture of Lang states that the image of the absolute Galois group in the adelic representation
contains an open subgroup of the group of homotheties. In [Bogomolov80], Bogomolov proved that for a fixed prime number , the group contains an open subgroup of the group of homotheties. Serre later showed that contains a fixed power of every admissible homothety. We will use the following version of his theorem.
Theorem 3.1 (Serre).
There is an integer such that, for any two coprime positive integers and , there exists an automorphism satisfying
This is [Wintenberger02], Théorème 3. See also [Serre86] p.136, Théorème 2 for the original statement, or [Hindry88], Lemme 12. ∎
Remark. The problem of finding an explicit in terms of - and of the field over which is defined - is still open and discussed in [Wintenberger02], Section 2. In order to simplify notations, and since is fixed, we will write for in the sequel.
Let be an irreducible non-torsion subvariety of . Serre’s theorem is our main tool to find a strict subvariety of that contains the torsion subset of . We let , where was defined in (2).
We distinguish several cases according to wether and how , and first tackle the simpler case where is not contained in .
If , there is a conjugate under the action of such that
The isogeny is defined over , so the extension is strict and there is a nontrivial field isomorphism such that
Since acts trivially on , the lemma follows. ∎
3.2. Scanning the field of definition
For the remaining of this section, we now assume that . In comparison with the toric case, some technical difficulties arise here because of the complexity of , and because of the gap between Lang’s conjecture and Serre’s theorem.
We start with a preliminary examination of , and we define two integers and that quantify more precisely the link between and . Since is a number field, there is an integer such that
We denote by the -adic valuation of an integer. For the remaining of the section, fix the smallest integer such that the latter inclusion holds, and
where . For , we consider the subset of the integers
This set contains because and . Let be the biggest integer not in , and
In particular, notice that both and are always negative. Finally, we set
We will use repeatedly the following computation based on the properties of binomial coefficients.
Let be two integers. If is an integer with :
If , then , and therefore
This inequality is still true for , so the proof of the lemma is complete. ∎
We will need to compare the -adic valuations of and .
The lemma follows directly from these inequalities:
This is trivially true if , so we can assume that . If , we compute
By choice of , . Hence, , and Lemma 3.3 shows that
Now, for all , the variety is defined over so and the stated inequality holds. ∎
Remark. With being fixed, we can associate in exactly the same way an integer to each translate of by an -torsion point . For a good choice of , we get that . After possibly replacing (resp. ) by (resp. ), we will now assume that . This will have no effect on our subsequent geometric construction, because the properties of that we want to prove are invariant under translation by a torsion point.
3.3. The torsion subset of
We are ready to locate the torsion of when . The following proposition gives an explicit description of an algebraic subset of that contains .
There are two automorphisms depending only on , such that if
Fix with exact order , and let
Since is odd, Bézout’s identity yields an odd positive integer and an integer such that
We consider different cases according to .
Case 1. Assume on the one hand that . Let
Since the integers and are coprime, we are in a position to use Theorem 3.1. This gives an automorphism such that
Looking at the action of on , we obtain:
where has order dividing . In particular , since
We immediately get
By construction, we also have
with , so the action of on does not depend on .
Case 2. Assume on the other hand that . We first examine the case where . Let
Remark that the prime divisors of also divide . Therefore and are coprime and Theorem 3.1 gives an automorpshim such that
We check that , so Lemma 3.3 shows that, for any integer ,
Hence, the action of on yields
where has exact order . So, we find that . Furthermore, we have that
So and we have , by definition of . We derive , and .
Case 3. Assume finally that . Let
which is an integer by the assumption. Again, and are coprime, and Theorem 3.1 ensures that there exists an automorphism such that
Notice that , so by Lemma 3.3, for any integer ,
From this, we derive
where . Hence
We check that the action of on does not depend on , since
Remark. The proof of the proposition shows that we can choose and such that
where the components of concerning only appear when . This completely describes the action of and on .
3.4. Pulling back from to .
At this point, the important condition that does not lie in the algebraic set is still missing in our construction. This will be fixed by pulling back from to and exploiting the classical properties of the stabilizer. Let be the preimage of by , i.e.
for chosen as in Proposition 3.5 (and the remark below the proof of this proposition).
First remark that
so because is an isogeny defined over . Thus
We now check that the inclusion is strict, and we cut the proof in three pieces corresponding to each type of component of .
Case 1. Suppose first that there is such that
Using for instance [Hindry88], Lemme 6, we compare the degrees of these two algebraic sets and find
which is a contradiction since is not a torsion subvariety of .
Case 2. This is where we really use the properties of our isogeny . Suppose that . Then we have that , and so
Case 3. After composing by , we are reduced to considering the possibility that there exist such that . We can find a torsion point such that . We may also assume that (see the remark following the proof of Proposition 3.5). Using Lemma 3.3, we see that
So is fixed by and . Furthermore, if , there is such that and . We compute
Since , we can apply Lemma 3.3 again to obtain, for ,
This means that , and so
Thus , and we finally have
which yields a contradiction. ∎
3.5. The case of curves
Let be an irreducible non-torsion curve. Then
We notice that since is non-torsion, its stabilizer is finite. Let and assume that . By Lemma 3.2, there is a conjugate of such that
We are thus in a position to use Bézout’s theorem:
and this bound is stronger than that stated in the proposition.
Remarks. If is not defined over , our proof gives a much stronger bound:
If we consider the case where is the Jacobian of , both and are defined over the same number field (in particular, the third union in the definition of disappears). If we consider the canonical embedding of the Jacobian, we have and a quick computation gives
4. Bounding the torsion through interpolation
We turn to the proof of our main theorem for general subvarieties of . Since a simple iterated application of Bézout’s theorem like in the case of curves would yield a bound far weaker than expected, we will follow a strategy based on the existence of a nice obstructing hypersurface through refined interpolation tools.
4.1. The interpolation machine
We first build a preliminary interpolation machine suited to our situation. This is mainly derived from Chardin and Philippon’s estimates for Hilbert functions.
Let be an irreducible subvariety of , as well as , and an integer.
If , there exists a hypersurface of such that , and .
If is non-torsion, there exists a hypersurface of such that , and .
Remark. The conclusion of (ii) implies that . In fact, this follows directly from the fact that is non-torsion, using the same argument as above in the proof of Lemma 3.6, Case 1.
where . Let . This is an equidimensional variety of dimension and degree . By Theorem 2.7, for any
Using both inequalities with , we obtain
Hence, there is a hypersurface of of degree such that and . The last inclusion implies that . Moreover, Lemma 2.5 gives , so we obtain:
concluding the proof of 1.
We now turn our attention to assertion 2. To simplify notations, we let
Recall here that given by (2) is assumed to be defined over , so that
Since is non-torsion, the variety is equidimensional of dimension and degree , where we denote . By Theorem 2.7, for any