SOME BOUNDS ON THE COEFFICIENTS OF COVERING CURVES

T.A. FISHER

Abstract. We compute bounds on the coefficients of the equations defining everywhere locally soluble n-coverings of elliptic curves over the rationals for n = 2, 3, 4. Our proofs use recent work of the author with Cremona and Stoll on the minimisation of genus one curves, together with standard results from the geometry of numbers. We use the same methods to give a criterion (satisfied by only a finite number of “small” elliptic curves) for ruling out the existence of elements of order 3 in the Tate-Shafarevich group.

1. Introduction

Let E/ℚ be an elliptic curve and n ≥ 2 an integer. The Selmer group S^{(n)}(E/ℚ) parametrises the everywhere locally soluble n-coverings π : C → E. By global class field theory the curve C admits a ℚ-rational divisor of degree n and hence can be written as either a double cover of ℙ^{1} (case n = 2) or a genus one normal curve C ⊂ ℙ^{n−1} (case n ≥ 3). The aim of a descent calculation is to compute the Selmer group S^{(n)}(E/ℚ) as an abelian group and to represent its elements by equations for the covering curves C. In view of the short exact sequence

(1.1)  0 → E(ℚ)/nE(ℚ) → S^{(n)}(E/ℚ) → � (E/ℚ)[n] → 0

this gives information about both the Mordell-Weil group E(ℚ) and the Tate-Shafarevich group � (E/ℚ). Indeed the covering curves can be used either to help search for points of infinite order in E(ℚ) or to exhibit explicit elements of � (E/ℚ).

There are two different approaches to explicit 2-descent on an elliptic curve. The number field method computes S^{(2)}(E/ℚ) as a subgroup of L^{×}/(L^{×})^{2} where L is a product of number fields. The Selmer group elements are then converted to binary quartics using a method that relies on an explicit version of the Hasse principle for conics. In contrast the invariant theory method bounds the coefficients of the required binary quartics, and then uses these bounds to make an exhaustive search. The invariant theory method was used by Birch and Swinnerton-Dyer in their pioneering computer calculations [BSD] and subsequently developed by Cremona in his program mwrank. The development of computer algebra packages able to compute the class group and units of number fields has since made the number field method equally suitable for computation.

Date: 29th July 2010.

The number field method has been generalised to p-descent (see [DSS], [SS], [CFOSS]) and is practical for p = 3 (and p = 5 in small examples). The method relies on an explicit version of the local-to-global principle for the p-torsion of the Brauer group of ℚ. The number field method also extends to 4-descent and 8-descent, as described in [MSS], [Wo], [S]. The invariant theory method in the case n = 3 was investigated in [DS], but does not appear to generalise in any practical way to n > 2.

The equations defining an n-covering C of E depend on a choice of coordinates on ℙ^{n−1}. It is obviously desirable to make a change of co-ordinates so that the equations have small integer coefficients. In practice this is achieved by the combination of two techniques, termed minimisation and reduction. In the minimisation stage spurious prime factors are removed from a suitably defined discriminant. In the reduction stage an integer unimodular change of co-ordinates is made to further reduce the size of the coefficients (without changing the discriminant). Minimisation and reduction are important for both the number field and invariant theory methods. In the number field method the equations computed typically have very large coefficients, and we need to minimise and reduce to get sensible answers. In the invariant theory method minimisation and reduction are used at the outset to obtain the bounds upon which the method relies.

In joint work with Cremona and Stoll [CFS] the author has described efficient algorithms for minimising and reducing n-coverings for n = 2, 3, 4. (The work on minimisation applies over an arbitrary local field.) It has been found in numerical examples that elements of the Tate-Shafarevich group typically have quite small coefficients and that the size of the coefficients tends to decrease with n. In this note we give some theoretical support for these observations. In fact we give bounds on the coefficients depending only on the naive height of E. In principle this generalises the invariant theory method to n = 3, 4 although the result is certainly not a practical algorithm. In view of this we concentrate on giving a single bound for all the coefficients and do not keep track of certain implied constants. Thus our treatment in the cases n = 2, 3 differs from that in [BSD], [DS].

In the cases n = 2, 3, 4 we represent Selmer group elements by equations of the following form.

Definition 1.1. A genus one model of degree n ∈ {2, 3, 4} is

(i) if n = 2, a binary quartic, i.e. a homogeneous polynomial of degree 4 in 2 variables, (ii) if n = 3, a ternary cubic, i.e. a homogeneous polynomial of degree 3

in 3 variables, (iii) if n = 4, a quadric intersection, i.e. a pair of homogeneous polyno-

mials of degree 2 in 4 variables.

We recall that the minimal discriminant of an elliptic curve E/ℚ is

Δ_{E} = (c_{4}^{3} − c_{6}^{2})/1728

where c_{4} and c_{6} are the usual quantities associated to a globally minimal Weierstrass equation for E. In Theorem 1.2 below we instead work with the naive height of E which we define as

H_{E} = max(|c_{4}|^{1/4}, |c_{6}|^{1/6}).

We write ||Φ||_{∞} for the maximum absolute value of a coefficient of a genus one model Φ. The notation f ≪ g should be understood to mean that f ≤ cg for some absolute constant c > 0.

Theorem 1.2. Let E/ℚ be an elliptic curve and let n ∈ {2, 3, 4}.

(a) Each ξ ∈ S^{(n)}(E/ℚ) can be represented by a genus one model Φ with integer coefficients and

||Φ||_{∞} ≪ H_{E}^{6}.

(b) If ξ is non-zero in S^{(n)}(E/ℚ) then this bound may be improved to

||Φ||_{∞} ≪ H_{E}^{4}.

(c) If the image of ξ in � (E/ℚ) has exact order n then

||Φ||_{∞} ≪ H_{E}^{6−n}.

We remark that Theorem 1.2(a) gives a proof that S^{(n)}(E/ℚ) is finite, and hence by (1.1) a proof of the weak Mordell-Weil theorem for n = 2, 3, 4. This proof differs from the usual proofs in that we work entirely over the rationals, i.e. we do not need to make any field extensions.

The formulae in Lemmas 3.11 and 3.12 of [CFS] suggest that the exponents of H_{E} in Theorem 1.2(a) and (b) might be best possible. We suspect that the exponent of H_{E} in Theorem 1.2(c) is also best possible in view of the models

n = 2 y^{2} = λ_{0}x^{4} + x^{2}z^{2} + λ_{1}z^{4}
n = 3 λ_{0}x_{0}^{3} + λ_{1}x_{1}^{3} + λ_{3}x_{3}^{3} − x_{0}x_{1}x_{2} = 0
n = 4  \{ \begin{matrix}λ_{0}x_{0}^{2} + x_{1}x_{3} − λ_{2}x_{2}^{2} & = 0 \\ λ_{1}x_{1}^{2} + x_{0}x_{2} − λ_{3}x_{3}^{2} & = 0 \end{matrix}

that arise in the context of descent by cyclic isogeny (see [F1, §1.2] for the cases n = 3, 4).

We expect that Theorem 1.2 generalises to the case n = 5. (See [F3] for the definition of a genus one model of degree 5.)

In favourable circumstances, the geometry of numbers can be used to construct a rational point on a smooth plane cubic. We turn this into a criterion for ruling out the existence of elements of order 3 in the Tate-Shafarevich group.

Theorem 1.3. Let E be an elliptic curve over ℚ with j-invariant j and minimal discriminant Δ_{E}. Let

B = min{|x| : x ∈ ℂ a root of (X − 3^{3})(X − 3^{5})^{3} + jX^{3} = 0}.

If |Δ_{E}| < \frac{1}{64}B^{3} then � (E/ℚ)[3] = 0.

Since B is bounded as a function of j this theorem applies to only finitely many elliptic curves. In Cremona’s tables [C], which go up to conductor 130000, we find 92 elliptic curves satisfying the condition of the theorem, and all of these have conductor less than 1000. Their ranks are distributed as follows

rank 0 1 2 # curves 49 41 2

There is no difficulty in verifying by 3-descent (see [SS]) that each of these curves has � (E/ℚ)[3] = 0. The interest of Theorem 1.3 instead lies in its method of proof, and in the hope that similar criteria might be found for ruling out elements of order n in � (E/ℚ) for other integers n.

Example 1.4. Let E be the elliptic curve

y^{2} + y = x^{3} + x^{2} − 2x.

Then Δ_{E} = 389, j = 2^{12}7^{3}/389 and \frac{1}{64}B^{3} = 528.57930586 . . .. Theorem 1.3 shows that � (E/ℚ)[3] = 0. In fact E(ℚ) ≅ ℤ^{2} and the (inverse pairs of) non-trivial elements of S^{(3)}(E/ℚ) ≅ (ℤ/3ℤ)^{2} are represented by the ternary cubics

F_{1}(x, y, z) = x^{2}z − xy^{2} − 2xyz + xz^{2} + y^{2}z + yz^{2}
F_{2}(x, y, z) = x^{2}z − xy^{2} + 2xyz − yz^{2} − z^{3}
F_{3}(x, y, z) = x^{2}y − xy^{2} − xz^{2} − y^{2}z − 2yz^{2}
F_{4}(x, y, z) = x^{2}y + xy^{2} − 2xyz + xz^{2} − y^{2}z − yz^{2}.

2. Background and overview

2.1. Invariants of genus one models. We work over a field K of characteristic zero and write \overline{K} for its algebraic closure. The space of genus one models of degree n = 2, 3, 4 is acted on by the group G_{n} defined as follows

G_{2} = 𝔾_{m} × GL_{2}  [μ, N] : F ↦ μ^{2}(F ∘ N)
G_{3} = 𝔾_{m} × GL_{3}  [μ, N] : F ↦ μ(F ∘ N)
G_{4} = GL_{2} × GL_{4}  [M, N] : (Q_{1}, Q_{2})^{T} ↦ M(Q_{1} ∘ N, Q_{2} ∘ N)^{T} .

Let det : G_{n} → 𝔾_{m} be the character defined by [μ, N] ↦ μ det N, respectively [M, N] ↦ det M det N. An invariant of weight k is a polynomial in the coefficients of a genus one model satisfying

(2.1)  I(gΦ) = det(g)^{k}I(Φ)

for all g ∈ G_{n}. The action of the centre of G_{n} shows that I is homogeneous of degree kn/(6 − n). In each of the cases n = 2, 3, 4 the ring of invariants is generated by invariants c_{4} and c_{6} of weights 4 and 6. See [F3, §7], [CFS] for explicit formulae. We put Δ = (c_{4}^{3} − c_{6}^{2})/1728. It is shown in [AKM^{3}P],

[F3] that Φ is non-singular (i.e. defines a smooth curve of genus one) if and only if Δ(Φ) ≠ 0, and that the Jacobian elliptic curve is

(2.2)  y^{2} = x^{3} − 27c_{4}(Φ)x − 54c_{6}(Φ).

Definition 2.1. Genus one models Φ_{1} and Φ_{2} are K-equivalent if they are in the same orbit for the action of G_{n}(K). They are properly K-equivalent if Φ_{2} = gΦ_{1} for some g ∈ G_{n}(K) with det g = 1.

Lemma 2.2. Non-singular genus one models Φ_{1} and Φ_{2} are properly \overline{K}- equivalent if and only if they have the same invariants.

Proof: The first implication is clear by (2.1). For the converse, we see by Propositions 4.6 and 4.7 in [F3] that every non-singular model is properly \overline{K}-equivalent to a model of the form

n = 2  y^{2} = x^{3}z + Axz^{3} + Bz^{4}
n = 3  y^{2}z = x^{3} + Axz^{2} + Bz^{3}
n = 4  x^{2} − zt = y^{2} − xt − Axz − Bz^{2} = 0.

It then suffices to note that these “Weierstrass models” are uniquely determined by their invariants. In fact c_{4} = −48A and c_{6} = −864B. □

A non-singular genus one model Φ defines both a smooth curve of genus one C and a regular 1-form ω on C. Writing F_{i} for the partial derivative of F with respect to x_{i} we have

n = 2  y^{2} = F (x_{0}, x_{1})  ω = x_{0}^{2}d(x_{1}/x_{0})/2y
n = 3  F (x_{0}, x_{1}, x_{2}) = 0  ω = x_{0}^{2}d(x_{1}/x_{0})/F_{2}
n = 4  F = G = 0  ω = x_{0}^{2}d(x_{1}/x_{0})/(F_{2}G_{3} − F_{3}G_{2}).

It is shown in [F3, Proposition 5.19] that if Φ_{2} = gΦ_{1} and γ : C_{2} → C_{1} is the morphism determined by g then

(2.3)  γ^{∗}ω_{1} = (det g)ω_{2}.

2.2. Galois cohomology. We consider pairs (C → S, ω) where C → S is a morphism from a smooth curve of genus one C to a Brauer-Severi variety S, and ω is a regular 1-form on C. An isomorphism between (C_{1} → S_{1}, ω_{1}) and (C_{2} → S_{2}, ω_{2}) is a pair of isomorphisms ϕ : C_{1} ≅ C_{2} and ψ : S_{1} ≅ S_{2} such that ϕ^{∗}ω_{2} = ω_{1} and the diagram

\xymatrix{C_{1} \ar[r] \ar[d]_{ϕ} & S_{1} \ar[d]^{ψ} \\ C_{2} \ar[r] & S_{2}}

commutes.

Let n ≥ 2 be an integer. Let E/K be an elliptic curve with invariant differential ω_{E}. We map E → ℙ^{n−1} via the complete linear system |n.0_{E}|. We recall that objects defined over K are called twists if they are isomorphic over \overline{K}.

Lemma 2.3. The twists of (E → ℙ^{n−1}, ω_{E}), up to K-isomorphism, are parametrised by H^{1}(K, E[n]).

Proof. This is [F2, Lemma 2.3]. □

The obstruction map, defined in [O], [CFOSS], is

Ob : H^{1}(K, E[n]) → Br(K)
(C → S, ω) ↦ [S].

In general this map is not a group homomorphism. Nonetheless we write ker(Ob) for the inverse image of the identity.

Lemma 2.4. Let E/K be an elliptic curve and let n ∈ {2, 3, 4}. Then the genus one models of degree n with the same invariants as a fixed Weierstrass equation for E, up to proper K-equivalence, are parametrised by ker(Ob) ⊂ H^{1}(K, E[n]).

Proof: A non-singular genus one model Φ defines a smooth curve of genus one C → ℙ^{n−1} and a regular 1-form ω on C. Conversely, every twist (C → S, ω) of (E → ℙ^{n−1}, ω_{E}) with S ≅ ℙ^{n−1} arises in this way. Let Φ_{E} be a genus one model defining (E → ℙ^{n−1}, ω_{E}). By (2.2) it has the same invariants as some Weierstrass equation for E. We see by (2.3) that Φ_{1} and Φ_{2} are properly equivalent if and only if they determine isomorphic pairs (C_{1} → ℙ^{n−1}, ω_{1}) and (C_{2} → ℙ^{n−1}, ω_{2}). Thus ker(Ob) parametrises the genus one models properly \overline{K}-equivalent to Φ_{E}, up to proper K-equivalence. By Lemma 2.2 the genus one models properly \overline{K}-equivalent to Φ_{E} are those with the same invariants as Φ_{E}. □

Remark 2.5. The subset ker(Ob) ⊂ H^{1}(K, E[n]) contains the identity and is closed under taking inverses. A binary quartic represents the identity if and only if it has a K-rational root. A ternary cubic, respectively quadric intersection, represents the identity if and only if it has a K-rational point of inflection, respectively hyperosculating point.

Taking Galois cohomology of the short exact sequence 0 → E[2] → E[4] → E[2] → 0 gives an exact sequence

E(K)[2]⟶H^{1}(K, E[2]) ⟶^{ι_{∗}} H^{1}(K, E[4]) ⟶^{[2]_{∗}} H^{1}(K, E[2]).

Lemma 2.6. The maps ι_{∗} and [2]_{∗} have the following interpretations.

(i) The binary quartic F (x, z) = ax^{4} + bx^{3}z + cx^{2}z^{2} + dxz^{3} + ez^{4} is mapped by ι_{∗} to the quadric intersection

x_{0}x_{2} − x_{1}^{2} = x_{3}^{2} − ax_{0}^{2} − bx_{0}x_{1} − cx_{1}^{2} − dx_{1}x_{2} − ex_{2}^{2} = 0.

(ii) The quadric intersection (Q_{1}, Q_{2}) where Q_{i}(x) = x^{T} A_{i}x for i = 1, 2 is mapped by [2]_{∗} to the binary quartic

F (x, z) = det(A_{1}x + A_{2}z).

Proof: (i) Let C_{2} be the curve defined by y^{2} = F (x, z) and C_{4} ⊂ ℙ^{3} the image of C_{2} under the 2-uple embedding

(x : y : z) ↦ (x^{2} : xz : z^{2} : y).

If C′_{2} and C′_{4} are related in the same way as C_{2} and C_{4} then each isomorphism (C_{2} → ℙ^{1}) ≅ (C′_{2} → ℙ^{1}) induces an isomorphism (C_{4} → ℙ^{3}) ≅ (C′_{4} → ℙ^{3}) compatible with the 2-uple embeddings. Hence twisting (C_{2} → ℙ^{1}) by ξ ∈ H^{1}(K, E[2]) has the effect of twisting (C_{4} → ℙ^{3}) by ι_{∗}ξ ∈ H^{1}(K, E[4]).

(ii) Let C_{4} be the curve Q_{1} = Q_{2} = 0 and C_{2} the curve y^{2} = F (x, z). Weil [We, Chapter II, Appendix III] constructs a morphism ω : C_{4} × C_{4} → C_{2} with the property that

ω(P, Q) = ω(P′, Q′) ⟺ P + Q ∼ P′ + Q′.

For fixed P ∈ C_{4} the map Q ↦ ω(P, Q) induces a map on Jacobians that is independent of the choice of P. This map is an isomorphism and we use it to identify the Jacobians of C_{4} and C_{2}. Then P ↦ ω(P, P) is a morphism that induces multiplication-by-2 on the Jacobians. Explicit formulae for this covering map are given in [AKM^{3}P], [MSS]. If C′_{4} and C′_{2} are related in the same way as C_{4} and C_{2} then each isomorphism (C_{4} → ℙ^{3}) ≅ (C′_{4} → ℙ^{3}) induces an isomorphism (C_{2} → ℙ^{1}) ≅ (C′_{2} → ℙ^{1}) compatible with the covering maps. Hence twisting (C_{4} → ℙ^{3}) by ξ ∈ H^{1}(K, E[4]) has the effect of twisting (C_{2} → ℙ^{1}) by [2]_{∗}ξ ∈ H^{1}(K, E[2]). □

2.3. Minimisation and reduction. We quote the following result on minimisation.

Proposition 2.7. Let n ∈ {2, 3, 4}. Let C be an everywhere locally soluble n-covering of an elliptic curve E/ℚ. Let c_{4} and c_{6} be the invariants of a minimal Weierstrass equation for E. Then C can be defined by an integer coefficient genus one model with invariants c_{4} and c_{6}, except in the case n = 2 where it may only be possible to find a model with invariants 2^{4}c_{4} and 2^{6}c_{6}.

Proof: This is [CFS, Theorem 1.1]. In [CFS] we gave a more general definition of genus one model of degree 2. The models considered here are obtained by completing the square. This has the effect of multiplying the invariants c_{4} and c_{6} by 2^{4} and 2^{6}. □

Our treatment of reduction differs from that in [CFS]. In that paper our goal was to find a practical algorithm for reducing, whereas here we are interested in bounding coefficients. In Section 3 we prove

Proposition 2.8. Let n ∈ {2, 3, 4}. Let Φ be a non-singular real genus one model of degree n with invariants c_{4} and c_{6}. Then Φ is properly ℝequivalent to a genus one model Φ′ with ||Φ′||_{∞} ≪ H^{(6−n)/n} where H = max(|c_{4}|^{1/4}, |c_{6}|^{1/6}).

Since c_{4} and c_{6} are polynomials of degrees 4n/(6 − n) and 6n/(6 − n) the exponent of H in Proposition 2.8 is best possible. Combining the last two propositions we immediately deduce

Theorem 2.9. Let n ∈ {2, 3, 4}. Let C be an everywhere locally soluble n-covering of an elliptic curve E/ℚ. Then C can be defined by an integer coefficient genus one model that is properly ℝ-equivalent to a genus one model Φ′ with ||Φ′||_{∞} ≪ H_{E}^{(6−n)/n}.

We write ||x|| = (∑ x_{i}^{2})^{1/2} for the usual Euclidean norm. In Section 4 we use the geometry of numbers to deduce Theorem 1.2 from Theorem 2.9. The key fact here is

Lemma 2.10 (Minkowski). Let Λ ⊂ ℝ^{n} be a rank n lattice with covolume 1. Then there are linearly independent vectors v_{1}, …, v_{n} ∈ Λ with ∏_{i=1}^{n} ||v_{i}|| ≤ γ_{n}^{n/2} where γ_{n}^{n} is Hermite’s constant.

Proof: See for example [PZ, p197]. In fact for n ≤ 4 we can take v_{1}, …, v_{n} a basis for Λ. □

The exact value of Hermite’s constant is known for n ≤ 8.

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ γ_{n}^{n} & 1 & 4/3 & 2 & 4 & 8 & 64/3 & 64 & 256 \end{matrix}

We use Lemma 2.10 to give upper bounds on all of the ||v_{i}||. For this we need lower bounds on some of the ||v_{i}||. The hypotheses in parts (a), (b) and (c) of Theorem 1.2 are used to give successively better lower bounds, and hence successively better upper bounds.

3. Normal forms for genus one models over the reals

In this section we prove Proposition 2.8.

Lemma 3.1. Let E/ℝ be an elliptic curve and n ≥ 2 an integer. (i) If n is odd or Δ_{E} < 0 then H^{1}(ℝ, E[n]) = 0. (ii) If n is even and Δ_{E} > 0 then H^{1}(ℝ, E[n]) ≅ (ℤ/2ℤ)^{2} and the ob-

struction map H^{1}(ℝ, E[n]) → Br(ℝ) has kernel of size 3.

Proof: We recall that E[n] ≅ (ℤ/nℤ)^{2} has a basis S, T with S ∈ E(ℝ) and

σ(T) = \{ \begin{matrix}−T & if Δ_{E} > 0 \\ S − T & if Δ_{E} < 0 \end{matrix}

where σ denotes complex conjugation. It is easy to compute H^{1}(ℝ, E[n]) using the rule

H^{1}(ℝ, A) = \frac{\{a ∈ A : a + σ(a) = 0\}}{\{b − σ(b) : b ∈ A\}}.

Now suppose n is even and Δ_{E} > 0. Then E(ℝ) ≅ ℤ/2ℤ × ℝ/ℤ and the exact sequence

0 → E(ℝ)/nE(ℝ) → H^{1}(ℝ, E[n]) → H^{1}(ℝ, E)[n] → 0

shows that ker(Ob) has size at least 2. Let ( , ) be the Tate pairing

H^{1}(ℝ, E[n]) × H^{1}(ℝ, E[n]) → Br(ℝ)

defined by the Weil pairing and cup product. It is shown in [O], [Z] that

(ξ, η) = Ob(ξ + η) − Ob(ξ) − Ob(η)

for all ξ, η ∈ H^{1}(ℝ, E[n]). Since the Tate pairing is non-degenerate, the obstruction map is not linear, and hence ker(Ob) has size 3. □

Let E/ℝ be an elliptic curve and let c_{4} and c_{6} be the invariants of a fixed Weierstrass equation. Lemma 2.4 identifies the proper ℝ-equivalence classes of genus one models with invariants c_{4} and c_{6} with ker(Ob) ⊂ H^{1}(ℝ, E[n]). Our strategy for proving Proposition 2.8 is therefore the following. According as we are in case (i) or (ii) of Lemma 3.1 we exhibit either 1 or 3 real genus one models with the given invariants. In case (ii) we then check that these models are not equivalent over the reals.

3.1. Binary quartics. As suggested in Lemma 3.1 we split into cases according to the sign of the discriminant.

Lemma 3.2. Let E/ℝ be an elliptic curve with positive discriminant. We fix a Weierstrass equation

(3.1)  y^{2} = (x − e_{1})(x − e_{2})(x − e_{3})

where e_{1}, e_{2}, e_{3} ∈ ℝ. Then every real binary quartic with the same invariants as (3.1) is properly ℝ-equivalent to exactly one of F_{1}, F_{2}, F_{3} where

F_{i}(x, z) = a_{i}(x^{4} + z^{4}) + 2b_{i}x^{2}z^{2}

and for i, j, k a cyclic permutation of 1, 2, 3 we put

a_{i} = (e_{i} − e_{j})/4, b_{i} = (e_{i} + e_{j} − 2e_{k})/4.

Proof: A direct calculation shows that the quartics F_{i}(x, z) have the same invariants as (3.1). Without loss of generality e_{1} < e_{2} < e_{3}. Since

4F_{i}(x, z) = (e_{i} − e_{j})(x^{2} − z^{2})^{2} + 4(e_{i} − e_{k})x^{2}z^{2}

it is clear that F_{1}(x, z) < 0 and F_{3}(x, z) > 0 for all (x : z) ∈ ℙ^{1}(ℝ), whereas F_{2}(x, z) = 0 has 4 roots in ℙ^{1}(ℝ). Hence the F_{i}(x, z) are not equivalent over the reals. □

The analogous result for negative discriminants is the following.

Lemma 3.3. Let E/ℝ be an elliptic curve with negative discriminant. We fix a Weierstrass equation

(3.2)  y^{2} = (x − e_{1})(x − e_{2})(x − e_{3})

where e_{1}, e_{2} ∈ ℂ are complex conjugates and e_{3} ∈ ℝ. Then every real binary quartic with the same invariants as (3.2) is properly ℝ-equivalent to

F (x, z) = a(x^{4} − z^{4}) + 2bx^{2}z^{2}

where

a = (e_{1} − e_{2})/4i, b = (e_{1} + e_{2} − 2e_{3})/4.

Proof: A direct calculation shows that the quartic F (x, z) has the same invariants as (3.2). □

The proof of Proposition 2.8 in the case n = 2 is completed by the following trivial lemma.

Lemma 3.4. Let e_{1}, e_{2}, e_{3} be the roots of f(x) = x^{3} − 27c_{4}x − 54c_{6}. Then max(|e_{1}|, |e_{2}|, |e_{3}|) ≪ H^{2} where H = max(|c_{4}|^{1/4}, |c_{6}|^{1/6}).

Proof: Since f(e_{i}) = 0 we have |e_{i}|^{3} ≪ max(|c_{4}e_{i}|, |c_{6}|). The result is immediate. □

3.2. Recall of analytic formulae. Before proceeding with the proof of Proposition 2.8 in the cases n = 3, 4 we recall some standard analytic formulae. For τ ∈ ℌ = {z ∈ ℂ : Im(z) > 0} and α ∈ ℚ we write q^{α} = e^{2πiατ}. The Dedekind η-function

(3.3)  η(τ) = q^{1/24} ∏_{n≥1}(1 − q^{n})

satisfies the functional equation

(3.4)  η(−1/τ) = \sqrt{\frac{τ}{i}} η(τ).

A useful formula in this context is the Jacobi triple product identity

(3.5)  ∏_{n≥1}(1 − q^{2n})(1 − q^{2n−1}z)(1 − q^{2n−1}z^{−1}) = ∑_{n∈ℤ}(−1)^{n}q^{n^{2}}z^{n}.

The spaces of modular forms of level 1 and weight k = 4, 6 are spanned by the Eisenstein series

E_{4}(τ) = 1 + 240 ∑_{n≥1} σ_{3}(n)q^{n} E_{6}(τ) = 1 − 504 ∑_{n≥1} σ_{5}(n)q^{n}

where σ_{m}(n) = ∑_{d|n} d^{m}. The discriminant modular form is

Δ(τ) = η(τ)^{24} = (E_{4}(τ)^{3} − E_{6}(τ)^{2})/1728.

The Eisenstein series E_{4} and E_{6} are related to the invariants c_{4} and c_{6} as follows.

Lemma 3.5. Let E be an elliptic curve over ℂ with Weierstrass equation

(3.6)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

Let Λ be the period lattice obtained by integrating dx/(2y + a_{1}x + a_{3}). If we choose a basis ω_{1}, ω_{2} for Λ so that τ = ω_{2}/ω_{1} ∈ ℌ then the invariants c_{4} and c_{6} of the Weierstrass equation (3.6) are given by c_{k} = (\frac{2π}{ω_{1}})^{k}E_{k}(τ).

Proof: The Weierstrass ℘-function

℘(z) = \frac{1}{z^{2}} + ∑_{0≠λ∈Λ} (\frac{1}{(z − λ)^{2}} − \frac{1}{λ^{2}} )

satisfies the equation

(3.7)  ℘′(z)^{2} = 4℘(z)^{3} − g_{2}℘(z) − g_{3}

where g_{2} = 60G_{4}(Λ), g_{3} = 140G_{6}(Λ). Moreover for k ≥ 4 we have

G_{k}(Λ) = ∑_{0≠λ∈Λ} \frac{1}{λ^{k}} = \frac{2ζ(k)}{ω_{1}^{k}}E_{k}(τ).

Hence g_{2} = \frac{1}{12}(\frac{2π}{ω_{1}})^{4}E_{4}(τ) and g_{3} = \frac{1}{216}(\frac{2π}{ω_{1}})^{6}E_{6}(τ). The uniformisation map ϕ with ϕ^{∗}(dx/(2y + a_{1}x + a_{3})) = dz is given by

ϕ : ℂ/Λ → E(ℂ)
z ↦ (℘(z) − \frac{1}{12}b_{2}, \frac{1}{2}℘′(z) − a_{1}(℘(z) − \frac{1}{12}b_{2}) − a_{3})

where b_{2} = a_{1}^{2} + 4a_{2}. A calculation comparing (3.6) and (3.7) now shows that c_{4} = 12g_{2} and c_{6} = 216g_{3}. □

3.3. Ternary cubics. Differentiating the Jacobi triple product identity (3.5) with respect to z and putting z = q we obtain

(3.8)  η(τ)^{3} = ∑_{n∈ℤ}(−1)^{n}nq^{(2n+1)^{2}/8} = q^{1/8} ∏_{n≥1}(1 − q^{n})^{3}.

Lemma 3.6. For k = 4, 6 we have

E_{k}(τ) = f_{k}(η(\frac{τ}{3})^{3}, \sqrt{27}η(3τ)^{3})/η(τ)^{k}

where

f_{4}(a, b) = a^{4} + \frac{4}{\sqrt{3}}a^{3}b + 2a^{2}b^{2} + \frac{4}{\sqrt{3}}ab^{3} + b^{4}
f_{6}(a, b) = a^{6} + 2\sqrt{3}a^{5}b + 5a^{4}b^{2} − 5a^{2}b^{4} − 2\sqrt{3}ab^{5} − b^{6}.

Proof: Let F_{k}(τ) = f_{k}(η(\frac{τ}{3})^{3}, \sqrt{27}η(3τ)^{3})/η(τ)^{k}. It is easily seen that the q-expansions of F_{4}(τ) and F_{6}(τ) each have leading term 1.

Let ζ_{n} = e^{2πi/n}. By (3.8) we have

η(\frac{τ}{3})^{3} − ζ_{24}^{−1}η(\frac{τ+1}{3})^{3} = (1 − ζ_{3}) ∑_{n≡1 mod 3}(−1)^{n}nq^{(2n+1)^{2}/24}

= (1 − ζ_{3}) ∑_{n∈ℤ}(−1)^{3n+1}(3n + 1)q^{3(2n+1)^{2}/8}

= 3(ζ_{3} − 1)η(3τ)^{3}.

Hence

(3.9)  \frac{η(\frac{τ+1}{3})^{3}}{η(τ + 1)} = \frac{η(\frac{τ}{3})^{3}}{η(τ)} + \sqrt{27}iζ_{3}^{2} \frac{η(3τ)^{3}}{η(τ)} .

It is readily verified that

f_{k}(a + iζ_{3}^{2}b, ζ_{3}b) = f_{k}(a, b).

Hence F_{k}(τ + 1) = F_{k}(τ). A straightforward calculation using the functional equation (3.4) shows that F_{k}(−1/τ) = τ^{k}F_{k}(τ). Since the space of modular forms of level 1 and weight k = 4, 6 is 1-dimensional it follows that E_{k} = F_{k}.

□

Lemma 3.7. Let E/ℝ be an elliptic curve with Weierstrass equation

(3.10)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

Let Λ = ℤω_{1} + ℤω_{2} be the period lattice obtained by integrating dx/(2y + a_{1}x + a_{3}). We may assume that ω_{1} ∈ ℝ_{>0} and τ = ω_{2}/ω_{1} ∈ ℌ with Re(τ) ∈ {0, 3/2}. Then every real ternary cubic with the same invariants as (3.10) is properly ℝ-equivalent to

F (x, y, z) = a(x^{3} + y^{3} + z^{3}) − 3(a + \sqrt{3}b)xyz

where

a = \frac{1}{\sqrt{27}} (\frac{2π}{ω_{1}} ) \frac{η(τ/3)^{3}}{η(τ)} b = (\frac{2π}{ω_{1}} ) \frac{η(3τ)^{3}}{η(τ)} .

Proof: Since Re(τ) ∈ {0, 3/2} it is clear that q^{1/3} is real, and hence a and b are real. For k = 4, 6 we compute

c_{k}(F) = 3^{3k/2}f_{k}(a, b) = (\frac{2π}{ω_{1}})^{k}f_{k}(η(\frac{τ}{3})^{3}, \sqrt{27}η(3τ)^{3})/η(τ)^{k} = (\frac{2π}{ω_{1}})^{k}E_{k}(τ).

It follows by Lemma 3.5 that F has the same invariants as (3.10). □

The proof of Proposition 2.8 in the case n = 3 is completed by

Lemma 3.8. For τ ∈ ℌ with Re(τ) ∈ {0, 3/2} we have

(3.11)  max (|\frac{η(τ/3)^{3}}{η(τ)} | , |\frac{η(3τ)^{3}}{η(τ)} |) ≪ max(|E_{4}(τ)|^{1/4}, |E_{6}(τ)|^{1/6}).

Proof: The functional equation (3.4) shows we are free to replace τ by −1/τ. Likewise (3.9) shows we may replace τ by τ + 1. So if the bound holds on some subset of ℌ, then it will hold on any SL_{2}(ℤ)-translate of that subset (possibly with a different implied constant).

We only need to establish the bound for Im(τ) large and Im(τ) small, since the result will then follow by a compactness argument. (Note that E_{4} and E_{6} have no common zeros in ℌ.) As Im(τ) → ∞ we have q → 0 and the result is clear. By the action of SL_{2}(ℤ) this implies the result for Im(τ) small. □

3.4. Quadric intersections. Putting z = −q, ±1 in the Jacobi triple product identity (3.5) we obtain functions

(3.12)  \begin{aligned}θ_{2}(τ) & = ∑_{n∈ℤ}q^{(2n+1)^{2}/4} = 2q^{1/4} ∏_{n=1}^{∞}(1 − q^{2n})(1 + q^{2n})^{2} \\ θ_{3}(τ) & = ∑_{n∈ℤ}q^{n^{2}} = ∏_{n=1}^{∞}(1 − q^{2n})(1 + q^{2n−1})^{2} \\ θ_{4}(τ) & = ∑_{n∈ℤ}(−1)^{n}q^{n^{2}} = ∏_{n=1}^{∞}(1 − q^{2n})(1 − q^{2n−1})^{2}. \end{aligned}

Lemma 3.9. For k = 4, 6 we have

E_{k}(τ) = f_{k}(θ_{2}(τ), θ_{3}(τ)) = (\frac{1}{2i})^{k}f_{k}(θ_{3}(\frac{τ}{4}), θ_{4}(\frac{τ}{4}))

where

f_{4}(a, b) = a^{8} + 14a^{4}b^{4} + b^{8}
f_{6}(a, b) = a^{12} − 33a^{8}b^{4} − 33a^{4}b^{8} + b^{12}.

Proof: Let F_{k}(τ) = f_{k}(θ_{2}(τ), θ_{3}(τ)). It is clear that F_{4}(τ) and F_{6}(τ) are power series in q with constant term 1. So to prove the first equality it suffices to show that F_{k}(−1/τ) = τ^{k}F_{k}(τ) for k = 4, 6.

The expressions for the θ_{j}(τ) as products allow us to rewrite them in terms of the Dedekind η-function:

θ_{2}(τ) = 2\frac{η(4τ)^{2}}{η(2τ)} ,  θ_{3}(τ) = \frac{η(2τ)^{5}}{η(τ)^{2}η(4τ)^{2}} ,  θ_{4}(τ) = \frac{η(τ)^{2}}{η(2τ)}.

By the functional equation (3.4) and the expressions for the θ_{j}(τ) as sums, we deduce

(3.13)  \begin{aligned}θ_{2}(−1/τ) & = \sqrt{\frac{τ}{2i}}θ_{4}(\frac{τ}{4}) = \sqrt{\frac{τ}{2i}}(−θ_{2}(τ) + θ_{3}(τ)) \\ θ_{3}(−1/τ) & = \sqrt{\frac{τ}{2i}}θ_{3}(\frac{τ}{4}) = \sqrt{\frac{τ}{2i}}(θ_{2}(τ) + θ_{3}(τ)). \end{aligned}

It is readily verified that

f_{k}(−a + b, a + b) = (2i)^{k}f_{k}(a, b).

Hence

F_{k}(−1/τ) = f_{k}(θ_{2}(−1/τ), θ_{3}(−1/τ))
= (\frac{τ}{2i})^{k}f_{k}(−θ_{2}(τ) + θ_{3}(τ), θ_{2}(τ) + θ_{3}(τ))
= τ^{k}F_{k}(τ).

Since the space of modular forms of level 1 and weight k = 4, 6 is 1dimensional it follows that E_{k} = F_{k}. The second expression for E_{k} is obtained by replacing τ by −1/τ and using (3.13). □

As suggested in Lemma 3.1 we split into cases according to the sign of the discriminant.

Lemma 3.10. Let E/ℝ be an elliptic curve with positive discriminant and with Weierstrass equation

(3.14)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

Let Λ = ℤω_{1}+ℤω_{2} be the period lattice obtained by integrating dx/(2y+a_{1}x+ a_{3}). We may assume that ω_{1} ∈ ℝ_{>0} and τ = ω_{2}/ω_{1} ∈ ℌ with Re(τ) = 0. Then every real quadric intersection with the same invariants as (3.14) is properly ℝ-equivalent to exactly one of (Q_{1}, Q_{2}), (Q′_{1}, Q′_{2}), (Q′′_{1}, Q′′_{2}) where

Q_{1} = a(x_{0}^{2} + x_{2}^{2}) − 2bx_{1}x_{3}  Q′_{1} = a(x_{0}^{2} − x_{2}^{2}) − 2bx_{1}x_{3}
Q_{2} = a(x_{1}^{2} + x_{3}^{2}) − 2bx_{0}x_{2}  Q′_{2} = a(x_{1}^{2} − x_{3}^{2}) − 2bx_{0}x_{2}
Q′′_{1} = b(x_{0}^{2} + x_{2}^{2}) − 2ax_{1}x_{3}
Q′′_{2} = b(x_{1}^{2} + x_{3}^{2}) − 2ax_{0}x_{2}

and

a = \frac{1}{2} \sqrt{\frac{π}{ω_{1}}} θ_{4}(τ/4) b = \frac{1}{2} \sqrt{\frac{π}{ω_{1}}} θ_{3}(τ/4).

Proof: In the notation of Lemma 3.9 all three quadric intersections have invariants 2^{8}f_{4}(a, b) and −2^{12}f_{6}(a, b). For k = 4, 6 we compute

(4i)^{k}f_{k}(a, b) = (\frac{iπ}{ω_{1}} )^{k}f_{k}(θ_{4}(\frac{τ}{4}), θ_{3}(\frac{τ}{4})) = (\frac{2π}{ω_{1}})^{k}E_{k}(τ).

It follows by Lemma 3.5 that these quadric intersections have the same invariants as (3.14). It remains to show that they are pairwise inequivalent over the reals.

Since Re(τ) = 0 we have q > 0 and hence b > a > 0. We put c = \sqrt[4]{b^{4} − a^{4}}. Then Q_{1} = Q_{2} = 0 has real point (in fact a hyperosculating point)

(x_{0} : x_{1} : x_{2} : x_{3}) = (\sqrt{b^{2} + c^{2}} : \sqrt{ab} : \sqrt{b^{2} − c^{2}} : \sqrt{ab}).

Rather more obviously Q′_{1} = Q′_{2} = 0 has real point

(x_{0} : x_{1} : x_{2} : x_{3}) = (\sqrt{2b/a} : 1 : 0 : 1).

On the other hand, since the quadratic form

Q′′_{1} + Q′′_{2} = \frac{b−a}{2} ((x_{0} + x_{2})^{2} + (x_{1} + x_{3})^{2}) + \frac{a+b}{2} ((x_{0} − x_{2})^{2} + (x_{1} − x_{3})^{2})

is positive definite, there are no real solutions to Q′′_{1} = Q′′_{2} = 0.

Finally we claim that (Q_{1}, Q_{2}) and (Q′_{1}, Q′_{2}) are not equivalent over the reals. Let A_{1}, A_{2} be the matrices of second partial derivatives of Q_{1}, Q_{2} and likewise for Q′_{1}, Q′_{2}. We compute

det(xA_{1} + zA_{2}) = −2^{4}(a^{2}x^{2} − b^{2}z^{2})(b^{2}x^{2} − a^{2}z^{2})
det(xA′_{1} + zA′_{2}) = 2^{4}(a^{2}x^{2} + b^{2}z^{2})(b^{2}x^{2} + a^{2}z^{2}).

The first of these quartics has four real roots, whereas the second has no real roots. This proves our claim □

The analogous result for negative discriminants is the following.

Lemma 3.11. Let E/ℝ be an elliptic curve with negative discriminant and with Weierstrass equation

(3.15)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

Let Λ = ℤω_{1}+ℤω_{2} be the period lattice obtained by integrating dx/(2y+a_{1}x+ a_{3}). We may assume that ω_{1} ∈ ℝ_{>0} and τ = ω_{2}/ω_{1} ∈ ℌ with Re(τ) = 1/2. Then every real quadric intersection with the same invariants as (3.15) is properly ℝ-equivalent to (Q_{1}, Q_{2}) where

Q_{1} = a(x_{0}^{2} − x_{2}^{2}) − 2bx_{1}x_{3}
Q_{2} = a(x_{1}^{2} − x_{3}^{2}) − b(x_{0}^{2} + x_{2})

and

a = \frac{1}{2} \sqrt{\frac{2π}{ω_{1}}} ζ_{8}^{−1}θ_{2}(τ) b = \frac{1}{2} \sqrt{\frac{2π}{ω_{1}}} θ_{3}(τ).

Proof: Since Re(τ) = 1/2 it is clear from (3.12) that a and b are real. In the notation of Lemma 3.9 the quadric intersection (Q_{1}, Q_{2}) has invariants 2^{8}f_{4}(ζ_{8}a, b) and 2^{12}f_{6}(ζ_{8}a, b). We compute

4^{k}f_{k}(ζ_{8}a, b) = (\frac{2π}{ω_{1}})^{k}f_{k}(θ_{2}(τ), θ_{3}(τ)) = (\frac{2π}{ω_{1}})^{k}E_{k}(τ).

It follows by Lemma 3.5 that (Q_{1}, Q_{2}) has the same invariants as (3.15). □

The proof of Proposition 2.8 in the case n = 4 is completed by

Lemma 3.12. For τ ∈ ℌ with Re(τ) ∈ {0, 1/2} we have

(3.16)  max(|θ_{2}(τ)|, |θ_{3}(τ)|) ≪ max(|E_{4}(τ)|^{1/8}, |E_{6}(τ)|^{1/12}).

and

(3.17)  max(|θ_{3}(\frac{τ}{4})|, |θ_{4}(\frac{τ}{4})|) ≪ max(|E_{4}(τ)|^{1/8}, |E_{6}(τ)|^{1/12}).

Proof: The first two equalities in (3.13) show that (3.16) is equivalent to (3.17) with τ replaced by −1/τ. The second two equalities in (3.13) show that (3.16) and (3.17) are equivalent. It is clear from the definitions of θ_{2} and θ_{3} that we may replace τ by τ + 1. So if either bound holds on some subset of ℌ then both bounds hold on any SL_{2}(ℤ)-translate of that subset (possibly with different implied constants).

We only need to establish the bounds for Im(τ) large and Im(τ) small, since the result will then follow by a compactness argument. (Note that E_{4} and E_{6} have no common zeros in ℌ.) As Im(τ) → ∞ we have q → 0 and the result is clear. By the action of SL_{2}(ℤ) this implies the result for Im(τ) small. □

4. Genus one models and the geometry of numbers

In this section we use the geometry of numbers to deduce Theorem 1.2 from Theorem 2.9.

4.1. Binary quartics. By Theorem 2.9 and Lemma 2.10 we have

Lemma 4.1. Let C be an everywhere locally soluble 2-covering of an elliptic curve E/ℚ. Then C can be defined by an integer coefficient binary quartic whose coefficient of x^{4−j}z^{j} is bounded in absolute value by Aμ_{1}^{4−j}μ_{2}^{j} where A ≪ H_{E}^{2} and μ_{1}μ_{2} ≪ 1.

The binary quartic representing C is non-singular, i.e. it has no repeated roots in ℙ^{1}(\overline{ℚ}). Under the hypothesis of Theorem 1.2(b) it has no ℚ-rational root (see Remark 2.5). Since n = 2 the bound claimed in Theorem 1.2(c) is the same as that in Theorem 1.2(b). The proof of Theorem 1.2 in the case n = 2 is completed by

Lemma 4.2. Let Φ be an integer coefficient binary quartic. Suppose that the coefficient of x^{4−j}z^{j} is bounded in absolute value by Aμ_{1}^{4−j}μ_{2}^{j}.

(i) If Φ has no repeated root in ℙ^{1}(ℚ) then ||Φ||_{∞} ≤ A^{3}(μ_{1}μ_{2})^{6}.

(ii) If Φ has no root in ℙ^{1}(ℚ) then ||Φ||_{∞} ≤ A^{2}(μ_{1}μ_{2})^{4}.

Proof: Without loss of generality μ_{1} ≤ μ_{2}.

(i) If Aμ_{1}^{3}μ_{2} < 1 then Φ has no x^{4} or x^{3}z terms and therefore a repeated root at (1 : 0). By hypothesis this does not happen. Therefore Aμ_{1}^{3}μ_{2} ≥ 1 and

||Φ||_{∞} ≤ Aμ_{2}^{4} ≤ A^{3}(μ_{1}μ_{2})^{6}.

(ii) If Aμ_{1}^{4} < 1 then Φ has no x^{4} term and therefore a root at (1 : 0). By hypothesis this does not happen. Therefore Aμ_{1}^{4} ≥ 1 and

||Φ||_{∞} ≤ Aμ_{2}^{4} ≤ A^{2}(μ_{1}μ_{2})^{4}.
□

4.2. Ternary cubics. By Theorem 2.9 and Lemma 2.10 we have

Lemma 4.3. Let C be an everywhere locally soluble 3-covering of an elliptic curve E/ℚ. Then C can be defined by an integer coefficient ternary whose coefficient of x^{i}y^{j}z^{k} is bounded in absolute value by Aμ_{1}^{i}μ_{2}^{j}μ_{3}^{k} where A ≪ H_{E} and μ_{1}μ_{2}μ_{3} ≪ 1.

The hypotheses of parts (b) and (c) of Theorem 1.2 are that C has no ℚ-rational point of inflection, respectively that C has no ℚ-rational point. The proof of Theorem 1.2 in the case n = 3 is completed by

Lemma 4.4. Let Φ be an integer coefficient ternary cubic defining a plane cubic curve C ⊂ ℙ^{2}. Suppose that the coefficient of x^{i}y^{j}z^{k} is bounded in absolute value by Aμ_{1}^{i}μ_{2}^{j}μ_{3}^{k}.

(i) If C is non-singular then ||Φ||_{∞} ≤ A^{6}(μ_{1}μ_{2}μ_{3})^{6}.

(ii) If C is non-singular and has no ℚ-rational point of inflection then

||Φ||_{∞} ≤ A^{4}(μ_{1}μ_{2}μ_{3})^{4}. (iii) If C has no ℚ-rational points then ||Φ||_{∞} ≤ A^{3}(μ_{1}μ_{2}μ_{3})^{3}.

Proof: Without loss of generality μ_{1} ≤ μ_{2} ≤ μ_{3}.

(i) If Aμ_{1}^{2}μ_{3} < 1 then Φ has no x^{3}, x^{2}y or x^{2}z terms and therefore C is singular at (1 : 0 : 0). If Aμ_{2}^{3} < 1 then Φ has no x^{3}, x^{2}y, xy^{2}, or y^{3} terms. This would imply that C contains the line z = 0 and is therefore singular. Accordingly we have Aμ_{1}^{2}μ_{3} ≥ 1 and Aμ_{2}^{3} ≥ 1. It follows by the identity

Aμ_{3}^{3}(Aμ_{1}^{2}μ_{3})^{3}(Aμ_{2}^{3})^{2} = A^{6}(μ_{1}μ_{2}μ_{3})^{6}

that ||Φ||_{∞} ≤ Aμ_{3}^{3} ≤ A^{6}(μ_{1}μ_{2}μ_{3})^{6}.

(ii) If Aμ_{1}μ_{2}^{2} < 1 then Φ has no x^{3}, x^{2}y or xy^{2} terms and therefore C meets the line z = 0 with multiplicity at least 3. This would imply that either C is singular or that (1 : 0 : 0) is a point of inflection. Accordingly we have Aμ_{1}μ_{2}^{2} ≥ 1. Exactly as in the proof of (i) we have Aμ_{1}^{2}μ_{3} ≥ 1. It follows by the identity

Aμ_{3}^{3}(Aμ_{1}μ_{2}^{2})^{2}Aμ_{1}^{2}μ_{3} = A^{4}(μ_{1}μ_{2}μ_{3})^{4}

that ||Φ||_{∞} ≤ Aμ_{3}^{3} ≤ A^{4}(μ_{1}μ_{2}μ_{3})^{4}.

(iii) If Aμ_{1}^{3} < 1 then (1 : 0 : 0) is a ℚ-rational point on C. Therefore Aμ_{1}^{3} ≥ 1 and ||Φ||_{∞} ≤ Aμ_{3}^{3} ≤ A^{3}(μ_{1}μ_{2}μ_{3})^{3}. □

4.3. Quadric intersections. By Theorem 2.9 and Lemma 2.10 we have

Lemma 4.5. Let C be an everywhere locally soluble 4-covering of an elliptic curve E/ℚ. Then C can be defined by an integer coefficient quadric intersection (Q_{1}, Q_{2}) whose coefficient of x_{j}x_{k} in Q_{i} is bounded in absolute value by Aλ_{i}μ_{j}μ_{k}, where A ≪ H_{E}^{1/2}, λ_{1}λ_{2} ≪ 1 and μ_{1}μ_{2}μ_{3}μ_{4} ≪ 1.

Suppose that rank(xQ_{1} + zQ_{2}) < 4 for some (x : z) ∈ ℙ^{1}(ℚ). Then by Remark 2.5 and Lemma 2.6(ii) the element ξ_{4} ∈ H^{1}(ℚ, E[4]) corresponding to C satisfies [2]_{∗}ξ_{4} = 0. Hence ξ_{4} = ι_{∗}(ξ_{2}) for some ξ_{2} ∈ H^{1}(ℚ, E[2]). Since C is everywhere locally soluble ξ_{2} has trivial obstruction, i.e. it is represented by a binary quartic. We can therefore represent C by a quadric intersection of the form specified in Lemma 2.6(i). In this case Theorem 1.2(b) follows from the result for n = 2. Since ξ_{2} and ξ_{4} have the same images in � (E/ℚ) the hypothesis of Theorem 1.2(c) is not satisfied.

The proof of Theorem 1.2 in the case n = 4 is completed by

Lemma 4.6. Let Φ = (Q_{1}, Q_{2}) be an integer coefficient quadric intersection defining a degree 4 curve C ⊂ ℙ^{3}. Suppose that the coefficient of x_{j}x_{k} in Q_{i} is bounded in absolute value by Aλ_{i}μ_{j}μ_{k}.

(i) If C is non-singular then ||Φ||_{∞} ≤ A^{12}(λ_{1}λ_{2})^{6}(μ_{1}μ_{2}μ_{3}μ_{4})^{6}.

(ii) If C is non-singular and there are no ℚ-rational singular quadrics in the pencil spanned by Q_{1} and Q_{2} then

||Φ||_{∞} ≤ A^{8}(λ_{1}λ_{2})^{4}(μ_{1}μ_{2}μ_{3}μ_{4})^{4}.

(iii) If C has no ℚ-rational points and there are no ℚ-rational singular quadrics in the pencil spanned by Q_{1} and Q_{2} then

||Φ||_{∞} ≤ A^{4}(λ_{1}λ_{2})^{2}(μ_{1}μ_{2}μ_{3}μ_{4})^{2}.

Proof: Without loss of generality λ_{1} ≤ λ_{2} and μ_{1} ≤ μ_{2} ≤ μ_{3} ≤ μ_{4}.

(i) We make the following observations:

• If Aλ_{2}μ_{1}μ_{3} < 1 then (1 : 0 : 0 : 0) is a singular point on C.

• If Aλ_{2}μ_{2}^{2} < 1 then C contains the line {x_{3} = x_{4} = 0}.

• If Aλ_{1}μ_{3}^{2} < 1 then Q_{1} has rank at most 2.

• If Aλ_{1}μ_{2}μ_{4} < 1 then Q_{1} has rank at most 2. We are given that C is non-singular, and so none of the above inequalities can hold. We further note that if both Aλ_{2}μ_{1}^{2} < 1 and Aλ_{1}μ_{1}μ_{4} < 1 then (1 : 0 : 0 : 0) is a singular point on C. We therefore split into the cases Aλ_{2}μ_{1}^{2} ≥ 1 and Aλ_{1}μ_{1}μ_{4} ≥ 1. In the first case it follows by the identity

(Aλ_{2}μ_{1}^{2})^{2}(Aλ_{2}μ_{2}^{2})(Aλ_{1}μ_{3}^{2})^{2}(Aλ_{1}μ_{2}μ_{4})^{2}(Aλ_{2}μ_{4}^{2}) = A^{8}(λ_{1}λ_{2})^{4}(μ_{1}μ_{2}μ_{3}μ_{4})^{4}

that ||Φ||_{∞} ≤ Aλ_{2}μ_{4}^{2} ≤ A^{8}(λ_{1}λ_{2})^{4}(μ_{1}μ_{2}μ_{3}μ_{4})^{4}. In the second case it follows by the identity

(Aλ_{2}μ_{1}μ_{3})^{2}(Aλ_{2}μ_{2}^{2})^{3}(Aλ_{1}μ_{3}^{2})^{2}(Aλ_{1}μ_{1}μ_{4})^{4}(Aλ_{2}μ_{4}^{2}) = A^{12}(λ_{1}λ_{2})^{6}(μ_{1}μ_{2}μ_{3}μ_{4})^{6}

that ||Φ||_{∞} ≤ Aλ_{2}μ_{4}^{2} ≤ A^{12}(λ_{1}λ_{2})^{6}(μ_{1}μ_{2}μ_{3}μ_{4})^{6}.

(ii) We replace the third and fourth observations in (i) by • If Aλ_{1}μ_{2}μ_{3} < 1 then Q_{1} has rank at most 3.

• If Aλ_{1}μ_{1}μ_{4} < 1 then Q_{1} has rank at most 3.

It follows by the identity

(Aλ_{2}μ_{1}μ_{3})^{2}(Aλ_{2}μ_{2}^{2})(Aλ_{1}μ_{2}μ_{3})^{2}(Aλ_{1}μ_{1}μ_{4})^{2}(Aλ_{2}μ_{4}^{2}) = A^{8}(λ_{1}λ_{2})^{4}(μ_{1}μ_{2}μ_{3}μ_{4})^{4}

that ||Φ||_{∞} ≤ Aλ_{2}μ_{4}^{2} ≤ A^{8}(λ_{1}λ_{2})^{4}(μ_{1}μ_{2}μ_{3}μ_{4})^{4}.

(iii) If Aλ_{2}μ_{1}^{2} < 1 then (1 : 0 : 0 : 0) is a ℚ-rational point on C. Therefore Aλ_{2}μ_{1}^{2} ≥ 1. We have already seen in (ii) that Aλ_{1}μ_{2}μ_{3} ≥ 1. It follows by the identity

(Aλ_{2}μ_{1}^{2})(Aλ_{1}μ_{2}μ_{3})^{2}(Aλ_{2}μ_{4}^{2}) = A^{4}(λ_{1}λ_{2})^{2}(μ_{1}μ_{2}μ_{3}μ_{4})^{2}

that ||Φ||_{∞} ≤ Aλ_{2}μ_{4}^{2} ≤ A^{4}(λ_{1}λ_{2})^{2}(μ_{1}μ_{2}μ_{3}μ_{4})^{2}.

□

5. A criterion for � (E/ℚ)[3] = 0

In this section we prove Theorem 1.3. We will need the following lemma whose proof is just an exercise in calculus.

Lemma 5.1. Let a, b ∈ ℝ and put c = (a^{2} + \sqrt{3}ab + b^{2})^{1/2}. Let

F (x, y, z) = a(x^{3} + y^{3} + z^{3}) − 3(a + \sqrt{3}b)xyz.

Then |F (x)| ≤ max(|a|, |b|, |c|)||x||^{3} for all x = (x, y, z) ∈ ℝ^{3}.

Proof: Let (x, y, z) be a local maximum of F on the sphere x^{2}+y^{2}+z^{2} = 1. Then we have

rank (\begin{matrix}F_{x} & F_{y} & F_{z} \\ x & y & z \end{matrix}) ≤ 1.

We compute

yF_{x} − xF_{y} = 3(x − y)(axy + (a + \sqrt{3}b)(x + y)z).

If x, y, z are distinct then a + \sqrt{3}b = a or −a/2. In the first case we have b = 0 and xy + yz + zx = 0. But then x + y + z = ±1 and F (x, y, z) = ±a. In the second case we have xy = yz = zx, and this contradicts that x, y, z are distinct.

If x = y = z then F (x, y, z) = ±b. So without loss of generality x ≠ y = z. Then

axy + (a + \sqrt{3}b)(x + y)y = 0.

If y = 0 we get F (x, y, z) = ±a. Otherwise

x = −(a + \sqrt{3}b)ξ
y = (2a + \sqrt{3}b)ξ

for some ξ ∈ ℝ. We compute

x^{2} + y^{2} + z^{2} = 9ξ^{2}(a^{2} + \frac{10}{9} \sqrt{3}ab + b^{2})
F (x, y, z) = 27ξ^{3}(a^{2} + \sqrt{3}ab + b^{2})(a^{2} + \frac{10}{9} \sqrt{3}ab + b^{2})

Eliminating ξ gives

|F (x, y, z)| = \frac{a^{2} + \sqrt{3}ab + b^{2}}{(a^{2} + \frac{10}{9} \sqrt{3}ab + b^{2})^{1/2}}

If ab ≥ 0 then |F (x, y, z)| ≤ |c|. Otherwise if |a| ≥ |b| we have

(a^{2} + \sqrt{3}ab + b^{2})^{2} − a^{2}(a^{2} + \frac{10}{9} \sqrt{3}ab + b^{2}) = (\frac{2}{\sqrt{3}}a + b)^{3}b ≤ 0

and hence |F (x, y, z)| ≤ |a|. The case |b| ≥ |a| is similar. □

Proof of Theorem 1.3: Let E/ℚ be an elliptic curve. We aim to show that (under suitable hypotheses) � (E/ℚ)[3] = 0. By (1.1) it is equivalent to show that every ξ ∈ S^{(3)}(E/ℚ) maps to zero in � (E/ℚ).

Let ξ ∈ S^{(3)}(E/ℚ). Then ξ corresponds to an everywhere locally soluble 3covering π : C → E. Our aim is to show that C(ℚ) ≠ ∅. By Proposition 2.7 we know that C can be defined by an integer coefficient ternary cubic f with the same invariants as a minimal Weierstrass equation for E. We fix a minimal Weierstrass equation for E and let τ = ω_{2}/ω_{1} ∈ ℌ be as in Lemma 3.7. Then f = F ∘ g for some g ∈ SL_{3}(ℝ) where

F (x, y, z) = a(x^{3} + y^{3} + z^{3}) − 3(a + \sqrt{3}b)xyz

and

a = \frac{1}{\sqrt{27}} (\frac{2π}{ω_{1}} ) \frac{η(τ/3)^{3}}{η(τ)} b = (\frac{2π}{ω_{1}} ) \frac{η(3τ)^{3}}{η(τ)} .

Let γ = a + iζ_{3}^{2}b and c = |γ| = (a^{2} + \sqrt{3}ab + b^{2})^{1/2}. The lattice Λ = g(ℤ^{3}) ⊂ ℝ^{3} has covolume 1. Hence by Lemma 2.10 there exists 0 ≠ x ∈ Λ with ||x||^{3} ≤ \sqrt{2}. If max(|a|, |b|, |c|) < 1/\sqrt{2} then by Lemma 5.1 we have

|F (x)| ≤ max(|a|, |b|, |c|) ||x||^{3} < 1.

Since F (x) = f(u, v, w) for some u, v, w ∈ ℤ it follows that F (x) = 0. Hence C(ℚ) ≠ ∅ and ξ maps to zero in � (E/ℚ).

It remains to show that the condition max(|a|, |b|, |c|) < 1/\sqrt{2} is equivalent to the hypothesis of the theorem. By (3.9) we have

γ = \frac{1}{\sqrt{27}} (\frac{2π}{ω_{1}} ) \frac{η(\frac{τ+1}{3})^{3}}{η(τ + 1)}, \overline{γ} = \frac{1}{\sqrt{27}} (\frac{2π}{ω_{1}} ) \frac{η(\frac{τ−1}{3})^{3}}{η(τ − 1)}.

We now put

α_{1}(τ) = −η(\frac{τ}{3})^{12}  α_{3}(τ) = η(\frac{τ+1}{3})^{12}
α_{2}(τ) = −3^{6}η(3τ)^{12}  α_{4}(τ) = η(\frac{τ−1}{3})^{12}

and claim that

(5.1) ∏_{i=1}^{4}(X − α_{i}(τ)) = (X − 3^{3}η(τ)^{12})(X − 3η(τ)^{12})^{3} + E_{4}(τ)^{3}η(τ)^{12}X.

It is routine to check using (3.3) and (3.4) that

α_{1}(τ + 1) = −α_{3}(τ)  α_{1}(−1/τ) = −τ^{6}α_{2}(τ)
α_{2}(τ + 1) = −α_{2}(τ)  α_{2}(−1/τ) = −τ^{6}α_{1}(τ)
α_{3}(τ + 1) = −α_{4}(τ)  α_{3}(−1/τ) = −τ^{6}α_{4}(τ)
α_{4}(τ + 1) = −α_{1}(τ)  α_{4}(−1/τ) = −τ^{6}α_{3}(τ).

Hence the square of each coefficient of the left hand side of (5.1) is a modular form of level 1. The claim is then proved by comparing the first few coefficients of the q-expansions. By Lemma 3.5 we have Δ_{E} = (\frac{2π}{ω_{1}})^{12}Δ(τ) and j = E_{4}(τ)^{3}/Δ(τ). Finally we compute

max(|a|, |b|, |c|) < \frac{1}{\sqrt{2}}

⟺ \frac{1}{\sqrt{27}}(\frac{2π}{ω_{1}}) max (|η(\frac{τ}{3})|^{3}, \sqrt{27}|η(3τ)|^{3}, |η(\frac{τ+1}{3})|^{3}) < \frac{1}{\sqrt{2}}|η(τ)|
⟺ 2^{6}|Δ_{E}| max{|x|^{3} : x a root of (5.1)} < 3^{18}|η(τ)|^{36}

⟺ 2^{6}|Δ_{E}| max{|x|^{3} : x a root of (X − 3^{3})(X − 3)^{3} + jX = 0} < 3^{18}

⟺ 2^{6}|Δ_{E}| < min{|x|^{3} : x a root of (X − 3^{3})(X − 3^{5})^{3} + jX^{3} = 0}.

This final condition is the hypothesis of the theorem. □

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University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

E-mail address: T.A.Fisher@dpmms.cam.ac.uk