Three-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field
For an abelian variety over a number field , we prove that the average rank of the quadratic twists of is bounded, under the assumption that the multiplication-by-3 isogeny on factors as a composition of 3-isogenies over . This is the first such boundedness result for an absolutely simple abelian variety of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field.
In dimension one, we deduce that if is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded. If is totally real, we moreover show that a positive proportion of twists have rank 0 and a positive proportion have -Selmer rank 1. These results on bounded average ranks in families of quadratic twists represent new progress towards Goldfeld’s conjecture – which states that the average rank in the quadratic twist family of an elliptic curve over should be – and the first progress towards the analogous conjecture over number fields other than .
Our results follow from a computation of the average size of the -Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny .
Let be an abelian variety over a number field . For any squareclass , the quadratic twist is another abelian variety over that becomes isomorphic to upon base change to (see Section 5). This recovers the usual notion of quadratic twist for elliptic curves: if is an elliptic curve with Weierstrass model , then has model , for any lift of to .
Quadratic twist families are the simplest possible families of abelian varieties—geometrically, they correspond to a single point in the moduli space. Nevertheless, we know very little about the arithmetic of such families. For example, for general abelian varieties , we know essentially nothing about the behavior of the ranks of the Mordell–Weil groups as varies. In the special case where is an elliptic curve over , Goldfeld conjectured that the average rank of is equal to , with of twists having rank 0 and having rank 1 . Even in this case, very little is known. Smith , building on work of Kane , has recently proved that in the very special case where , the average rank is at most . To date, this is the only case where the average rank of has been proven to be bounded.
In this paper, we prove the boundedness of the average rank of for a large class of abelian varieties , namely, those with an isogeny that factors as a composition of 3-isogenies over . As a consequence, over , we give the first known examples of elliptic curves not having full rational 2-torsion for which the average rank of is bounded, specifically, those having a rational subgroup of order 3. Over general number fields , we give the first known examples of elliptic curves of any kind for which the average rank of is bounded. We also provide the first known examples of absolutely simple abelian varieties over having dimension greater than one for which the average rank of is bounded, and indeed we give such examples over any number field and in arbitrarily large dimension.
The proofs of these results are accomplished through a computation of the average size of a family of Selmer groups, namely, the -Selmer groups associated to a quadratic twist family of 3-isogenies of abelian varieties. There have been a number of recent results on the average sizes of Selmer groups in large universal families of elliptic curves or Jacobians of higher genus curves (see, e.g., [2, 3, 5, 6, 39, 40, 44]) obtained via geometry-of-numbers methods. It has been a recurring question as to whether such methods could somehow be adapted to obtain analogous results in much thinner (e.g., one-parameter) families. The current work represents the first example of such methods being used to determine the average size of a Selmer group in a family of quadratic twists.
We now describe our results more precisely. Suppose admits an isogeny over . One can define a Selmer group , which is a direct generalization of the -Selmer group attached to the multiplication-by- isogeny on an elliptic curve (see Section 7). For each , there is also an isogeny and a corresponding Selmer group . Our main theorem computes the average size of as varies, in the case where has degree 3.
For any place of , we write for the -adic completion of . From the results of , one naturally expects that the average number of non-trivial elements of is governed by an Euler product whose factor at is the average size of the local -Selmer ratio
as varies over . This is exactly what we prove, but unlike in [1, Thm. 2], the Euler product turns out to be finite. This allows us to formulate our results as follows.
We call a subset of a local condition. If is infinite, then we give the discrete measure, and if is finite, we give the measure obtained by taking the usual (normalized) Haar measure of the preimage of in . This choice of measure gives rise to a natural notion of the average of over , which we denote by . In fact, it turns out that , regardless of , for every finite prime , where is the conductor of (Theorem 6.2). Hence this local average is equal to for all but finitely many primes.
Recall the definition of the global Selmer ratio:
We say that a subset is defined by local conditions if , where each is a local condition; if for all but finitely many , then we say that is defined by finitely many local conditions. We define the height of by
For any subset defined by finitely many local conditions, we may then define the global average of over all by
which is a rational number; here, . If , then consists of the squareclasses of all squarefree integers of absolute value less than , recovering the usual ordering of quadratic twists over .
Our main result is:
Let be a -isogeny of abelian varieties over a number field , and let be a non-empty subset of defined by finitely many local conditions. When the abelian varieties , , are ordered by the height of , the average size of is .
Theorem 2.1 has a number of applications to ranks of quadratic twists of abelian varieties.
Suppose is an abelian variety over a number field with an isogeny that factors as a composition of -isogenies over . Then the average rank of , , is bounded.
Theorem 2.2 gives the first known examples of absolutely simple abelian varieties of dimension greater than one whose quadratic twists have bounded rank on average. As an example, Theorem 2.2 applies to the Jacobian of the genus three curve over , or indeed over any number field . For a robust set of examples of abelian surfaces over to which Theorem 2.2 applies, see  and . In Section 12, we provide a large class of such examples of absolutely simple Jacobians of arbitrarily large dimension by considering the trigonal curves .
Theorem 2.1 also yields the first known examples of absolutely simple abelian varieties over a number field of dimension greater than one with a positive proportion of twists having rank 0. One example is the base change of above to the cyclotomic field , for which we prove that at least of twists have rank 0 (Theorem 12.5).
Any abelian variety obtains full level-three structure upon base change to a sufficiently large number field, so that the multiplication-by-three isogeny is the composition of 3-isogenies. We thus obtain the following immediate corollary of Theorem 2.2, which is new even in the case that is an elliptic curve.
Let be an abelian variety over a number field . Then there exists a finite field extension such that the average rank of the quadratic twist family , , of the base change is bounded.
When is an elliptic curve, the hypothesis in Theorem 2.2 is simply that admits a 3-isogeny, or equivalently, an -rational subgroup of order 3. Accordingly, we can be much more precise in this case. To state the result, we introduce the logarithmic Selmer ratio : the global Selmer ratio lies in , and we take .
Suppose is an elliptic curve admitting a -isogeny and let be defined by finitely many local conditions. Then the average rank of , , is at most .
Theorem 2.4 yields the first examples of elliptic curves over that do not have full rational two-torsion whose quadratic twists are known to have bounded average rank. The case of elliptic curves over with full rational two-torsion is the aforementioned work of Kane , which builds on prior work of Heath-Brown  and Swinnerton-Dyer .
Theorem 2.4 also yields the first examples of elliptic curves over number fields other than whose quadratic twists are known to have bounded average rank. While Goldfeld’s conjecture  on average ranks is only stated for quadratic twists of elliptic curves over , it has a natural generalization to elliptic curves over number fields (for example, see [24, Conjecture 7.12] and the ensuing discussion). Theorem 2.4 provides the first progress toward this general version of Goldfeld’s conjecture. We note that when the twists are ordered in a non-standard way (essentially by the number of prime factors of and the largest such factor), the boundedness of average rank has been demonstrated by Klagsbrun, Mazur, and Rubin  when .
The exact value of the upper bound in Theorem 2.4 depends on the primes for which has bad reduction and can be computed explicitly for any given curve . For a “typical” elliptic curve with a 3-isogeny, the bound on the average 3-Selmer rank, and thus average rank, given in Theorem 2.4 is roughly on the order of , where is the norm of the conductor of . In particular, the upper bound on the average rank in Theorem 2.4 can become arbitrarily large as the curves considered become more complicated.
Nonetheless, we are still able to prove that there are many twists of small rank in all but the most pathological families. Define, for each , the subset
and let denote the density of within .
Let be an elliptic curve over admitting a -isogeny . Then
The proportion of twists having rank is at least ; and
The proportion of twists having -Selmer rank is at least .
If we assume the conjecture that is even for all , then part (b) implies that a proportion of at least twists have rank .
For all but a pathological set of 3-isogenies , the sets and are non-empty and hence have positive density. In fact, over many fields, we show that these pathologies do not arise at all.
Let be an elliptic curve over and suppose admits a -isogeny.
If is totally real, then a positive proportion of have rank and a positive proportion of have -Selmer rank .
If has at most one complex place and at least one real place, then a positive proportion of have -Selmer rank .
If is imaginary quadratic, then a positive proportion of have rank or .
Furthermore, if has complex places, then there exists an elliptic curve over admitting a -isogeny and for which for all .
Recently and independently, for elliptic curves over with a -isogeny, Kriz and Li  have given a different proof of the fact that a positive proportion of twists have rank 0 and rank 1. In fact, their beautiful results in the rank 1 case are unconditional, but the lower bounds on the proportions of rank 0 and rank 1 are smaller than ours, and their method does not give a bound on the average rank. Previously, there have been a number of special cases of such rank 0 and rank 1 results proved by various authors, using both analytic and algebraic methods; see [13, 21, 26, 29, 45, 46].
The proportions provided by Theorem 2.5 will naturally be largest when every is either in or . The elliptic curve of smallest conductor over for which this occurs is the curve having Cremona label 19a3. Since the parity of is easily seen to be equidistributed in quadratic twist families over , we deduce that, in this special situation, half of the squareclasses lie in and half lie in . Thus, by Theorem 2.4, the average rank of is at most 7/6, and, by Theorem 2.5, at least of twists have rank 0 and at least have -Selmer rank 1.
Over general number fields, the parity of is not necessarily equidistributed in quadratic twist families, and so we can obtain larger proportions of rank 0 or -Selmer rank 1 curves in certain families. For instance, suppose is an elliptic curve over a number field that has complex multiplication by an imaginary quadratic field contained in , and that is not inert in . Then the curve is isogenous to a curve with , and admits a 3-isogeny to , where is an ideal of norm 3. We show in Section 11 that every then lies in for this isogeny . By Theorem 2.5, we thus obtain:
Suppose that is an elliptic curve over a number field such that is an imaginary quadratic field in which is not inert. Then the average rank of for is at most , and at least of twists have rank .
In the setting of Theorem 2.7, Goldfeld’s minimalist philosophy  would predict that of twists have rank since they all have even parity. In Section 13, we give an example of a non-CM elliptic curve with a -isogeny whose twists all have even parity and for which we derive the same conclusion as Theorem 2.7. We also provide an example of a curve whose twists all have odd parity and for which we may deduce that of twists have -Selmer rank , which we take as strong evidence in favor of the conjecture that of twists of this curve should have rank .
Finally, we note that there are a number of other applications of Theorem 2.1, e.g., to the existence of elements of various orders in Tate–Shafarevich groups, which we pursue in forthcoming work.
Methods and organization
In Sections 2–6, we develop a new correspondence between -Selmer elements (in fact, arbitrary -coverings) and certain binary cubic forms with coefficients in the base field . This bijection recovers work of Selmer  and Satgé  in the very special case when is an elliptic curve over with -invariant 0. Our correspondence is new for elliptic curves with , and for higher dimensional abelian varieties having a 3-isogeny . In Section 7, we show, moreover, that integral models exist for -Selmer elements, i.e., -Selmer elements correspond to integral binary cubic forms. This may be thought of as a generalization of the case proven by the first and last authors and Elkies in , though the general case treated here indeed exhibits many important differences and subtleties in the ‘minimization’ results.
Our parametrization of -Selmer elements by integral binary cubic forms has the property that -Selmer elements for different yield integral binary cubic forms having discriminants that lie in different classes in as well; moreover, we prove that these discriminants are squarefree away from a fixed finite set of primes. This property is what allows us to use results on counting binary cubic forms to get a handle on the count of -isogeny Selmer elements across a family of quadratic twists. Indeed, in Section 8, we combine this new parametrization of -Selmer elements with the geometry-of-numbers techniques developed in recent work of the first author, Shankar, and Wang , which enables one to count integral binary cubic forms over a number field satisfying properties of the type described. This yields Theorem 2.1. We prove Theorem 2.2 in Section 9 as a consequence, and also prove Theorems 2.4 and 2.5. In Sections 10–11, we consider some further applications to ranks of elliptic curves and, in particular, prove Theorems 2.6 and 2.7. In Section 12, we show that the Jacobians of a large family of trigonal curves fall under the scope of Theorem 2.2. Finally, we use Section 13 to highlight some examples where our results yield particularly interesting conclusions.
3. Orbits of binary cubic forms over a Dedekind domain
We recall from [1, §2] some facts about orbits of binary cubic forms. Let be the lattice of integer-matrix binary cubic forms, i.e., forms
with . Equivalently, is the space of integer symmetric trilinear forms. The group acts naturally on by linear change of variable, and we define the discriminant by
For any ring , we write . The action of on satisfies
for all and . In particular, the discriminant is -invariant. For any , we write for the set of with .
Now let be a Dedekind domain of characteristic not 2 or 3, and let be its field of fractions. For any non-zero , we define the ring and let be its total ring of fractions.
[1, Thm. 12] Let be non-zero. Then there is a bijection between the orbits of on and equivalence classes of triples , where is a fractional -ideal, , and , satisfying the relations , is the principal fractional ideal in , and in . Two triples and are equivalent if there exists such that , , and . Under this correspondence, the stabilizer in of is isomorphic to , where is the ring of endomorphisms of .
We call triples satisfying the above conditions -triples.
When is a field, so that , the previous result simplifies quite a bit. We write to denote the kernel of the norm map . We also write for the kernel of the norm map , where is the restriction of scalars of the group scheme .
There is a bijection between and the set of -orbits of . Moreover, the stabilizer inside of any is isomorphic to .
The bijection sends the class of in to the orbit of the binary cubic form corresponding to the -triple , where is any choice of cube root of . Explicitly, this is the orbit of the cubic form on , where is the image of in [1, Thm. 12]. ∎
Since can be replaced by for any , the -orbits of discriminant are in bijection with the -orbits of discriminant . The bijection is given by where We call the action of on the twisted -action.
If is a discrete valuation ring and if is the maximal order in , we obtain:
Suppose is a discrete valuation ring with fraction field . Assume that the -algebra is the maximal order in . Then the set of -orbits on is in bijection with the unit subgroup . In particular, every rational -orbit of discriminant whose class lies in this unit subgroup contains a unique integral -orbit.
4. Three-isogenies and orbits of binary cubic forms over a field
Again assume that is a field of characteristic not 2 or 3, and fix a 3-isogeny of abelian varieties over . In this section, we relate the -orbits of binary cubic forms over to the arithmetic of . After outlining the general theory, we make more explicit the case of elliptic curves.
Write for the group scheme over , and write for the Galois group of the separable closure of . As the group has order 3, there is an étale quadratic -algebra such that the action of on factors through . If the action is non-trivial, this is unique. If the action is trivial, then we take to be the split algebra . We may write for some . Now set , with , and note that is only determined by up to squares in . We call the ‘mirror algebra’ of . The relevance of this mirror comes from the following proposition.
There is an isomorphism of group schemes
and hence an induced isomorphism
where denotes the kernel of the norm .
Since there is a unique rank 3 group scheme over that becomes constant over , it is enough to show that the kernel of the norm map has this property. Over , this kernel is the subgroup of , where is a primitive third root of unity. The two factors are indexed by the two embeddings and of into , and the action of is given by . A simple computation then shows that is fixed by the subgroup , since is the third quadratic algebra in the biquadratic extension over . For a more geometric proof of the proposition, see [1, Prop. 18]. ∎
There is a natural bijection between and the -orbits on . Moroever, the stabilizer in of any is isomorphic to .
Let us specialize to the case where and are elliptic curves. An elliptic curve with a 3-isogeny admits a model of the form
with , , and in . The points generate the kernel . In particular, the quadratic -algebra is independent of the chosen model of , and is the same defined above. The curve admits a dual 3-isogeny , and a model for is [14, 1.3]
We phrase everything below in terms of because it is cleaner. In particular, the set is in bijection with -equivalences classes of cubic forms of discriminant , and not . On the other hand, the group also parameterizes isomorphism classes of -coverings, i.e. maps of curves that become isomorphic to over . Because it may be useful in other applications, we give a clean description of these -coverings, using the bijection in Theorem 4.2:
Let be a -covering corresponding to under the bijection of Theorem 4.2. Then is isomorphic to the cubic curve
in . Here, is the scaled Hessian of .
Let be a representative for the class corresponding to . From [14, Thm. 4.1, Rem. (1)], we deduce the following equation for in with coordinates :
where is a cube root of and is the image of in . Note that replacing by , for any , gives an isomorphic curve. Thus, we may assume , by the proof of Corollary 3.3. One computes that , so we in fact have . Another computation shows that the Hessian of is , so the equation for has the desired form. ∎
5. -soluble orbits of binary cubic forms
As before, let be a 3-isogeny of abelian varieties over a field . There is then a bijection between the elements of and the -orbits on the set of binary forms of discriminant , where is defined as in the previous section.
As in the elliptic curve case, the group classifies -coverings , where is a torsor for . The orbits in the image of the Kummer map
correspond to soluble -coverings , i.e., -coverings with . We let denote the set of binary cubic forms corresponding, under the bijection of Theorem 4.2, to a class in the image of the Kummer map. If , we say that is -soluble.
For each , we have the quadratic twists and and a 3-isogeny . If is an elliptic curve with model then a model for the quadratic twist is
and is the map given by taking the quotient by the points . For higher dimensional abelian varieties, one defines , , and via descent, using the cocycle corresponding to the extension ; see [38, III.1.3] for example. Note that it makes sense to twist the 3-isogeny by since is preserved by the inversion automorphism on . We often omit the subscript from when the context makes it clear—for example, when writing .
The key fact we will use about is that
as -modules. In particular, if is the mirror algebra of , then is the mirror algebra for . This can be read directly off the models in the elliptic curve case.
By Theorem 4.2 applied to , the set of -orbits on the set of binary cubic forms over of discriminant is in bijection with . This allows us to define a notion of -solubility for binary cubic forms over of any discriminant, compatible with the already defined notion of -solubility for forms of discriminant :
A binary cubic form is -soluble if it corresponds under the bijection of Theorem 4.2 to an element in the image of , where . The set of all -soluble is denoted by .
For any , we have . In other words, this notion of -solubility depends only on the quadratic twist family of and the associated family of 3-isogenies, , and not on the isogeny itself.
Our main theorem parametrizing elements of , , by -soluble binary cubic forms over is as follows.
There is a natural bijection between the -orbits on having discriminant and the elements of the group . Under this bijection, the identity element of corresponds to the unique -orbit of reducible binary cubic forms over of discriminant , namely the orbit of
The image of the Kummer map for is isomorphic to , so the first statement follows. For the second part of the corollary, we use the explicit description of the bijection given in (the proof of) Corollary 3.3. If , the corresponding cubic form is . This is reducible over if and only if (modulo cubes in ), since is the unique element in of trace-zero up to -scaling. If this is the case, then , so we must have , i.e. is the class of the identity in . ∎
6. -soluble orbits over local fields
Now assume that is a prime and is a finite extension of with ring of integers . Fix a 3-isogeny of abelian varieties over . The 3-isogeny determines a notion of -solubility, as defined in the previous section, on the space of binary cubic forms.
In this section we show that, in certain circumstances, -solubility of implies that is -equivalent to an integral form . We find that the conditions under which we can guarantee integrality are related to the value of the Selmer ratio
defined in the introduction.
As before, let be the mirror quadratic -algebra attached to . Assume from now on that lies in . We first consider the case of good reduction.
If and has good reduction, then the Kummer map induces an isomorphism
where is the ring of integers of and
Moreover, in this case.
Under the hypotheses, the image of the Kummer map
is exactly the subgroup of unramified classes, i.e., those that become trivial when restricted to the inertia subgroup of . In the case of elliptic curves, this is [11, Lem. 4.1]. For the general case, see [12, Prop. 2.7(d)]. On the other hand, by local class field theory, the unramified classes map onto under the isomorphism of Proposition 4.1, which proves the first claim.
For the second claim, first note that may be taken to be a unit in in the good reduction case. This follows from the criterion of Ogg-Néron-Shaferevich. For an elementary proof in the elliptic curve case, note that divides the discriminant of the elliptic curve (4.1). It follows that . We then compute
Suppose that , that has a quadratic twist of good reduction, and that is a not in the square of the maximal ideal in . Then any -soluble is -equivalent to an integral form . Moreover, we have .
First, suppose that has good reduction. By the Ogg-Néron-Shaferevich criterion, the kernel is unramified, so that is unramified over (since ). Since , it follows that is a unit in , and is the maximal order in . By Proposition 3.5, the orbit of corresponds under the bijection of Theorem 5.3 to a class in . Thus by Proposition 6.1, the orbit of contains an integral orbit, and .
Next, assume that has bad reduction. By assumption, it must be a ramified twist of an abelian variety with good reduction, i.e., for some in the maximal ideal of . It follows that the mirror algebra is ramified over . But then the group is trivial, and is -equivalent to the integral form .
Finally, we need to show that in this bad reduction case. The numerator in the definition of equals 1 since
On the other hand, the denominator equals , which also equals 1. Indeed, the field , over which the Galois action on trivializes, is ramified over (since the mirror -algebra is ramified, and since ). Thus is trivial. ∎
7. -Selmer groups and locally -soluble orbits over a global field
Now let be a number field. Write for the completion of at a place . If is an isogeny of abelian varieties over , the -Selmer group is the subgroup of of classes that are locally in the image of the Kummer map
for every place of . Equivalently, these are the classes locally in the kernel of the map
for every place , i.e. the classes corresponding to principal homogeneous spaces with an -point for every place .
Now assume is a 3-isogeny over . If is a place of , write for the base change . We have defined the subset of -soluble cubic forms over . For each , we have also defined the subset of -soluble cubic forms over . We let denote the set of locally -soluble binary cubic forms, i.e. the set of such that for all places of .
Finally, fix so that is the mirror algebra associated to . The following theorem now follows immediately from Theorem 5.3.
Let be a -isogeny of abelian varieties over a number field , and fix . Then there is a natural bijection between the -orbits of of discriminant and the elements of the -Selmer group corresponding to the isogeny Under this bijection, the identity element of corresponds to the unique orbit of reducible binary cubic forms of discriminant