Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0
1 Introduction
Any elliptic curve over is isomorphic to a unique curve of the form , where and for all primes : whenever . The (naive) height of the elliptic curve is then defined by
In a previous paper [7], we showed that the average rank of all elliptic curves, when ordered by their heights, is finite. This was accomplished by proving that the average size of the 2Selmer group of elliptic curves, when ordered by height, is exactly 3. It then followed from the latter result that the average rank of all elliptic curves is bounded above by 1.5.
In this article, we prove an analogous result for the average size of the 3Selmer group:
Theorem 1
When all elliptic curves are ordered by height, the average size of the Selmer group is at most .
The above result is also seen to imply the boundedness of the average rank of elliptic curves. Indeed, Theorem 1 immediately yields the following improved bound on the average rank of all elliptic curves:
Corollary 2
When all elliptic curves over are ordered by height, their average Selmer rank is at most ; thus their average rank is also at most .
Theorem 1 also gives the same bound of on the average 3rank of the TateShafarevich group of all elliptic curves, when ordered by height.
We will in fact prove a stronger version of Theorem 1, namely:
Theorem 3
When elliptic curves in any family defined by finitely many local conditions, are ordered by height, the average size of the Selmer group is at most .
Thus the average size of the 3Selmer group remains at most 4 even when one averages over any subset of elliptic curves defined by finitely many local conditions. We will actually prove Theorem 3 for an even larger class of families, including some that are defined by certain natural infinite sets of local conditions (such as the family of all semistable elliptic curves).
Theorem 3, and its abovementioned extensions, allow us to deduce a number of additional results on ranks that could not be deduced solely through understanding the average size of the 2Selmer group, as in [7]. First, by combining our counting techniques with the remarkable recent results of Dokchitser–Dokchitser [19] on the parity of ranks of Selmer groups, we prove:
Theorem 4
When all elliptic curves are ordered by height, a positive proportion of them have rank .
In the case of rank 1, if we assume the finiteness of the TateShafarevich group, then we also have:
Theorem 5
Assume X( is finite for all . When all elliptic curves are ordered by height, a positive proportion of them have rank .
Next, combining our counting arguments with the important recent work of Skinner–Urban [30] on the Iwasawa Main Conjectures for , we obtain:
Theorem 6
When all elliptic curves are ordered by height, a positive proportion of them have analytic rank ; that is, a positive proportion of elliptic curves have nonvanishing Lfunction at .
As the elliptic curves of analytic rank 0 that arise in Theorem 6 form a subset of those that we construct in Theorem 4, we conclude:
Corollary 7
A positive proportion of elliptic curves satisfy BSD.
Our previous results on the average size of the 2Selmer group were obtained through counting integral binary quartic forms, up to equicalence, having bounded invariants. The connection with elliptic curves is that the process of 2descent has a classical interpretation in terms of rational binary quartic forms; this connection was indeed behind the beautiful computations of Birch and SwinnertonDyer in [8]. The process of 2descent through the use of binary quartic forms, as in Cremona’s remarkable mwrank program, remains the fastest method in general for computing ranks of elliptic curves.
In order to prove an analogous upper bound for the average size of 3Selmer groups, we apply our counting techniques in [7], appropriately modified, to the space of integral ternary cubic forms. The group naturally acts on , and the ring of polynomial invariants for this action turns out to have two independent generators, traditionally denoted and , having degrees 4 and 6 respectively.
These invariants may be constructed as follows. For a ternary cubic form , let denote the Hessian of , i.e., the determinant of the matrix of second order partial derivatives of :
(1) 
Then is itself a ternary cubic form and, moreover, it is an covariant of , i.e., for , we have . An easy computation gives
(2) 
for certain rational polynomials and in the coefficients of , having degrees 4 and 6 respectively; note that (2) uniquely determines , and also uniquely determines up to sign. The sign of is fixed by the requirement that the discriminant of a ternary cubic form be expressible in terms of and by the same formula as for binary quartic forms, namely
(3) 
These polynomials and are evidently invariant, and in fact they generate the full ring of polynomial invariants over (see, e.g., [31]).
Now, for ternary cubic forms over the integers, the general work of Borel and HarishChandra [9] implies that the number of equivalence classes of integral ternary cubic forms, having any given fixed values for these basic invariants and (so long as and are not both equal to zero), is finite. The question thus arises: how many classes of integral ternary cubic forms are there, on average, having invariants , as the pair varies?
To answer this question, we require a couple of definitions. Let us define the height of a ternary cubic form by
(as usual, the constant factor on is present for convenience and is not of any real importance). Thus is a “degree 12” function on the coefficients of . We may then order all classes of ternary cubic forms by their height , and we may order all pairs of invariants by their height .
As with binary quartic forms, we wish to restrict ourselves to counting ternary cubic forms that are irreducible in an appropriate sense. Being simply irreducible—i.e., not having a smaller degree factor—is more a geometric condition rather than an arithmetic one. We wish to have a condition that implies that the ternary cubic form is sufficiently “generic” over . The most convenient notion (also for the applications) turns out to be what we call strong irreducibility.
Let us say that an integral ternary cubic form is strongly irreducible if the common zero set of and its Hessian in (i.e., the set of flexes of in ) contains no rational points. (Note that this implies that is irreducible in the usual sense, i.e., it does not factor over ). We prove:
Theorem 8
Let denote the number of equivalence classes of strongly irreducible ternary cubic forms having invariants equal to and . Then:
In order to obtain the average size of , as varies, we first need to know which pairs can actually occur as the invariants of an integral ternary quartic form. For example, in the case of binary quadratic and binary cubic forms, the answer is wellknown: there is only one invariant—the discriminant—and a number occurs as the discriminant of a binary quadratic (resp. cubic) form if and only if it is congruent to or (mod 4).
In the binary quartic case, we proved in [7] that a similar scenario occurs; namely, an is eligible—i.e., it occurs as the invariants of some integer binary quartic form—if and only if it satisfies any one of a certain specified finite set of congruence conditions modulo 27 (see [7, Theorem 1.7]).
It turns out that the invariants that can occur (i.e., are eligible) for an integral ternary cubic must also satisfy these same conditions modulo 27. However, there is also now a strictly larger set of possibilities at the prime 2. Indeed, the pairs that occur for ternary cubic forms need not even be integral, but rather lie in in this set that actually occur are then defined by certain congruence conditions modulo 64 on and , in addition to the same congruence conditions modulo on and that occur for binary quartic forms. ; and the pairs
In particular, the set of integral pairs that occur as invariants for integral ternary cubic forms is the same as the set of all pairs that occur for integral binary quartic forms! We prove:
Theorem 9
A pair occurs as the pair of invariants of an integral ternary cubic form if and only if satisfies one of the following congruence conditions modulo : , the pair

and (f) and

and (g) and

and (h) and

and (i) and

and (j) and
and satisfies one of the following congruence conditions modulo :

and

and

and

and
From Theorem 9, we conclude that the number of eligible pairs , is a certain constant times ; by Theorem 8, the number of classes of strongly irreducible ternary cubic forms, per eligible , is therefore a constant on average. We have: , with
Theorem 10
Let denote the number of equivalence classes of strongly irreducible integral ternary cubic forms having invariants equal to and . Then:
The fact that this class number is a finite constant on average is indeed what allows us to show that the size of the 3Selmer group of elliptic curves too is bounded by a finite constant on average.
We actually prove a strengthening of Theorem 10; namely, we obtain the asymptotic count of ternary cubic forms having bounded invariants that satisfy any specified finite set of congruence conditions (see §2.4, Theorem 20). This strengthening turns out to be crucial for the application to 3Selmer groups (as in Theorem 1), as we now discuss.
Recall that, for any positive integer , an element of the Selmer group of an elliptic curve may be thought of as a locally soluble covering. An covering of is a genus one curve together with maps and , where is an isomorphism defined over , and is a degree map defined over such that the following diagram commutes:
Thus an covering may be viewed as a “twist over of the multiplicationby map on ”. Two coverings and are said to be isomorphic if there exists an isomorphism defined over , and an torsion point , such that the following diagram commutes:
A soluble covering is one that possesses a rational point, while a locally soluble covering is one that possesses an point and a point for all primes . Then we have, as groups, the isomorphisms:
Now, counting elements of leads to counting ternary cubic forms for the following reason. There is a result of Cassels (see [12, Theorem 1.3]) that states that any locally soluble covering posseses a degree divisor defined over . If , we thus obtain an embedding of into , thereby yielding a ternary cubic form, welldefined up to equivalence! Conversely, given any ternary cubic form having rational coefficients and nonzero discriminant, there exists a degree mapping defined over from the plane cubic defined by the equation to the elliptic curve , where is the Jacobian of and is given by the equation
(4) 
Note that (4) gives another nice interpretation for the invariants and of a ternary cubic form .
To carry out the proof of Theorems 1 and 3, we do the following:

Given , choose an integral ternary cubic form for each element of , such that

gives the desired 3covering;

the invariants of agree with the invariants of the elliptic curve (at least away from 2 and 3);


Count these integral ternary cubic forms via Theorem 11. The relevant ternary cubic forms are defined by infinitely many congruence conditions, so a suitable sieve has to be performed.
As we are primarily aiming for upper bounds, we are able to use a particularly simple sieve in the last step (compare with [7]) in order to prove Theorems 1 and 3.
This paper is organized as follows. In Section 2, following the methods of [7], we determine the asymptotic number of equivalence classes of strongly irreducible integral ternary cubic forms having bounded height; in particular, we prove Theorems 8, 9, and 10. The primary method is that of reduction theory, allowing us to reduce the problem to counting integral points in certain finite volume regions in . However, the difficulty in such a count, as usual, lies in the fact that these regions are not compact, but rather have cusps going off to infinity. By studying the geometry of these regions via the averaging method of [7], we are able to cut down to the subregions of the fundamental domains that contain predominantly and all of the strongly irreducible points. The appropriate volume computations for these subregions are then carried out to obtain the desired result.
In Section 3, we then describe the precise correspondence between ternary cubic forms and elements in the Selmer groups of elliptic curves. We show, in particular, that nonidentity elements of the 3Selmer group correspond to strongly irreducible ternary cubic forms. We then apply this correspondence, together with the counting results of Section 2 and a simple sieve (which involves the determination of certain local mass formulae for 3coverings of elliptic curves over ), to prove that the average size of the Selmer groups of elliptic curves, when ordered by height, is at most 4; we thus prove Theorems 1 and 3.
2 The number of integral ternary cubic forms having bounded invariants
Let denote the space of all ternary cubic forms having coefficients in . The group acts on on the left via linear substitution of variable; namely, if and , then
For a ternary cubic form , let denote the Hessian covariant of , defined by (1), and let and denote the two fundamental polynomial invariants of as in (2). As noted earlier, these polynomials and are invariant under the action of and, moreover, they are relative invariants of degrees and , respectively, for the action of on ; i.e, and for and .
The discriminant of a ternary cubic form is a relative invariant of degree 12, as may be seen by the formula . We define the height of by
Note that the height is also a degree relative invariant for the action of on .
It is easy to see that the action of on preserves the lattice consisting of integral ternary cubic forms. In fact, it also preserves the two sets and consisting of those integral ternary cubics that have positive and negative discriminant, respectively.
As before, let us say that an integral ternary cubic form is strongly irreducible if the corresponding cubic curve in has no rational flex. Our purpose in this section is to prove the following restatement of Theorem 8:
Theorem 11
For an invariant set , let denote the number of equivalence classes of strongly irreducible elements in having height bounded by . Then
2.1 Reduction theory
Let (resp. ) denote the set of elements in having positive (resp. negative) discriminant. We first construct fundamental sets in for the action of on , where is the subgroup of all elements in having positive determinant.
To this end, let be a ternary cubic form in having nonzero discriminant, and let denote the cubic curve in defined by the equation . The set of flexes of is given by the set of common zeroes of and in , and hence the number of such flexes is 9 by Bezout’s Theorem. As both and have real coefficients, the flex points of are either real or come in complex conjugate pairs. Therefore, since the total number of flex points is odd, possesses at least one real flex point.
This implies, in particular, that any ternary cubic form over is equivalent to one in Weierstrass form, i.e., one in the form
(5) 
for some . It can be checked that the ternary cubic form in (5) has invariants and equal to and , respectively. Thus, since and are relative invariants of degrees and , respectively, two ternary cubic forms and over are equivalent if and only if there exists a positive constant such that and . It follows that a fundamental set (resp. ) for the action of on (resp. ) may be constructed by choosing one ternary cubic form for each and such that and (resp. ). We may thus choose
The key fact that we will need about these fundamental sets is that the coefficients of all the ternary cubic forms in the are uniformly bounded. Note also that if is any fixed compact subset then, for any , the set is also a fundamental set for the action of on , and again the coefficients of all the forms in are uniformly bounded, independent of .
We also require the following fact whose proof we postpone to Section 3:
Lemma 12
Let be any ternary cubic form having nonzero discriminant. Then the size of the stabilizer in of is equal to .
Let denote a fundamental domain in for the left action of on that is contained in a standard Siegel set. We may assume that , where
here is a measurable subset of dependent only on and is an absolute constant.
We see from Lemma 12 that for any , the multiset is essentially the union of fundamental domains for the action of on . More precisely, we see that an equivalence class of is represented exactly times in , where .
However, let us say that a ternary cubic form is totally irreducible if it is strongly irreducible and its Jacobian (i.e., the Jacobian of the plane cubic curve associated to ) does not possess a nontrivial rational 3torsion point. Then we have the following fact, whose proof will be given in Section 3.
Lemma 13
The stabilizer in of a totally irreducible integral ternary cubic form is trivial.
Furthermore, we will also prove in §2.5 that the number of orbits of points in that are strongly irreducible but not totally irreducible, and have height less than , is negligible (i.e., ).
For , let and denote the multisets defined by
Then by Lemmas 12 and 13, that we see that, up to a negligible number of forms that are not totally irreducible, the quantity is equal to the number of strongly irreducible integral ternary cubic forms contained in .
Counting strongly irreducible integer points in a single such is difficult because the domain is not compact. As in [7], we simplify the counting by averaging over lots of such domains, i.e., by averaging over a continuous range of elements lying in a certain fixed compact subset of .
2.2 Averaging and cutting off the cusp
Let be a compact invariant subset that is the closure of some nonempty open set in , such that every element in has determinant greater than . For any invariant set , let denote the set of strongly irreducible elements of . Then, up to an error of from the bound (proved in Lemma 22) on points that are strongly but not totally irreducible, we have:
(6) 
where and . Given , let denote the unique point in that is equivalent to . Then we have
(7) 
For a given , by Lemma 12 there exist three elements satisfying . Hence
Since is an invariant measure on , we see that
Therefore, if , then
(8)  
(9)  
(10)  
(11) 
where the last equality follows because we have assumed that is invariant and .
For , let us write . We then have
(12) 
To further simplify the right hand side of (12), we require the following lemma which states that contains no strongly irreducible points if or is large enough (i.e., when we are in the “cuspidal regions” of the fundamental domains):
Lemma 14
Let be a constant that bounds the absolute values of the , , , and coefficients of all the forms in . Then the set contains no strongly irreducible integral ternary cubic forms if or if .
Proof: It is easy to see that if , then the absolute values of the coefficients of , , and of any ternary cubic form in are all less than . Therefore any integral ternary cubic form in must have its , , and coefficients equal to , and such a form has a rational flex at and so is not strongly irreducible.
Similarly, if , then any integral ternary cubic form in has its , , and coefficients equal to , and such a form too always has a flex at , and so is not strongly irreducible.
Now let denote the set of all integral ternary cubic forms that are not totally irreducible. Then we have the following lemma which states that the number of reducible points—i.e., points in —that are in the “main body” of the fundamental domains is negligible.
Lemma 15
Let denote the set of elements that satisfy and . Then
We defer the proof of Lemma 15 to Section 2.6.
To estimate the number of integral ternary cubic forms in , we use the following proposition due to Davenport [15].
Proposition 16
Let be a bounded, semialgebraic multiset in having maximum multiplicity , and that is defined by at most polynomial inequalities each having degree at most . Let denote the image of under any upper or lower triangular, unipotent transformation of . Then the number of integer lattice points counted with multiplicity contained in the region is
where denotes the greatest dimensional volume of any projection of onto a coordinate subspace obtained by equating coordinates to zero, where takes all values from to . The implied constant in the second summand depends only on , , , and .
Therefore, by Equation (12), Lemmas 14 and 15, and Proposition 16, we see that up to an error of , we have
(13) 
As every element of was assumed to have determinant greater than , the set is empty if . The integral of the error term in the integrand of (13) is computed to be
Meanwhile, the integral of the main term in the integrand of (13) is equal to
Since is independent of and , Equation (13) then implies that
(14) 
where .
Thus to prove Theorem 11, it remains only to compute the volume .
2.3 Computing the volume
Let and let denote the set of those points in having height less than . Then since , where is a fundamental domain for the left action of on , we have .
For each with (resp. ), the set (resp. ) contains exactly one point having invariants and . There is thus a natural measure on both the sets , given by . Now define the usual subgroups , and of as follows:
It is wellknown that the natural product map maps injectively on to a full measure set in and that this decomposition gives a Haar measure on defined by .
We have the following proposition:
Proposition 17
For any measurable function on , we have
(15) 
By Lemma 12, we see that is a fold cover of a full measure set of ; the proposition can then be verified by a Jacobian computation. However, we present a more general proof of Proposition 17 that does not depend on the specific form of the sets . Specifically, we prove the following proposition which will also be useful to us in the sequel.
Proposition 18
Let be the Haar measure on obtained from the decomposition. Let be the Euclidean measure on , the vector space of all ternary cubic forms with complex coefficients. Let be any subset of that for each pair