Elliptic curves with a rational 2-torsion point ordered by conductor and the boundedness of average rank

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the large families considered by the aforementioned authors. We prove the analogous counting theorem for elliptic curves with so-called square-free index as well as for curves with suitably bounded Szpiro ratios. We note that our assumptions on the size of the Szpiro ratios is less stringent than would be expected by the naive generalization of their approach.


Introduction
In this paper we consider the problem of counting elliptic curves and estimating their average rank in certain thin families ordered by their conductor.The families we consider will be sub-families of the family E 2 of elliptic curves with a rational 2-torsion point (or equivalently, those that admit a degree 2 isogeny over Q) and having good reduction at 2 and 3.The latter is a technical assumption that simplifies the exposition.For this, we use the work of J. Mulholland on characterizing the reduction types at 2 and 3 of elliptic curves with rational 2-torsion [12].
Aside from the work [15], there has been a recent explosion of interest in enumerating elliptic curves ordered by their conductors, inspired by work of He, Lee, Oliver, Pozdnyakov [7] and Zubrilina [20].
We may assume without loss of generality, that our 2-torsion point is marked and so it suffices to consider the parametrization The discriminant of the curves in this family is given by The conductor of the curve is the quantity C(E a,b ) given in terms of p-adic valuation by 1 if E a,b has multiplicative bad reduction at p 2 if E a,b has additive bad reduction at p.
Our goal in this paper is to count certain subfamilies of elliptic curves of our families in E 2 ordered by conductor, as well as estimating the average rank with respect to such an ordering.This is motivated by recent work of A. N. Shankar, A. Shankar, and X. Wang [15] on counting elliptic curves in large families having bounded conductor.They also showed that the average size of the 2-Selmer group in the families they consider is at most 3.
Before we state our theorems, let us make a comparison with the approach and results in [15].In their treatment they deal with families which are conjectured to have positive density.In order to make progress they make a key assumption which we do not need: they assumed that the j-invariant j(E) of their elliptic curves E is bounded by O(log |∆(E)|).In particular, this allows them to remove the well-known archimedean difficulties of counting elliptic curves by discriminant.Another idea critical to their argument is to embed their families into the space of binary quartic forms.As a corollary they are able to count the corresponding 2-Selmer elements with a bit of extra work.Since we do not require this embedding, and indeed the strength of our results depend on our direct treatment of the elliptic curves under consideration, we do not obtain an analogous count of 2-Selmer elements.In fact we expect such a count to be useless for our purposes: most of the 2-Selmer elements we encounter for the family E 2 are expected to correspond to elements of the Shafarevich-Tate group X; see work of Klagsbrun and Lemke-Oliver [10] as well as recent work of Bhargava and Ho [3].Therefore in order to obtain an estimate for the average rank we instead look at the 3-Selmer group, employing the parametrization of 3-Selmer elements of curves in our family due to Bhargava and Ho [3].
There is a significant difference between the results of [15] and our results.In particular, Shankar, Shankar, and Wang obtain theorems where counting by conductor produces the same order of magnitude as counting by discriminant: in other words, on average the conductor is only marginally smaller than the discriminant in the cases they consider.For us, however, we obtain substantially larger counts when ordering our curves by conductor rather than discriminant: this phenomenon is again caused by the special shapes of our discriminants.Our previous paper [17] with C. Tsang provides another example of this phenomenon.
To motivate our results, let us discuss the analogous results of [15] in more detail.First we recall, as discussed in [15], that their strategy involves first counting elliptic curves by their discriminant first.This is where they needed to assume that the j-invariants of the curves E under consideration are bounded by log |∆(E)|.This means that for this subset of curves the problem of counting by discriminant and counting by naive height are essentially equivalent.They then require one of two assumptions.The cleaner of the two assumptions is that the quotient ∆(E)/C(E), which they call the index, is square-free.The second assumption, which is not disjoint from the first, is a bound on the so-called Szpiro ratio defined by (1.4) Their second assumption is the requirement that β E ≤ κ for some κ < 7/4.
In our situation we do not require any restrictions on the size of the j-invariant.This is because unlike the large family case it is possible to count the number of elliptic curves with rational 2-torsion by discriminant precisely; see Theorem 1.9 in [17].On the other hand some restrictions on the p-adic valuation of the discriminant akin to the assumptions in [15] mentioned above are necessary.
Crucial to our arguments is the existence of a canonical degree-2 isogeny for each curve E a,b in E 2 , defined by: (1.5) φ : The conductor of an elliptic curve is invariant under isogeny, so However, the discriminant is not in general invariant under isogeny.It is thus more natural to consider the Szpiro ratios of E a,b and E −2a,a 2 −4b simultaneously.
We will require the notion of the conductor polynomial for an elliptic curve E a,b ∈ E 2 : (1.6) We also define Observe that The analogues of the non-archimedean conditions in [15] for our situation will be: (1) Either the conductor polynomial C(E a,b ) is cube-free, equivalently that ind(E) (as defined by (1.7)) is square-free; or (2) The average of the Szpiro ratios β(E a,b ) and β(E −2a,a 2 −4b ) is less than 155/68.The value of the number 155/68 > 9/4 is significant because 9/4 is the natural analogue of the constant 7/4 as an upper bound for the Szpiro ratio in [15], obtained using geometry of numbers.Therefore the positivity of Ψ = 155/68 − 9/4 represents progressing beyond the simple application of geometry of numbers present in [15].
To see that 9/4 is the analogue of 7/4 in our case, note that in any large family the average value of the Szpiro ratio is expected to be 1.In the family E 2 , however, due to the presence of the square term the average value of the Szpiro ratio is expected to be 3/2 instead.We then see that 7/4 − 1 = 3/4 and 3/2 + 3/4 = 9/4, justifying 9/4 as the natural boundary of the methods of [15] when applied in the present setting.
We define, for κ < 155 68 , the family (1.8) ) is cube-free}.The first theorem in this paper is the following, which gives a an asymptotic formula for the number of curves in the families E 2,κ and E * 2 , is the following: Theorem 1.1.Let 1 < κ < 155/68 be a positive number.Then we have Note that the constant 1/64 in the statement of Theorem 1.1 comes from the assumption that our curves have good reduction at 2 and 3, which contributes 1/32, the remainder coming from the real density.This assumption is introduced to avoid having to deal with the conductor polynomial being divisible by arbitrarily large powers of 2, 3 while the conductor remains bounded at 2, 3.
We expect E 2,κ to satisfy the asymptotic formula in Theorem 1.1 for all κ > 1.The abc-conjecture implies that there are only finitely many elliptic curves with β E > 6.This then shows that if we replace E 2,κ with E 2 that the second asymptotic formula in (1.1) will hold as well.The p-adic densities present in the asymptotic formulae arise from the densities of elliptic curves in E 2 over Q p with fixed Kodaira symbol.These densities are computed in Section 2.
We compare Theorem 1.1 with the main theorems in [15].The value κ < 9/4 obtained in [15] is from fine tuning the geometry of numbers method pioneered by Bhargava in [2] and extended to the space of binary quartic forms by Bhargava and Shankar in [4].If we apply only geometry of numbers methods in the present paper, we will obtain the analogous quantity κ < 9/4.To push beyond this barrier we will require an application of a theorem of Browning and Heath-Brown in [5].This theorem relies on the p-adic determinant method of Heath-Brown [8], refined by Salberger in [14].One also requires certain linear programming bounds in order to make good use of this method; this is similar to joint work of the author and B. Nasserden in [13].
Our next theorem follows from adapting the methods in [3] on counting 3-Selmer elements of curves in E 2 : Theorem 1.2.When elliptic curves in E 2 are ordered by their conductor polynomials, the average size of their 3-Selmer groups is bounded.
We remark that Theorem 1.2 is weaker than Theorem 1.2 in [15] in that we do not have an exact count of the average, and we are only able to obtain boundedness for E 2 when ordered by the conductor polynomial.The reason is that the parametrization of 3-Selmer elements of E 2 obtained by Bhargava and Ho [3] uses a much more complicated coregular space of very high dimension, and as such we are unable to apply the p-adic determinant method which at present only provides superior results in the case of curves and surfaces.
1.1.Uniformity estimates.In order to prove Theorem 1.1 we will need to prove certain tail estimates.Indeed, we will require the following theorems: Theorem 1.3 (Uniformity estimate for curves with cube-free conductor polynomial).For all δ > 0 there exists a positive number κ such that For curves for which (β E + β φ(E) )/2 is bounded, we have the following: Theorem 1.4 (Uniformity estimate for curves with bounded average Szpiro ratio).Suppose that 1 < κ < 155/68 and θ > 0. Then there exists κ ′ , depending only on κ and θ, such that A summary of these uniformity estimates in the case of large families is given in [15]; we refer the reader to the aforementioned paper for historical progress.As mentioned in [15], the main difficulty in proving Theorems 1.3 and 1.4 is that the size of the conductor polynomial can be very large for curves with bounded conductor.This necessitates some new ideas.
Departing from the ideas given in [15], our new input is that the shape of our conductor polynomial allows us to turn the counting problem into one about counting integer points on a family of quadrics over P 2 or counting over sublattices of Z 2 defined by congruence conditions.Depending on the size of the parameters involved the bounds are stronger from one interpretation over the other.To do this we rely on a uniform estimate, essentially sharp, of counting integer points having bounded coordinates due to Browning and Heath-Brown [5].This is an application of the p-adic determinant method of Heath-Brown [8], refined by Salberger to the global determinant method; see [14] and [18] for a summary.1.2.Outline of the paper.In Section 2 we will characterize the possible Kodaira symbols of curves in E 2 and compute their relative densities.In Section 3 we compute the real density of curves in E 2 and prove the third part of Theorem 1.1.In Section 4 we prove the first two parts of Theorem 1.1 assuming the uniformity estimates given by Theorems 1.3 and 1.4.In Sections 5 and 6 we prove the aforementioned uniformity estimates.Finally, in Section 7 we prove Theorem 1.2.
Acknowledgements.We thank C. Tsang for discussions during the early development phase of this paper.We also thank the referee for extensive comments that lead to several major improvements to the paper.

Kodaira symbols for curves in E 2
The Kodaira symbols of curves in E 2 , for primes p ≥ 5 based on the Table 1 of [15], can be significantly simplified.This is because our minimal Weierstrass model is already minimal with respect to every prime p ≥ 5 because the constant coefficient is zero.In particular, we see at once that all of the symbols requiring a power of p to exactly divide the constant coefficient and for p to divide the x 2 and x-coefficients are not possible for curves in E 2 .This leaves only I n , n ≥ 1, I * 0 , III, and III * as possible Kodaira symbols in our family for p ≥ 5.
For p = 2, 3 separate treatment is necessary.Recall that we defined our E 2 to consist of curves with good reduction at 2, 3. Therefore, it suffices to take advantage of the work of Mulholland in [12] to identify congruence conditions on a, b which give good reduction at 2, 3.
2.1.Good reduction at 2 and 3. We use the following statement, which follows from Theorems 2.1 and 2.3 in [12].Proposition 2.1.Suppose that E a,b is given as in (1.1).
We now proceed to compute the relative density of curves with good reduction.In the case of good reduction at 2, the congruence conditions are determined by (Z/32Z) 2 .The first case in (1) of Proposition 2.1 contributes a density of 1/16, and the second case contributes 1/128, for a total of 9/128.
For the density of (a, b) such that E a,b has good reduction at 3, we note that the first case contributes 2/9 to the density.The second case we must allow for the possibility that a is divisible by an arbitrarily large power of 3. Let k ≥ 1 be given, and suppose 3 k exactly divides a.The number of pairs (a, b) Summing over k, we obtain a density of Combining the considerations of good reduction at 2 and 3, we conclude that the density of pairs (a, b) ∈ Z 2 such that E a,b has good reduction at 2 and 3 is (9/128)(4/9) = 1/32.The constraint that p 7 ∤ ∆(E a,b ) implies that 2ℓ + min{2k, ℓ} ≤ 6.
We also have k ≥ 1, ℓ ≥ 2. If ℓ > 2 then min{2k, ℓ} > 2, and therefore 2ℓ + min{2k, ℓ} > 6. Hence we must have ℓ = 2 and k = 1.This implies that C(E a,b ) is exactly divisible by p 4 .In this case we note that the conductor Therefore, the congruence information for a prime of Kodaira symbol If we put a = p k a 0 with p ∤ a 0 , then our conductor polynomial is equal to We then see that p 2 exactly divides the conductor polynomial, and hence exactly divides the conductor C(E a,b ).Therefore all of the congruence information is contained in the ring (Z/p 2 Z) 2 , and we have that If we put a = p k a 0 with p ∤ a 0 , then our conductor polynomial is equal to [15], it follows that p 6 exactly divides the conductor polynomial.Note that in this case the conductor C(E a,b ) is divisible by p 2 .Thus the congruence condition is contained in the Z-module (Z/p 4 Z) 2 , and E a,b (Z/p 4 Z) has Kodaira symbol III * if and only if a ≡ 0 (mod p 2 ) and b ≡ 0 (mod p 3 ), b ≡ 0 (mod p 4 ).Thus there are p 2 (p − 1) choices for (a, b), and the relative density is (p − 1)/p 6 .
2.5.Contribution to the conductor for semi-stable primes.Recall that for an elliptic curve E/Q, E has multiplicative bad reduction at p if and only if E has semi-stable bad reduction at p. Now the exponent k ≥ 1 can be arbitrarily large.Over the Z-module (Z/p k+1 Z) 2 a pair (a, b) ∈ (Z/p k+1 Z) 2 corresponds to an elliptic curve E a,b having semi-stable bad reduction at p if and only if p divides exactly one of b and c = a 2 − 4b.If b ≡ 0 (mod p k ), b ≡ 0 (mod p k+1 ) then we must have a ≡ 0 (mod p).Thus there are p k (p − 1) choices for a and p − 1 choices for b.If c ≡ 0 (mod p k ), c ≡ 0 (mod p k+1 ) then we cannot have a ≡ 0 (mod p), since otherwise b ≡ 0 (mod p) also which is not allowed.Therefore for any a co-prime to p we may choose a unique b (mod p k+1 ) such that c ≡ 0 (mod p k ) and c ≡ 0 (mod p k+1 ).There are again p k (p − 1) 2 such choices.Moreover it is clear that the two sets of possibilities are disjoint.It follows that there are 2p k (p − 1) 2 possibilities, and the relative density is 2(p − 1) 2 /p k+2 .

The family E 2 ordered by conductor polynomial
Recall that our family E is given by the equation (1.1).Analogous to our construction [17], we introduce the conductor polynomial of E a,b as Let A ∞ (Z) be the Lebesgue measure of the set The region A(Z) has four long cuspidal regions, defined by y = 0 and x 2 = y respectively.Nevertheless, we compare the areas of A(Z) and A * (Z) defined below: and show that they have comparable areas for Z large.We note that the set A * (Z) is in fact bounded.To see this, note that for its Lebesgue measure.We then have .
We put A ∞ (Z) = m(A(Z)).By Lemma 3.1, the area of the bounded region A * (Z) is comparable to A ∞ (Z), so it suffices to compute A ∞ (Z) which scales homogeneously with the parameter Z.In particular, we have To compute A ∞ (Z), we note the symmetry about the y-axis and thus assume without loss of generality that x > 0. Call the corresponding region A + (Z).We then partition A + (Z) into three regions: , We now compute m(A 1 ).To do this, for fixed 0 < x < √ 2Z 1 4 we have y lies in the interval defined by the equation y( it follows that As in [17], we will compute the above area using elliptic integrals.Make the substitution x = √ 2Z Next we compute m(A 2 ).Similar to the considerations above we find that Again making the substitution To evaluate this integral we apply integration by parts and obtain This implies that A similar calculation reveal that It follows that .
We may use a refined version of Davenport's lemma, due to Barroero and Widmer [1].Before we state the proposition, we introduce some notation in [1].Here we consider, as in [1], a parametrized family Z ⊂ R m+n of subsets Z T = {x ∈ R n : (T, x) ∈ Z}.We consider a lattice Λ ⊂ R n .The goal of the following proposition is to estimate the cardinality |Λ ∩ Z T | as the parameter T ranges over an infinite set.They obtained the following, the form of which was stated in [15]: Proposition 3.2.Let m and n be positive integers and let Λ ⊂ R n be a lattice.Denote the successive minima of Λ by λ i , i = 1, • • • , n.Let Z ⊂ R n be a definable family in an o-minimal structure, and suppose the fibres Z T are bounded.Then there exists a positive number c Z , depending only on Z, such that where V j (Z T ) is the sum of the j-dimensional volumes of the orthogonal projections of Z T onto every jdimensional coordinate subspace of R n .
Suppose now that we are given a set S ⊂ Z 2 defined by congruence conditions modulo some integer n > 0. Then we may break S up into a union of n 2 ν(S) translates of the lattice nZ × nZ, where ν(S) denotes the volume of the closure of S in Ẑ2 .Applying Proposition 3.2 to each of these translates and summing gives us the following result: Proposition 3.3.Let S ⊂ Z 2 be a set of pairs (a, b) defined by congruence conditions on a, b modulo some positive integer n.Then we have Proof.This is a direct consequence of Proposition 3.2 and the observation that to count the correct points in A(X) we need to impose the condition y ≡ 0 (mod 4), which leads to the leading term √ 2/2 in the front.See also Proposition 3.12 in [17].
We now prove the third part of Theorem 1.1.Let us put N m (X) be the number of curves E a,b ∈ E 2 in A(X) such that p 2 |a and p 3 |b for each p dividing m.It follows that Here the last estimate comes from the fact that by using the isogeny φ we may assume that |b| ≤ X 1/2 , and hence m 3 divides |b| implies that |m| ≤ X 1/6 .We can optimize δ by solving This is sufficient for the proof of the third part of Theorem 1.1.

Main counting theorems assuming uniformity estimates
We follow the same approach as in [15], and note that our uniformity estimates given by Theorem 1.3 imply The last line follows from our uniformity estimates.
We introduce some notation, following [15].For each prime p ≥ 5 let Σ p be a non-empty subset of possible reduction types.We say that Σ = (Σ p ) p is a collection of reduction types.We will assume that each Σ p contains the good and multiplicative reduction types for all p ≥ 5.
We then perform another inclusion-exclusion sieve to evaluate each summand on the right-hand side of the expression above.For each prime p let for every (a, b) ∈ Z 2 .Put ν * (nq, Σ) to be the product over all primes p of the integral of χ Σp,nq over Z 2 p .Then for nq < X δ we have where the final equality follows by reversing the inclusion-exclusion sieve in (4.1) (see (44) in [15]).
For each prime p ≥ 5 and integer k ≥ 0, put ν * (p k , Σ) for the p-adic density of the set of all (a, b) ∈ Z 2 such that E a,b ∈ E * (Σ) and ind p (E a,b ) = p k .The constant λ * (n, Σ) is a product over all p of local densities: It follows that λ * (n, Σ) is a multiplicative function of n, and hence The computation of ν * (p k ) then follows from the calculations in Section 2. In the case of E * 2 , we have that p | ind(E a,b ) if and only if p is semi-stable.Further, our imposition implies that ind(E a,b ) is square-free in this case, so k ≤ 1.We thus obtain the density For the general case, we have that the bulk of the contribution comes from the semi-stable primes.The density is then seen to be .
The contribution from the other Kodaira symbols occurs for k = 0, 2, 4. Combined with the contribution.This gives the total contribution .
This suffices for the proof of Theorem 1.1.

Counting curves with cube-free conductor polynomial
In this section we prove the necessary uniformity estimates in the sieve given in Section 4 in the case when the conductor polynomial C(E a,b ) is cube-free.
In order to take advantage of these results, we note that C(E a,b ) being cube-free implies the curve E a,b has no primes of Kodaira symbol I * 0 and III * .Let P = p 1 • • • p k be the product of a finite number of primes, and we shall restrict our attention to those curves E ∈ E * 2 which have Kodaira symbol III at each of the primes dividing P , and no other primes.We then have that a = P u, b = P v for some u, v ∈ Z and Further, the integers v and w = P u 2 − 4v are co-prime to P .Our cube-free condition then implies we that we may express v, w in the form: (5.2) w = P u 2 − 4v = w 0 w 2 1 , gcd(w 0 , w 1 ) = 1, w 0 , w 1 square-free.Thus we obtain a quadratic curve over P 2 defined by the equation We then see that the conductor is equal to We now prove Theorem 1.3.The proof will follow from Theorem 5.1 below.
We give some further preliminaries before stating Theorem 5.1.Our bound on the conductor is equivalent to
We shall prove the following result, which implies Theorem 1.3: Theorem 5.1.Suppose t i , i = 1, 2, 3, 4 are given as in (5.7), satisfy (5.8), and 4δ < t The proof requires two lemmata which counts points inside sub-lattices of Z 2 and points of bounded height on quadric curves in P 2 effectively.In particular we require the following result due to Browning and Heath-Brown [5].It provides an essentially sharp upper bound for the number of rational points of bounded height on a quadric curve in P 2 .Proposition 5.2 (Corollary 2, [5]).Let Q ∈ Z[x 1 , x 2 , x 3 ] be a non-singular quadratic form with matrix M .Let ∆ = | det(M )| and write ∆ 0 for the greatest common divisor of the 2 × 2 minors of M .Then we have where τ (•) is the divisor function.
The strength of this Proposition lies in its uniformity with respect to the height of the coefficients of Q, in that it is entirely independent of the height except on the determinant of M .A similar result, due to Heath-Brown [9], holds for sub-lattices of Z 2 .Lemma 5.3.Let Λ ⊂ Z 2 be a lattice.Then for all positive real numbers R 1 , R 2 the number of primitive Proof.See Lemma 2 in [9].Now to prove Theorem 5.1, we will consider various conditions on the t i 's each of which gives satisfactory bounds for R(T 1 , T 2 , T 3 , T 4 ), and together cover all cases except for t 1 + t 2 ≤ 1 − 2δ.We first prove the following proposition: Proposition 5.4.Suppose that there exists δ 1 > 0 such that either (5.11) Then there exists κ > 0 such that Proof.Without loss of generality, we may suppose that Then we may choose v 0 , v 1 , w 0 in O(T 1 T 2 T 3 ) = O X 3/4−δ1 ways.Having done so, (5.3) becomes where w 0 , v 0 , v 1 are fixed and u, w 1 are bounded by a power of X.The left hand side is a quadratic form, and thus this equation has O(τ , and choosing κ = 2δ 1 say gives us the desired result. We now consider the quadruples (t 1 , t 2 , t 3 , t 4 ) for which (5.11) fails.In fact we consider a slightly larger set of quadruples.In particular, we choose some δ 2 > 0, arbitrarily small, so that (5.12) We now show that fixing v 0 , w 1 and treating (5.3) as counting integral points in O ε (X ε ) lattices of the shape {(x, y) ∈ Z 2 : x ≡ ωy (mod 4v 2 1 ), |x| ≍ T 1/2 1 T 3 , |y| ≍ T 4 } will allow us to recover u = x, w 1 = y which then gives v 0 = (P u 2 − w 0 w 2 1 )/4v 2 1 .Applying Lemma 5.3 gives the bound We summarize this as: Proposition 5.5.Suppose that t 1 , t 2 , t 3 , t 4 satisfies (5.12) and one of (5.14) or (5.15).Then This then implies t 3 + t 4 ≤ 4δ 2 , so are excluded from the theorem as long as δ 2 < δ.In the second case we have (5.16) By (5.8), the latter inequality implies and therefore (5.12) gives We note that the condition (5.17) and (5.8) imply that t 1 ≤ 1/2 + 2δ 2 .To see this, by (5.8) we have If t 1 > 1/2 + 2δ 2 , then in turn we find that This in turn violates (5.17).We thus conclude from (5.17) that (5.18) But then (5.16) implies that Feeding these estimates back into (5.8)gives Next (5.12)implies that and feeding this back into (5.18)gives We summarize this as: Proposition 5.6.Suppose that (5.12) and (5.16) hold.Then

Now observe that
We now seek to apply Corollary 2 in [5] to (5.3) or Proposition with the variables satisfying (5.6).Indeed, we will have The last inequality comes from the observation that P, w 0 , v 0 are pairwise co-prime.Proposition 5.2 then gives the bound (5.22) , equation (5.22) and (5.8) give the bound for any ε > 0. Using Multiplying by T 1 , T 2 we obtain 4 (T 1 T 2 ) 1/4 Z ε .By (5.18) and (5.19) we have Therefore the total contribution is at most (5.23) and summing P ≪ X 1/2 gives a total contribution of by choosing δ 2 sufficiently small with respect to ε.This completes the proof of Theorem 5.1.
We remark that the estimate (5.23) gives us more room than we need, and this can be used to good effect to control the number of curves with large Szpiro ratio.

Curves with large Szpiro constant
Recall that for an elliptic curve E, the Spizro ratio is defined to be (6. 1) Our goal in this section is to count curves in E 2 having bounded conductor and such that the average of the Szpiro constants of E and φ(E) is as large as possible.For a given E = E a,b ∈ E 2 , put P I * 0 (E), P III (E), P III * (E) for the rational primes for which E has Kodaira symbols I * 0 , III, III * respectively.Recall that Next we put Σ(E) for the set of prime powers p ki i for 1 ≤ i ≤ m and q ℓj j , 1 ≤ j ≤ n such that p ki i exactly divides b and Here . and The condition of C(E a,b ) ≤ X then implies that (6.2) .
We may use the fact that elements in E 2 are naturally connected by a rational 2-isogeny mapping which preserves the conductor and essentially swaps the roles of b, c.
, where u, v are square-free integers co-prime to P 1 , P 2 respectively.It follows that our bound condition becomes Using the symmetry between u, v, we may assume that |u| ≤ |v| and therefore |u| ≤ Y /P 1 P 2 .
6.1.Application of linear programming.Put (6.6) ) Further put , and γ III * = log P I * 0 log X Then the average of the Szpiro ratios of the pair E a,b , E −2a,a 2 −4b is .
An application of linear programming bounds yields: Then Theorem 1.4 then follows from Proposition 6.2 and the following proposition: Proposition 6.3.We have .
Let us describe how to prove Theorem 1.4.Let E 2,κ (θ; X) be the set given by (1.10), namely Proposition 6.2 then asserts that Thus the proof of Theorem 1.4 requires us to estimate the size of the two sets on the right.The set on the left is easy to handle, as we will see in the following lemma: Proof.We further partition the set in the lemma into two ranges, namely We estimate the size S 1 as follows.Using the fact that our Szpiro ratio is bounded by 2κ, we find that given To estimate the size of S 2 , we cut into dyadic ranges X ψ /2, X ψ with In this range, there are O(X ψ log X) choices to choose P 1 , P 2 (namely, we select a square-free integer m satisfying X ψ ≪ m ≪ X ψ , and then take divisors).Again, our bound on the Szpiro ratio implies there are at most (2⌈κ/ψ⌋ + 1) 4 choices for Q 1 , Q 2 .Thus, this gives a total contribution of O X ψ (log X) 2 contributions.Summing over dyadic ranges, we find that each dyadic interval contributes O X δ (log X) 3 possible curves.We conclude that Thus, for δ small, this is comfortably an error term.
It remains to prove Propositions 6.1 and 6.2.We will prove the latter first.
Note that Thus, Lemma 6.1 immediately implies that Theorem 1.4 holds with for all δ sufficiently small.
We will now prove Proposition 6.1.
6.3.Proof of Proposition 6.1.In this subsection we follow the strategy of [15], and fix sets of primes P = P I * 0 ∪ P III ∪ P III * corresponding to the Kodaira symbols in the subscripts and primes Σ = P 1 ∪ P 2 corresponding to primes having multiplicative bad reduction.We are then interested in counting the number of elements in (6.18) We therefore obtain the decomposition of N (P, Σ)(X) as a sum (6.19) where N u (Z) is the number of solutions to the inequality . We will establish the following estimate for N (P, Σ)(X): Proposition 6.5.Let N (P, Σ)(X) be given as in (6.19).We then have the bounds: and Proof.We estimate N u (Q 2 Y P −1 1 |u| −1 ) in several different ways, depending on the relative sizes of the quantities involved.

First note that
Further we have (6.22) Thus a is constrained to a union of intervals of total length O( Q 2 Y /P 1 |u|).It follows that Next we consider the possibility that √ Y This is equivalent to This condition implies

If this holds then we obtain the estimate
.
If on the other hand In the range one has to consider the possibility, as above, that .
which is the same bound as (6.23).Otherwise In the range (6.25) we have that a 2 is constrained by .
Dividing by P 2 Q 2 to account for the congruence, this means that for a fixed u satisfying (6.25), we have Summing over the range (6.25) then gives the bound for some δ > 0. The condition It follows that Y 3/4 (P 1 P 2 ) 3/4 (P 2 Q 2 ) 1/2 ≤ Y 3/4 (P 1 P 2 ) 5/4 .Thus, the first two lines of (6.20) can be replaced with If (6.26) holds then we obtain the bound (6.28) Summing over dyadic intervals gives Proposition 6.1.
6.4.Proof of Proposition 6.3.To push beyond the 9/4 barrier, we note that the optimal solution given by Proposition 6.2 with r = 0 comes from α 1 , α 2 , β 1 , β 2 all nearly equal to 1/4.This means that the curves at the boundary all satisfy P i ≍ Q i , so most of the exponents in Q i are equal to one.To measure this, we impose the condition that (6.29) That is, the primes of multiplicative bad reduction with higher multiplicity mostly divide the discriminant at most twice.
Put R i for the product of all primes p such that p 2 ||b, c respectively, and put S i so that P i Q i = R 2 i S i for i = 1, 2. Then (6.29) implies that we have We then obtain Note that rad(S 1 S 2 ) ≥ 1, and since S 1 S 2 ≤ X 6ν , it follows that To finish the proof, it suffices to note that Theorem 5.1 and its proof are given in terms of an auxiliary parameter Z. Replacing Z with X 1+6ν P −2 in (5.23) gives (6.32) X 1+6ν P 2 Choosing ε appropriately then gives Proposition 6.3.

3-Selmer elements of elliptic curves with a marked 2-torsion point
We follow the parametrization obtained by Bhargava and Ho [3] for 3-Selmer elements of elliptic curves in the family E 2 with a marked rational 2-torsion point.In particular, they proved that the 18-dimensional space of triples of 3 × 3 symmetric matrices represent elements of the 3-Selmer groups of elliptic curves with a marked 2-torsion point.Further they showed that 3-Selmer elements admit integral representatives, so we may take triples of integral 3 × 3 symmetric matrices instead.Moreover we must take equivalence classes of a GL 3 (Z) × GL 3 (Z)-action obtained by letting GL 3 act simultaneously on the three ternary quadratic forms defined by each of the matrices in the triple, and on the triple itself.In order to obtain a faithful action we must mod out by a certain element of order 3, which gives us an action by the group SL 2 3 (Z)/µ 3 .
Using a product of two Siegel fundamental domains for the action of SL 3 (Z) on SL 3 (R) one then obtains a fundamental domain for the action of SL 2 3 (Z) on V 3 , the space of triples of 3 × 3 symmetric matrices.Further, they proved that the problematic regions with regards to geometry of numbers, namely the so-called cusps, contain only irrelevant points.Using the now-standard "thickening and cutting off the cusp" method of Bhargava (see [2], [3], [4], etc.), one then obtains an expression of the shape Here N (S; X) counts the number of G(Z)-equivalence classes of irreducible elements in S having height bounded by X, and where G 0 is a compact, semi-algebraic left K-invariant subset of the group G(R) = GL 2 3 (R)/µ 3 which is the closure of a non-empty, connected open set and such that every element has determinant at least one and R = R (i) denotes a connected and bounded set representing real-orbits of G(R) acting on V 3 (R).Here I ′ (α) is a measurable subset of [−1/2, 1/2] 3 dependent only on α ∈ A ′ and c > 0 is an absolute constant.Now the set ναG 0 R is then seen as the image of ναG 0 acting on R, where ναG 0 is the left-translation of G 0 by ν ∈ N ′ and α ∈ A ′ .
A key observation, implicit in previous works on this subject but seemingly not written down explicitly, is that obtaining an asymptotic formula for N (S; X) boils down to an acceptable estimate for the number of integral points inside E(ν, α, X).Note that the set ναG 0 R is independent of the choice of height H.The height function used by Bhargava and Ho in [3] has degree 36: it is the defined to be the maximum of |a 2 | 6 , |a 4 | 3 where a 2 , a 4 are degree 6 and degree 12 homogeneous polynomials in the entries of B ∈ V 3 (R) respectively.For us the height is given by the conductor polynomial, which we recall is given by (3.1).For a = a 2 , b = a 4 the conductor polynomial evidently has degree 24 instead of 36.
The key observations are the following: we can use the structure of the action of G(R) on V to transform the problem of counting in a bounded region in V to a corresponding problem of counting in a region in R 2 , where our earlier arguments of counting elliptic curves in the related region apply.As long as the counting problem for Selmer elements is compatible with the error estimates obtained in the essentially purely geometric reduction argument, we are able to obtain a suitable count of the total number of Selmer elements and thus obtain our desired outcome.
We remark that the condition |C(B)| < X does not define a bounded set in terms of a 2 , a 4 as is the case with Bhargava and Ho's choice of height function in [3].However, since C(B) is defined in terms of the invariant functions a 2 , a 4 , that the value of C(B) homogeneously expands with B. In particular, for any g ∈ G(R) the value of C(g •B) depends only on deg g and C(B), and not on the u, t, x parameters in (7.1).
The key lemma we require, which is proved in its essential form as Proposition 7.3 in [3], is the following: .
Proof.In order to obtain this Proposition we essentially follow along the same lines as in the proof of Propositions 7.3 and 7.6 in [3].However, some remarks are in order.First, the homogeneous expansion condition still holds, regardless of the choice of height function.In particular |C(B)| ≤ X and the assumption that |b 111 | ≥ 1 imply (25) in [3], namely the existence of an absolute constant J such that for each entry b ijk of B we have . This condition implies that the volume computations and projection arguments in [3] apply equally well in our setting.In particular, the proof essentially follows along the same lines as given in [3].The key difference is that in the proofs of Proposition 7.3-7.6 in [3] one must replace the parameter k = 36 with k = 24; all other details are the same.
An additional issue is that we are looking to exclude the points satisfying a 4 = 0 or a 2 2 − 4a 4 = 0. We have the following lemma to address this issue: Lemma 7.2.For B ∈ Z 3 ⊗ Sym 2 (Z 3 ), let a 2 , a 4 be the invariant polynomials given in [3].Then the number of elements B ∈ F hR ∩ V (Z) with |C(B)| < X and a 4 (a 2 2 − 4a 4 ) = 0 is O X .
Proof.To solve the equation a 4 = 0 or a 2 2 − 4a 4 = 0 we fix all but one coefficient.As in [3] this amounts to O X Next, we require a change-of-measure formula.This is proved in [3] as well, namely it is their Proposition 7.7.We state this again for completeness.Proposition 7.3 (Proposition 7.7, [3]).There exists a rational number J such that, for any measurable function φ on V (R), we have In particular, Proposition 7.3 implies d a.

2. 2 .
Contributions to the conductor for type-I * 0 primes.Put a = p k a 0 and b = p ℓ b 0 .Then our conductor polynomial takes the form C(E a,b ) = p ℓ b 0 (p 2k a 2 0 − 4p ℓ b 0 ).

2 . 3 .
and only if a ≡ 0 (mod p) and b ≡ 0 (mod p 2 ), b ≡ 0 (mod p 3 ).Thus there are p 2 (p − 1) choices for (a, b) ∈ (Z/p 3 Z) 2 , and the relative density is (p − 1)/p 4 .Contributions to the conductor for type-III primes.If a given curve E a,b ∈ E 2 has Kodaira symbol III at a prime p, then we must have p|b exactly.We may then put b = pb 0 where p ∤ b 0 .Our curve then has the equation E a,b :

2 . 4 .
E a,b has Kodaira symbol III at p if and only if a ≡ 0 (mod p) and b ≡ 0 (mod p), b ≡ 0 (mod p 2 ).There are p(p − 1) such possibilities, and the relative density is (p − 1)/p 3 .Contributions to the conductor for type-III * primes.If a given curve E a,b ∈ E 2 has Kodaira symbol III * at a prime p, then we must have p 3 |b exactly.We may then put b = p 3 b 0 where p ∤ b 0 .Our curve then has the equation E a,b :

χ
Σp,nq : Z 2 p → R be the characteristic function of the set of all (a, b) ∈ Z 2 p that satisfy the reduction type specified by Σ p and satisfy nq| ind(E a,b ).Let us put χ p = 1 − χ Σp,nq and define χ k := p|k χ p for square-free integers k.Then we have (4.2) p χ Σp,nq (a, b) = k µ(k)χ k (a, b)

3 4
−κ where ν * (nq, Σ) is the product over all primes p of the p-adic integral of χ Σp,nq .For each n, put λ i (n, Σ) for the volume of the closure in Ẑ2 of the set of all (a, b) ∈ Z 2 such that E a,b belongs to G = E 2 (Σ) and E a,b has index n.Returning to (4.1), we obtain

Proposition 7 . 1 .
Let h take a random value in G 0 uniformly with respect to the Haar measure dg.Then the expected number of elements B ∈ F hR ∩ V (Z) such that |C(B)| < X, |a 4 |, |a 2 2 − 4a 4 | ≥ 1, and b 111 = 0 is equal to Vol(R X ) + O X 17 24 b) choices, where the prime indicates that precisely one term is missing in the product.By choosing to remove the largest weight we can guarantee that ′ b w(b) ≤ 1.Having chosen 17 of the variables, the remaining variable (regardless of size) can be chosen in at most 24 ways by solving the corresponding polynomial equation in a single variable.
δ for any δ > 0. The third line of (6.21) can then be replaced with