Quadratic twists of abelian varieties with
Let be a totally real number field and a principally polarized abelian variety with real multiplication by the ring of integers of a totally real field. Assuming admits an -linear 3-isogeny over , we prove that a positive proportion of the quadratic twists have rank 0. We also prove that a positive proportion of have rank , assuming the groups are finite. If is the Jacobian of a hyperelliptic curve , we deduce that a positive proportion of twists have no rational points other than those fixed by the hyperelliptic involution.
As part of his study of the modular Jacobians , Mazur showed that there are geometrically simple abelian varieties of arbitrarily large dimension having finite Mordell-Weil group. The existence of rank 0 abelian varieties also follows from results of Gross-Zagier, Kolyvagin-Logachev, and Bump-Friedberg-Hoffstein [5, 14, 15]. These examples are obtained by considering quadratic twists of simple quotients of , for any .
Of course, one expects much more than the mere existence of such abelian varieties. Presumably, a significant proportion of all abelian varieties should have rank 0. Unfortunately, it is hard to prove quantitative results in this direction. Indeed, it was only recently that Bhargava and Shankar proved that a positive proportion of elliptic curves over have rank 0 . Geometry-of-numbers methods have since been deployed to study various algebraic families of abelian varieties of higher dimension [1, 2, 23, 26], yet the following question still seems to be open.
Is there a non-trivial, algebraic family of abelian varieties of dimension over for which a positive proportion of members are geometrically simple and have rank
In this note, we use geometry-of-numbers to provide many such examples. Our families turn out to be quadratic twist families of simple quotients of , but our approach is via Selmer groups and does not use any automorphic input.
In fact, recent work of the author, with Bhargava, Klagsbrun, and Lemke Oliver, already answered Question 1 over certain number fields. For instance, let be the Jacobian of the genus 3 curve , and let be any number field containing the cyclotomic field . For each , let be the th quadratic twist of . By studying the 3-Selmer group of , we showed that at least of the twists have rank 0 [3, Thm. 11.5]. One can construct similar examples using other abelian varieties with complex multiplication (CM). Indeed, our proof leverages the fact that has an endomorphism of degree 3 over . Since an abelian variety cannot have CM over , this approach does not readily give examples of the desired type over .
This paper considers a different class of abelian varieties with extra endomorphisms: those with real multiplication These have the advantage that their extra endomorphisms may be defined over . The disadvantage is that they do not admit endomorphism of degree 3. To compensate for this, we will impose a bit of level structure.
1.1. Real multiplication and -linear isogenies
Let be a number field, and a polarized abelian variety over of dimension . Let be a totally real number field of degree over , and let be a subring of finite index in the ring of integers . We say has real multiplication (RM) by if there is an embedding such that the polarization is -linear with respect to the induced actions of on and . These abelian varieties are sometimes said to be of -type.
Any elliptic curve has RM by . But among abelian varieties of dimension greater than 1, those with RM are quite special. Special in the sense that they lie on proper closed subvarieties of the corresponding moduli space, but also because their arithmetic parallels the arithmetic of elliptic curves in certain respects. For example, if , then it follows from a result of Ribet  that is a quotient of a modular Jacobian for some . Conversely, the simple quotients of have RM, so this gives a steady source of examples of RM abelian varieties.
We will consider abelian varieties with RM by and with an additional bit of level structure. We say admits an -linear -isogeny if there is another abelian variety with RM by , and an -equivariant 3-isogeny over . The kernel of is then an -module (scheme) which is annihilated by an ideal of index 3 in . We call the kernel ideal of .111This terminology is used in , but with a slightly different meaning.
1.2. Main results
For each , we write for the quadratic twist by the character corresponding to the extension . There is a natural height function on , given by
the product being over finite places of . This gives an ordering of the quadratic twist family.
Our first result answers Question 1 affirmatively, and provides a large class of examples.
Let be a totally real number field. Let be a polarized abelian variety over with by , and with . Suppose admits an -linear -isogeny whose kernel ideal is an invertible -module. Then a positive proportion of twists , , have rank .
In Section 7 we give examples of satisfying the hypotheses of Theorem 1.1. For now, we simply note that is automatically invertible if is principal or if the index is not divisible by 3. Moreover, in the latter case, necessarily admits a polarization of degree prime to .
Let be a simple quotient of of dimension . Then of twists have rank , and of twists have rank .
Note that the rank of is a multiple of , since has RM. On the analytic side, is the product of automorphic -functions, all with the same root number. Thus, Goldfeld’s minimalist philosophy and the conjecture of Birch and Swinnerton-Dyer lead directly to conjecture 1.2.
Our next result concerns the quadratic twists with rank , the smallest possible non-zero rank. It is conditional on the finiteness of the Tate-Shafarevich group, but we prove an unconditional result on the ranks of certain Selmer groups defined in Section 4.
Let and be as in Theorem 1.1, and let be the natural -isogeny. Then a positive proportion of twists , for , satisfy . If is finite for all , then a positive proportion of the twists have rank .
In the elliptic curve case , Theorems 1.1 and 1.3 were proven by Bhargava, Klagsbrun, Lemke Oliver, and the author [3, Thm. 1.6]. And indeed, a key ingredient in the proofs of Theorem 1.1 and 1.3 is the general result [3, Thm. 1.1], which determines the average size of the Selmer groups , where is the quadratic twist family of any 3-isogeny of abelian varieties. The other two main ingredients are an analysis of Selmer groups in ‘-isogeny chains’ of RM abelian varieties, and a collection of arithmetic duality theorems.
We have been careful to allow non-principal polarizations and non-maximal rings in order to cover many examples of RM abelian varieties which arise naturally. One place to look for such examples is the modular Jacobians over of prime level . If , then has a point of order 3, and by a result of Emerton, there is at least one optimal quotient with a point of order 3. If moreover , then is unique. In Section 7, we prove:
Suppose or , and let be the optimal quotient of with a rational point of order . Assume has a polarization of degree prime to . Then a positive proportion of quadratic twists , for squarefree, have rank . If is finite for all , then a positive proportion of twists have rank .
The proof uses Mazur’s analysis of the Eisenstein ideal to verify the invertibility hypothesis in Theorem 1.1. The condition on the polarization degrees of is presumably satisfied for most such optimal quotients; we give some examples in Section 7. It is possible that when , the results on twists of rank can be made unconditional by extending the methods of Kriz-Li [16, Thm. 7.1] to higher dimensional modular abelian varieties.
Our final result on ranks is an explicit upper bound on the average rank of .
Let be as in Theorem 1.1, and let be the corresponding family of -linear -isogenies. Then the average rank of the quadratic twists , , is at most , where is the absolute log-Selmer ratio of .
Theorem 1.5 is explicit in the sense that the stated upper bound can be easily computed if one knows the Tamagawa numbers of and , and all their quadratic twists, along with the root number of . With this data, one can also give an explicit lower bound on the proportions of rank 0 (resp. -Selmer rank 1) quadratic twists. In general, these Tamagawa numbers are not easy to compute when .
1.3. Rational points on curves of genus
Theorem 1.1 also has consequences for the study of rational points in quadratic twist families of curves of genus . For such families, it is natural to ask about the average size of , since . The notion of quadratic twist does not make sense for an arbitrary curve of genus , but it does make sense for hyperelliptic curves. In such families, one expects very few rational points other than the ‘trivial’ rational points, i.e. the rational points fixed by the hyperelliptic involution:
Conjecture 1.6 (Granville ).
If be a smooth hyperelliptic curve over of genus , then for of , the set consists only of fixed points for the hyperelliptic involution.
More generally, suppose is a smooth projective curve over with an involution that induces on the Jacobian . Then for each , there is a quadratic twist and whose Jacobian is . If has a rational point, then it embeds in via the Abel-Jacobi map. Moreover, any rational point in fixed by gives rise to a rational point in which is fixed by the involution . Following Conjecture 1.6, one expects to have no rational points other than these fixed points, on average. This automatically holds if has rank 0 and if , and the latter condition holds for all but finitely many . Thus, Theorem 1.1 immediately gives the following partial result towards Conjecture 1.6.
Let be a totally real field, and suppose is the family of quadratic twists of a smooth projective curve over with involution . Assume has by and admits an -linear -isogeny whose kernel ideal is an invertible -module. Then for a positive proportion of , the only rational points on are the fixed points of .
In Section 7 we give some genus 2 examples of such curves, with both simple Jacobians and rational fixed points. This gives progress towards Conjecture 1.6 for many curves which were previously inaccessible. By considering curves with local obstructions or with a map to an elliptic curve, one can find families of curves satisfying the conclusion of Theorem 1.7, but these tricks do not help when has rational fixed points and simple Jacobian.
The author thanks Manjul Bhargava, Pete Clark, Victor Flynn, Eyal Goren, Ben Howard, Zev Klagsbrun, Robert Lemke Oliver, Barry Mazur, Bjorn Poonen, Alice Silverberg, Drew Sutherland, and Yuri Zarhin for helpful conversations.
2. Abelian varieties with real multiplication
Let be a number field. Also, let be a totally real number field of degree over , and let be a subring of finite index in the ring of integers .
2.1. Real multiplication and isogenies
A -dimensional abelian variety over has real multiplication by if there is an algebra embedding .
Let be an abelian variety with real multiplication (RM) by , and fix , so that we may think of elements of as endomorphisms of . Our goal is to study the ranks of the quadratic twists , for in . Recall that is the twist of by the character corresponding to ; here .
The abelian variety has RM by , for all . More precisely, the embedding induces an embedding .
Since the automorphism commutes with action of , we see that the action of on descends to . ∎
Since is fixed, we can safely consider as an endomorphism of and as an endomorphism of , without any ambiguity.
In order to say something about the ranks of the twists , we suppose from now on that admits an -linear 3-isogeny. In other words, we assume there exists an abelian variety which also has RM by over and a 3-isogeny over which is -equivariant.
The kernel is an -module scheme of order 3, and hence is annihilated by an ideal in of index 3. We call the kernel ideal of . Since , the subgroup of all -torsion points on is isomorphic to over , and . The quotient also has RM by , and the natural -isogeny is -linear as well.
Conversely, if is an ideal of index 3, and if the -isogeny factors through a 3-isogeny , then has RM by and is necessarily -linear with kernel ideal .
Let be the 3-isogeny over such that . By Lemma 2.2, there are 3-isogenies and , such that , for each . Note that
2.2. Polarizations and duality
A polarization is a homomorphism over which, over , takes the form for some ample line bundle . If is an isomorphism, we call it a principal polarization.
In order to study the ranks of the twists , we further impose:
Assume admits an -linear 3-isogeny over , and that
is a totally real field.
admits an -linear polarization over of degree prime to 3.
The kernel ideal of is an invertible -module.
In particular, the polarized abelian variety has RM by , as defined in the introduction. These assumptions allow us to glean extra information on the group schemes , and .
The group scheme is self-dual.
Since is an invertible -module, there exists an ideal of which is coprime to and such that , for . It follows that is also isomorphic to . Since is prime to 3, we may fix an isogeny of degree prime to 3 such that the composition lies in . By -linearity of , the following diagram is commutative:
Since and are prime to 3, the kernel maps isomorphically onto , and hence is self-dual. ∎
For any , the group schemes and are Cartier dual.
We identify group schemes such as and with their corresponding -modules. Note that isomorphism classes of group schemes of order 3 over are in bijection with quadratic characters . Moreover, the Cartier dual of corresponds to the character , where is the character of . Thus by (2.1), we may reduce to the case and drop the subscripts.
Let and be the quadratic characters corresponding to and . These are the Jordan-Holder factors of . Since is self-dual, we have the equality of multi-sets
since the right hand side is the set of Jordan-Holder factors of the dual of . As is totally real, we have , and so we must have . Thus and are Cartier dual. ∎
In fact, Proposition 2.6 holds over any not containing .
From the proof of the propositions, we obtain the following corollaries.
There is an isogeny of degree prime to , which induces an isomorphism . In particular, there is an isomorphism .
The polarization induces an isogeny of degree prime to .
Using the notation from the proof of Proposition 2.5, the composition , sends isomorphically onto , and so the composition
gives the desired isogeny. ∎
The isogeny is equal to its own dual.
We give an alternate construction of . Let . The -linearity of exactly means that is fixed by the Rosati involution on associated to . Thus, the homomorphism is symmetric, i.e. equal to its own dual. It follows that over , we have for some line bundle on .
We claim that is of the form , for some line bundle on . Indeed, by the theory of descent of line bundles on abelian varieties, it is enough to show that the (cyclic) group scheme lies in the kernel of , which is of course true since . By degree considerations, it follows that the symmetric isogeny has degree . By construction, we have . Since , , and are all defined over , the map descends to a morphism over , which is the isogeny defined earlier. ∎
The symmetric isogeny is not necessarily a polarization, even if and is principal (in which case ). For example, if with , then is not a polarization. On the other hand, if with , then is a (principal) polarization. The difference is that is totally positive, while is not. See [12, Thm. 2.10].
2.3. Isogeny chains
Let be the order of the class in . Then , for some . We will also regard as an endomorphism of . If is any -ideal, then the quotient has RM by . We may therefore recursively define
Note that and
so that for all .
For , we define to be the natural -isogeny with kernel . In particular, and . We also have for all .
For each , there exists an isogeny of degree prime to , which maps isomorphically onto .
We may immediately reduce to the case , so that . Since is an invertible -module, we may choose an invertible -ideal which is prime to 3 and in the same ideal class of in . Then and the quotients and have degrees prime to 3. The commutativity of the diagram
shows that sends isomorphically onto , as desired. ∎
Each -isogeny factors as a composition of -isogenies, and the endomorphism factors as a composition of -isogenies.
For each , there are isomorphisms and .
Finally, note that the results in this section apply equally well if we replace and by and (as well as replacing the isogenies and by and ), for any .
3. Local Selmer ratios
For the first part of this section, we let be any isogeny of abelian varieties over a number field . We will recall the definitions of the local and global Selmer ratios attached to . One expects (and in certain special cases, one can prove) that these numbers dictate the average behavior of the ranks of the Selmer group , as varies through an algebraic family of isogenies. Assume, for simplicity, that the degree of is for some prime . Eventually we will take .
For each place of , write for the completion at . There is an induced homomorphism of groups , and the local Selmer ratio is defined to be
which lies in .
We also define the global Selmer ratio , where the product is over all places of , including the archimedean ones. The following proposition and its corollary show that this is a finite product, and hence the global Selmer ratio is well-defined [25, Lem. 3.8].
If is a finite place of , then
where is the Tamagawa number of at , and is the normalized absolute value of the determinant of the Jacobian matrix of partial derivatives of the map induced by on formal groups over , evaluated at the origin. In particular, for some integer .
If is a finite place of not above , then . Hence if has good reduction at and , then .
The local Selmer ratio at archimedean places of is easy to compute:
If is an archimedean place of and if , then .
In this case, is both a 2-group and an -group, so is trivial. ∎
We will need some formal properties of local and global Selmer ratios.
Let be a place of and let and be isogenies of abelian varieties over . Then .
[20, I.7.2]. ∎
Let and be isogenies of abelian varieties over . Then .
Suppose and are isogenies over for some place of . Then .
Let be the ring of integers in a totally real field, and suppose has by . Let be an element of norm for some . Suppose also that is a place of dividing . Then .
The case is proved in [25, Prop. 3.9], and the case of general is proved in exactly the same way. Note that it is important here that is an isogeny, which follows from the fact that the embedding sends to . ∎
3.2. RM isogeny chains
We return to the setup of the previous section, so that is an -linear 3-isogeny of abelian varieties with RM by over . We also impose Assumption 2.4. Recall that is the kernel ideal of and is the natural -isogeny. Also recall from Section 2.3 the endomorphism as well as the chain of -isogenies for .
Proposition 2.6 allows us to compute various local and global Selmer ratios in this setting.
If is an archimedean place of , then , for all .
It is enough to prove the statement for . There are only two group schemes of rank 3 over , namely and , and they are dual to each other. Thus, by Proposition 2.6, one of and is isomorphic to and the other is isomorphic to (over ). So is either an extension of by or an extension of by . In the former case, we clearly have , which shows that , by Lemma 3.3. In the latter case, we have
over . Since , we see that , and we again have . ∎
For each and every place of , we have .
This follows from Proposition 2.12. ∎
We have and .
For the proof, we introduce some notation. For any isogeny of abelian varieties over , and for any place of , we define , where the product is over the places of over .
We have .
4. Poitou-Tate duality and parity of the -Selmer rank
An isogeny of abelian varieties over a field , gives rise to a short exact sequence
of group schemes over , which we will also view as a short exact sequence of -modules. Taking the long exact sequence in group cohomology gives the Kummer map . If is a number field, then the -Selmer group is defined to be the subgroup of consisting of cohomology classes whose restriction to lies in the image of the Kummer map , for all places of .
We now place ourselves in the context of the previous sections, so that is an abelian variety over a totally real field satisfying Assumption 2.4. For each we have the 3-isogenies and , whose composition is the -isogeny . Each of the Selmer groups , and is a finite dimensional vector space over .
The purpose of this section is to collect some results which show that the global Selmer ratio encodes important information concering the -ranks of these three Selmer groups. The key input is Poitou-Tate global duality. In the case of elliptic curves, all results in this section are due to Cassels .
For and as above, we have
For each place of , the subgroup of local conditions in defining is orthogonal under local Tate duality to the local conditions in defining , by [7, Prop. B.1]. Thus, by Wiles’ duality formula [21, Thm. 8.7.9], we have
On the other hand, by Corollary 2.8, we have , , and , from which we deduce the desired formula. ∎
The next result is presumably well-known to experts, but we could not find it in the literature. This is a generalization of a result Cassels used to prove the non-degeneracy of the Cassels-Tate pairing for elliptic curves. Recall that the Cassels-Tate pairing for is a pairing [20, Prop. I.6.9]
Using the isogeny from Corollary 2.9, we construct a bilinear pairing
defined by .
Suppose . Then is in the image of if and only if for all .
Theorem 4.2 will be used in the proof of the following theorem.
If , then .
A diagram chase gives the following well-known five-term exact sequence
Combining this with Theorem 4.1, we reduce to showing that
is even. But by Theorem 4.2, the Cassels-Tate pairing on restricts to a non-degenerate pairing on this finite group. Moreover, the pairing is anti-symmetric by [22, Cor. 6]. Since any anti-symmetric pairing on an -vector space is also alternating, it follows from the non-degeneracy that has even -rank, as desired. ∎
Finally, we note that the theorems in this section apply equally well if we replace and by and (as well as replacing the isogenies and by and ) for any .
5. The ranks of the quadratic twists
Recall the height function on from the introduction:
where the product is over the finite primes of . This gives a way of ordering , and lets us define the average value of a function defined on :
We define the density of a subset to be
where is the characteristic function of .
We say is defined by finitely many local conditions if there exists for each place of subsets , such that for all but finitely many and such that
with the intersection taking place inside . The average of a function on (ordered by the height function ) is denoted .
Suppose is a degree isogeny of abelian varieties over a number field , and let be the corresponding quadratic twist family of -isogenies, for . If is a subset defined by finitely many local conditions, then the average size of , for is .
Now assume that is abelian variety over a totally real field satisfying Assumption 2.4. For , we define the subsets
Then the sets are defined by finitely many local conditions, and for any fixed , either is empty or it has positive density. We also write for .
Our first result gives a concrete bound on the average rank of the quadratic twists . To state the result cleanly, we define the absolute log-Selmer ratio .
Suppose has dimension . Then the average Mordell–Weil rank of the quadratic twists , for , is at most .
Note that is a finitely generated -module, and that has no 3-torsion for all but finitely many . For such , the rank of is equal to . Recall that is the order of the ideal class of in , and is the generator of the th power of the kernel ideal of . We have , so it is enough to show that the average -module rank of is at most . Since , and since for all (Corollary 2.14), the rank of as an -module is at most . So it is enough to show that the average size of is at most
To prove this, we fix and show that the average -rank of for is at most . We will suppose ; the proof in the case is similar. Then by Theorem 5.1, the average size of for is . For any , we have the inequality , so it follows that the average -rank of for is at most . By Corollary 3.12, the average size of for is , so the average -rank of is at most .
Using the exact sequence from (4.1),
we deduce that the average -rank of is bounded by as desired. ∎
To show that a positive proportion of quadratic twists have -Selmer ranks 0 (resp. 1), we will need the following proposition.
The sets and have positive density in .
We will prove the theorem for ; the proof for is similar. We need to construct a set , defined by finitely many local conditions, such that