FINDING RATIONAL POINTS ON ELLIPTIC CURVES USING 6-DESCENT AND 12-DESCENT

TOM FISHER

Abstract. We explain how recent work on 3-descent and 4-descent for elliptic curves over ℚ can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the Stein-Watkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.

1. Introduction

Let E be an elliptic curve over ℚ. An n-descent calculation on E provides us with n-covering curves π_{α} : C_{α} → E for α running over a finite indexing set A, with the property that

⋃_{α∈A} π_{α}(C_{α}(ℚ)) = E(ℚ).

The usual choice of indexing set A is the n-Selmer group S^{(n)}(E/ℚ) which sits in a short exact sequence

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

Given α ∈ S^{(n)}(E/ℚ) there are two possibilities: either π_{α}(C_{α}(ℚ)) is a coset of nE(ℚ) in E(ℚ), in which case α is the image of this coset by δ, or C_{α}(ℚ) is empty, in which case α maps to a non-trivial element of the Tate-Shafarevich group � (E/ℚ).

It has long been known that n-descent can help in the search for generators of the Mordell-Weil group E(ℚ). Indeed the theory of heights (see for example [12]) suggests that if we write our n-coverings as curves of degree n with small coefficients, then a point of (logarithmic) height h on E(ℚ) should come from a point of height approximately h/(2n) on C_{α}(ℚ) for suitable α. This is not a precise statement (the height is only bounded up to the addition of a constant whose behaviour with respect to n is unknown) but the idea seems to work well in practice.

We would therefore like to perform n-descent calculations for n as large as possible. Until recently n-descent has only been practical for general^{1} elliptic curves in the case n = 2. Methods for 4-descent and 8-descent have been developed in the PhD theses of Siksek [18], Womack [24] and Stamminger [20]. Joint work of the author with Cremona, O’Neil, Simon and Stoll [5] has now made 3-descent practical, and in a few preliminary examples also 5-descent. The algorithms for 2-descent, 3-descent and 4-descent have been contributed to the computer algebra system Magma [15].

Date: 23rd November 2007.

The natural question therefore arises as to how we can combine an m-covering and n-covering for m and n coprime to give an mn-covering. At the level of abelian groups it is trivial that

S^{(mn)}(E/ℚ) ≅ S^{(m)}(E/ℚ) × S^{(n)}(E/ℚ).

However, if we are to represent the Selmer group elements as covering curves, then it is not so clear how one should proceed.

Suppose we are given an m-covering π_{m} : C_{m} → E and an n-covering π_{n} : C_{n} → E. Then the curves C_{m} and C_{n} are torsors under E, and an mn-covering is given by

C_{mn} = \frac{C_{m} × C_{n}}{E}

where we quotient out by the diagonal action of E. An alternative would be to take fibre product

C_{mn} = C_{m} ×_{E} C_{n}

with respect to the covering maps π_{m} and π_{n}. As far as we can see, neither of these constructions is suitable for practical computation. We have therefore taken a different approach based on representations of the Heisenberg group.

Unfortunately our approach does not work for arbitrary coprime integers m and n, but only when each of m and n is plus or minus a square modulo the other. This includes the case of consecutive integers. In this case, we specify an embedding of E in ℙ(Mat_{n,n+1}) as a curve of degree n(n + 1), in such a way that when E acts on itself by translation, the n-torsion points act as left multiplication by n × n matrices, and the (n + 1)-torsion points act as right multiplication by (n + 1) × (n + 1) matrices. We can then twist E by a pair of cocycles taking values in E[n] and E[n + 1] to obtain the required n(n + 1)- covering C_{n(n+1)} as a curve in ℙ(Mat_{n,n+1}). Moreover, it turns out that the covering map C_{n(n+1)} → C_{n+1} is defined by the n × n minors.

1i.e. we make no assumption on the Galois module structure of E[n].

We give a precise statement of these results in §2. We employ two different methods of proof. The first, described in §3, uses representations of the Heisenberg group, and leads to results for arbitrary n. The second, described in §§4,5, uses the invariant theory of binary quartics and ternary cubics, and gives practical formulae specific to 6-descent and 12-descent.

In §6 we give some details of our implementation of 6-descent and 12-descent in Magma. Using 12-descent, we now expect to be able to find rational points on an elliptic curve over ℚ up to logarithmic height 600 (provided the coefficients of the original elliptic curve are not too large). The main bottleneck comes in the 3-descent, where we must compute the class group and units of each number field generated by the co-ordinates of a 3-torsion point of E. (There is usually just one such field, and it has degree 8.) Fortunately, since our final answer comes in the form of a rational point, there is no need to perform these intermediate calculations rigorously.

Stein and Watkins [21] have constructed a database of elliptic curves that is expected to contain most elliptic curves over ℚ of prime conductor N with N < 10^{10}. We are able to show that every curve in their database (of prime conductor) has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer. Prior to our involvement, this had been reduced by Cremona and Watkins to a list of 35 curves of analytic rank 2 for which one generator of small height (less than 34) was known, but a second generator of large height (greater than 220) remained to be found. In each case we were able to find the second generator using either 6-descent or 12-descent.

We give two numerical examples in §7. In the first we use 6-descent to find a pair of non-zero integers x and y for which both

x^{2} − 809xy + y^{2} and x^{2} + 809xy + y^{2}

are squares. We find that the smallest solution is given by x and y with 534 and 537 decimal digits respectively. Our second example is the last in the list of 35 curves mentioned above. In this case we use 12-descent to find a generator of height 642.63.

Acknowledgments

I would like to thank Steve Donnelly for sharing his initial thoughts on this problem, and Mark Watkins for suggesting suitable test data, including the examples in §7. All computer calculations in support of this work were performed using Magma [15].

2. Computing twists

Let k be a field of characteristic zero, with algebraic closure \overline{k}. We fix an elliptic curve E over k with identity O. Let n ≥ 2 be an integer. A base diagram [E → ℙ^{n−1}] of level n is a morphism defined over k determined by the complete linear system |n.O|. Thus any two base diagrams differ by an element of PGL_{n}(k).

More generally we consider diagrams [C → S] where C is a torsor under E and S is a variety (both defined over k) and the map C → S is a morphism defined over k. Two such diagrams [ϕ_{1} : C_{1} → S_{1}] and [ϕ_{2} : C_{2} → S_{2}] are isomorphic if there is an isomorphism of torsors α : C_{1} ≅ C_{2} and an isomorphism of varieties β : S_{1} ≅ S_{2} satisfying ϕ_{2} ∘ α = β ∘ ϕ_{1}. We define a Brauer-Severi diagram [C → S] to be a twist of the base diagram. Then S is a Brauer-Severi variety, and the morphism C → S is that determined by a complete linear system |D|, where the divisor D is linearly equivalent to all its Galois conjugates, but need not itself be defined over k.

We recall from [5, Paper I, §1.3] that the Brauer-Severi diagrams are parametrised, as twists of a fixed base diagram, by the Galois cohomology group H^{1}(k, E[n]). Moreover there is an obstruction map

Ob_{n} : H^{1}(k, E[n]) → Br(k)[n]

taking the class of [C → S] to the class of [S]. In general this map is not a group homomorphism. We are interested in the elements of H^{1}(k, E[n]) with trivial obstruction, equivalently those that are represented by diagrams of the form [ϕ_{n} : C → ℙ^{n−1}]. With the convention that points of ℙ^{n−1} are written as column vectors, we define the “character” associated to ϕ_{n} to be the unique morphism of k-group schemes χ_{n} : E[n] → PGL_{n} such that

ϕ_{n}(T + P) = χ_{n}(T)ϕ_{n}(P)

for all T ∈ E[n](\overline{k}) and P ∈ C(\overline{k}).

In §3 we use representations of the Heisenberg group to prove

Theorem 2.1. Let m and n be coprime integers satisfying

(1)  u^{2}n ≡ ±1 (mod m) and v^{2}m ≡ ±1 (mod n),

for some integers u and v. Let χ_{m}, χ_{n} and χ_{mn} be the characters associated to base diagrams of level m, n and mn. Then there is a morphism of k-group schemes

Ξ : PGL_{m} × PGL_{n} → PGL_{mn}

such that

Ξ(χ_{m}(S), χ_{n}(T)) = χ_{mn}(uS + vT)

for all S ∈ E[m](\overline{k}) and T ∈ E[n](\overline{k}).

Let [C → ℙ^{n−1}] be a twist of the base diagram [E → ℙ^{n−1}]. By definition this means that there is a commutative diagram

\xymatrix{C \ar[r] \ar[d]_{α} & ℙ^{n−1} \ar[d]^{β} \\ E \ar[r] & ℙ^{n−1}}

where α and β are isomorphisms defined over \overline{k}. We say that the matrix B_{n} ∈ PGL_{n}(\overline{k}) representing β is a flex matrix for [C → ℙ^{n−1}]. In §6.2 we give some algorithms for computing flex matrices, starting from equations for C. Conversely, it is clear that we can recover equations for C from a flex matrix (by starting with equations for E and making the relevant substitution).

Our next result explains how Theorem 2.1 is used to construct an mn-covering from an m-covering and an n-covering.

Proposition 2.2. In the setting of Theorem 2.1, suppose that ξ ∈ H^{1}(k, E[m]) and η ∈ H^{1}(k, E[n]) are represented by Brauer-Severi diagrams [C_{m} → ℙ^{m−1}] and [C_{n} → ℙ^{n−1}] with flex matrices B_{m} and B_{n}. Then uξ + vη ∈ H^{1}(k, E[mn]) is represented by a Brauer-Severi diagram [C_{mn} → ℙ^{mn−1}] with flex matrix B_{mn} = Ξ(B_{m}, B_{n}).

Proof: Let σ ∈ Gal(\overline{k}/k). Since σ(B_{m})B_{m}^{−1} = χ_{m}(ξ_{σ}), σ(B_{n})B_{n}^{−1} = χ_{n}(η_{σ}) and Ξ is defined over k, we have

σ(B_{mn})B_{mn}^{−1} = Ξ(σ(B_{m})B_{m}^{−1}, σ(B_{n})B_{n}^{−1})
= Ξ(χ_{m}(ξ_{σ}), χ_{n}(η_{σ}))
= χ_{mn}(uξ_{σ} + vη_{σ}).

Thus B_{mn} represents a change of co-ordinates on ℙ^{mn−1} taking E to a curve C_{mn} defined over k. Then [C_{mn} → ℙ^{mn−1}] is the twist of [E → ℙ^{mn−1}] by uξ + vη ∈ H^{1}(k, E[mn]). □

Remark 2.3. The proposition shows that if ξ and η have trivial obstruction, then so does uξ + vη. In fact, standard properties of the obstruction map (see [5, Paper I], [17], [25]) already show that

(2)  Ob_{mn}(uξ + vη) = u^{2}n Ob_{m}(ξ) + v^{2}m Ob_{n}(η)

for m and n coprime. Interestingly, the hypothesis (1) of Theorem 2.1 is that the coefficients on the right hand side of (2) are ±1.

We are mainly interested in the case m and n are consecutive integers. We can therefore either take u = v = 1 in Theorem 2.1, or use the refined version of the theorem we give next.

We write ℙ(Mat_{a,b}) for the projective space of dimension ab − 1 formed from the vector space of a × b matrices. Taking n × n minors defines a rational map

μ : ℙ(Mat_{n,n+1}) − → ℙ^{n}; A ↦ (…: (−1)^{i} det(A^{{i}}) : …)

where A^{{i}} is A with the ith column deleted. The following refinement of Theorem 2.1 is proved alongside the original theorem in §3.

Theorem 2.4. Let [ϕ_{n} : E → ℙ^{n−1}] and [ϕ_{n+1} : E → ℙ^{n}] be base diagrams of levels n and n+1, with associated characters χ_{n} and χ_{n+1}. Then there is a base diagram

[ϕ_{n,n+1} : E → ℙ(Mat_{n,n+1})]

of level n(n + 1), with associated character χ_{n,n+1} given by

χ_{n,n+1}(S + T) : A ↦ χ_{n}(S)Aχ_{n+1}(T)

for all S ∈ E[n](\overline{k}) and T ∈ E[n + 1](\overline{k}). Moreover if [n] : E → E is the multiplication-by-n map then the diagram

\xymatrix{E \ar[r]^{ϕ_{n,n+1}} \ar[d]_{[n]} & ℙ(Mat_{n,n+1}) \ar@{-->}[d]^{μ} \\ E \ar[r]^{ϕ_{n+1}} & ℙ^{n}.}

commutes.

We obtain equations for an n(n + 1)-covering from an n-covering and an (n + 1)-covering, by first finding the base diagram ϕ_{n,n+1} of Theorem 2.4, and then twisting by the flex matrices B_{n} and B_{n+1}. These twists may be performed one after the other. In fact twisting by B_{n} first gives the following generalisation of Theorem 2.4.

Theorem 2.5. Let [ϕ_{n} : C ⟶^{|D|} ℙ^{n−1}] be a Brauer-Severi diagram and [ϕ_{n+1} : E → ℙ^{n}] a base diagram, of levels n and n + 1, with associated characters χ_{n} and χ_{n+1}. Then there is a Brauer-Severi diagram

[ϕ_{n,n+1} : C ⟶^{|(n+1)D|} ℙ(Mat_{n,n+1})]

of level n(n + 1), with associated character χ_{n,n+1} given by

χ_{n,n+1}(S + T) : A ↦ χ_{n}(S)Aχ_{n+1}(T)

for all S ∈ E[n](\overline{k}) and T ∈ E[n + 1](\overline{k}). Moreover if π : C → E is the n-covering map then the diagram

\xymatrix{C \ar[r]^{ϕ_{n,n+1}} \ar[d]_{π} & ℙ(Mat_{n,n+1}) \ar@{-->}[d]^{μ} \\ E \ar[r]^{ϕ_{n+1}} & ℙ^{n}}

commutes.

Proof: Theorem 2.4 is the special case where (C, [D]) = (E, [n.O]). We twist by B_{n} to obtain the general result. □

The advantage of using Theorem 2.5 (instead of Theorem 2.4) is that we then only need to twist by B_{n+1} (rather than both B_{n} and B_{n+1}) to obtain the desired n(n + 1)-covering. In §5 we use invariant theory to give an alternative proof of Theorem 2.5 in the cases n = 2, 3. In particular we obtain explicit formulae for ϕ_{2,3} and ϕ_{3,4} which we then use in our implementations of 6-descent and 12-descent.

3. The Heisenberg group

We continue to work over a field k of characteristic 0. Let E be an elliptic curve with identity O, and n ≥ 2 an integer. We recall that if D is a divisor on E of degree n then the Riemann-Roch space L(D) has dimension n. Let V_{n} = L(n.O)^{∗}. Then there is a “coordinate free” base diagram [E → ℙ(V_{n})] with associated character χ_{n} : E[n] → PGL(V_{n}).

Definition 3.1. (i) The theta group Θ_{n} is the inverse image of χ_{n}(E[n]) in GL(V_{n}). It sits in a commutative diagram of k-group schemes with exact rows:

[figure: 0; 𝔾_{m}; Θ_{n}; E[n]; χ_{n}; 0; 0; 𝔾_{m}; GL(V_{n}); PGL(V_{n}); 0.]

(ii) The Heisenberg group H_{n} is the inverse image of χ_{n}(E[n]) in SL(V_{n}). It sits in a commutative diagram of k-group schemes with exact rows:

[figure: 0; μ_{n}; H_{n}; E[n]; χ_{n}; 0; 0; μ_{n}; SL(V_{n}); PGL(V_{n}); 0.]

Remark 3.2. It is well known (see e.g. [13]) that over k = \overline{k} we may choose a basis for V_{n} such that Θ_{n} is generated by

(\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & ζ_{n} & 0 & ⋯ & 0 \\ 0 & 0 & ζ_{n}^{2} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & ⋯ & ζ_{n}^{n−1} \end{matrix}) , (\begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix}) ,

and the scalar matrices. (Here ζ_{n} ∈ k is a primitive nth root of unity.) In particular, taking commutators in Θ_{n} defines a non-degenerate pairing e_{n} : E[n] × E[n] → μ_{n}, which turns out to be the Weil pairing.

The group H_{n}(\overline{k}) is a non-abelian group of order n^{3} with centre μ_{n}(\overline{k}).

Definition 3.3. A representation ρ : H_{n}(\overline{k}) → GL_{d}(\overline{k}) has central character [r] if ρ(ζ) = ζ^{r}I_{d} for all ζ ∈ μ_{n}(\overline{k}).

Lemma 3.4. Let r be an integer coprime to n. Then

(i) Every representation of H_{n}(\overline{k}) with central character [r] is a direct sum of irreducible n-dimensional representations. (ii) Up to equivalence, there is a unique n-dimensional representation of H_{n}(\overline{k}) with central character [r].

Proof: Let ρ be a representation of H_{n}(\overline{k}) with central character [r]. If σ, τ ∈ H_{n}(\overline{k}) lift S, T ∈ E[n](\overline{k}) then ρ(σ)ρ(τ)ρ(σ)^{−1} = e_{n}(S, T)^{r}ρ(τ) and so Tr(ρ(τ)) = e_{n}(S, T)^{r} Tr(ρ(τ)). Since r is coprime to n, the character of ρ vanishes outside the centre of H_{n}(\overline{k}). Then orthogonality relations in the character table show that ρ is irreducible if and only if it has dimension n. This proves (i) and the uniqueness in (ii). Existence is clear in the case r = 1. In general we take the rth tensor power and apply (i). □

To prove Theorem 2.1 we need to construct a morphism Ξ that is defined over k. We therefore study representations ρ : H_{n}(\overline{k}) → GL_{d}(\overline{k}) that are Galois equivariant, equivalently those that induce a morphism of k-group schemes H_{n} → GL_{d}. We call these representations of H_{n}.

By construction, V_{n} is an n-dimensional representation of H_{n} with central character [1]. So by Lemma 3.4(i) it is irreducible. We might hope to construct other irreducible n-dimensional representations of H_{n} by any one of the following standard methods.

(i) Take a subspace or quotient of a tensor power of V_{n}.

(ii) Replace V_{n} by one of its Galois conjugates.

(iii) Precompose ρ : H_{n} → GL(V_{n}) with an automorphism of H_{n}.

We see no way of using (i) in the proof of Theorem 2.1, other than in the case of V_{n}^{∗} = ∧^{n−1}V_{n}. (The problem is that there is no analogue of Lemma 3.4(i) with H_{n} replaced by GL_{n}.) Our restriction to Galois equivariant representations rules out the use of (ii).

To use (iii) we must first describe the (Galois equivariant) automorphisms of H_{n}. Each automorphism of H_{n} induces an automorphism of E[n]. We may identify Aut(E[n]) as the centraliser of the image of Galois in Aut\overline{k}(E[n]) ≅ GL_{2}(ℤ/nℤ). So in general the only automorphisms of E[n] are the maps T ↦ aT for a ∈ (ℤ/nℤ)^{×}. We show that each of these maps lifts to an automorphism of H_{n}. (Without our insistence on Galois equivariance, this would be trivial.)

Lemma 3.5. For each a ∈ (ℤ/nℤ)^{×} there is a morphism of k-group schemes ψ_{a} : H_{n} → H_{n} making the diagram

\xymatrix{0 \ar[r] & μ_{n} \ar[r] \ar[d]^{a^{2}} & H_{n} \ar[r] \ar[d]^{ψ_{a}} & E[n] \ar[r] \ar[d]^{a} & 0 \\ 0 \ar[r] & μ_{n} \ar[r] & H_{n} \ar[r] & E[n] \ar[r] & 0}

commute.

Proof: The [−1]-map on E lifts to a matrix ι ∈ GL_{n}(k). Conjugation by ι gives the map ψ_{−1}. In general we first define

λ : H_{n} → μ_{n}; h ↦ ιhι^{−1}h

and then put

ψ_{a} : H_{n} → H_{n}; h ↦ λ(h)^{a(a−1)/2}h^{a}.

Since λ(xy) = xy(ιxι^{−1})(ιyι^{−1}) = λ(x)λ(y)xyx^{−1}y^{−1} for all x, y ∈ H_{n}, it is easy to check that ψ_{a} is a group homomorphism. Galois equivariance is clear from the construction. □

Let V_{n}^{(a)} be the representation of H_{n} given by

H_{n} × V_{n} → V_{n}; (h, v) ↦ ψ_{a}(h)v.

It is an irreducible n-dimensional representation of H_{n} with central character [a^{2}]. Taking these representations and their duals we obtain all the representations of Lemma 3.4(ii) with r ≡ ±a^{2} (mod n).

We write τ_{P} : E → E for translation by P ∈ E(\overline{k}). We recall from [16, §23], (see also [5, Paper I]), that the theta group Θ_{n} may be described as pairs

Θ_{n}(\overline{k}) = \{ (f, T) ∈ \overline{k}(E)^{×} × E[n](\overline{k}) | div(f) = τ_{T}^{∗}(n.O) − n.O \} with group law

(3)  (f, S) ∗ (g, T) = ((τ_{T}^{∗}f)g, S + T).

There is a natural action of Θ_{n} on V_{nN}^{∗} = L(nN.O) given by

(4)  (f, T) : h ↦ τ_{−T}^{∗} (h/f^{N}).

We use this notation to relate the Heisenberg groups H_{n} for different levels n.

Proposition 3.6. Let m and n be coprime integers. Then there is an isomorphism of k-group schemes

\begin{matrix}H_{m} × H_{n} & ≅ & H_{mn} \\ ((f, S), (g, T)) & ↦ & (f^{n}, S) ∗ (g^{m}, T). \end{matrix}

For the proof we need two lemmas.

Lemma 3.7. Let M_{T} ∈ Θ_{n}(\overline{k}) be a lift of T ∈ E[n](\overline{k}). If T has exact order r then

det(M_{T}) = (−1)^{n(n−1)/r}M_{T}^{n}.

Proof: By Remark 3.2 we know that M_{T} is similar to

λ Diag(1, ζ_{r}, …, ζ_{r}^{n−1})

for some λ ∈ \overline{k}^{×} and ζ_{r} a primitive rth root of unity. Then det(M_{T}) = ζ_{r}^{n(n−1)/2}M_{T}^{n} = (−1)^{n(n−1)/r}M_{T}^{n} as required. □

Lemma 3.8. Let N be a positive integer. Then there is a commutative diagram of k-group schemes

\xymatrix{Θ_{n} \ar[d]_{det} \ar[r]^{α} & Θ_{nN} \ar[d]^{det} \\ 𝔾_{m} \ar[r]^{N^{2}} & 𝔾_{m}.}

where α : (f, T) ↦ (f^{N}, T).

Proof: It is clear from (3) that α is a group homomorphism. Now let x ∈ Θ_{n} and assume n is odd. Using Lemma 3.7 we compute

det(α(x)) = α(x)^{nN} = α(det x)^{N} = (det x)^{N^{2}}.

The calculation for n even is similar. □

Proof of Proposition 3.6: Let (f, S) and (g, T) be elements of Θ_{m}(\overline{k}) and Θ_{n}(\overline{k}). The commutator of (f^{n}, S) and (g^{m}, T) in Θ_{mn}(\overline{k}) is both an mth root of unity and an nth root of unity, and therefore trivial. So there is a morphism of k-group schemes

\begin{matrix}Θ_{m} × Θ_{n} & → & Θ_{mn} \\ ((f, S), (g, T)) & ↦ & (f^{n}, S) ∗ (g^{m}, T). \end{matrix}

By Lemma 3.8 we can restrict to a map H_{m} × H_{n} → H_{mn}. Since m and n are coprime, this restriction is clearly an isomorphism. □

We recall that H_{n} acts on L(nN.O) as specified in (4).

Lemma 3.9. The H_{n}-invariant subspace of L(nN.O) is trivial unless n divides N, in which case it has dimension N/n.

Proof: By Lemma 3.8 there is a group homomorphism H_{n}(\overline{k}) → H_{nN}(\overline{k}) given by (f, T) ↦ (f^{N}, T). By the proof of Lemma 3.4 the character of L(nN.O) is trivial outside the centre of H_{nN}(\overline{k}). The same is therefore true when L(nN.O) is viewed as a representation of H_{n}(\overline{k}). We are done by the orthogonality relations in the character table. □

Proposition 3.10. Let m and n be coprime integers. Suppose that n ≡ a^{2} (mod m) and m ≡ b^{2} (mod n) for some integers a and b. Then there is a k-isomorphism of H_{mn}-representations

π : V_{m}^{(a)} ⊗ V_{n}^{(b)} ≅ V_{mn}.

Proof: We recall that V_{m}^{(a)} is an irreducible H_{m}-representation with central character [a^{2}]. Likewise V_{n}^{(b)} is an irreducible H_{n}-representation with central character [b^{2}]. Then Proposition 3.6 makes V_{m}^{(a)} ⊗ V_{n}^{(b)} an irreducible H_{mn}-representation with central character [1]. Indeed (ζ_{m}, ζ_{n}) ∈ H_{m} × H_{n} acts on V_{m}^{(a)} ⊗ V_{n}^{(b)} as ζ_{m}^{a^{2}}ζ_{n}^{b^{2}} and on V_{mn} as ζ_{m}^{n}ζ_{n}^{m}. The required isomorphism π exists by Lemma 3.4(ii). Finally, since we work with Galois equivariant representations, we can choose an isomorphism π that is defined over k. □

Proof of Theorem 2.1: We first treat the case u^{2}n ≡ 1 (mod m) and v^{2}m ≡ 1 (mod n). Let a and b be inverses for u and v modulo m and n respectively. Then the map π : V_{m} ⊗ V_{n} → V_{mn} constructed in Proposition 3.10 satisfies

π(ψ_{a}(f, S)v_{m} ⊗ ψ_{b}(g, T)v_{n}) = ((f^{n}, S) ∗ (g^{m}, T))π(v_{m} ⊗ v_{n})

for all (f, S) ∈ H_{m}, (g, T) ∈ H_{n}, v_{m} ∈ V_{m}, v_{n} ∈ V_{n}. Passing to ℙ(V_{mn}) we obtain

π(χ_{m}(aS)v_{m} ⊗ χ_{n}(bT)v_{n}) = χ_{mn}(S + T)π(v_{m} ⊗ v_{n}).

Hence

π ∘ (χ_{m}(aS) ⊠ χ_{n}(bT)) = χ_{mn}(S + T) ∘ π

where

⊠ : PGL(V_{m}) × PGL(V_{n}) → PGL(V_{m} ⊗ V_{n})

is the natural map. The theorem now holds on defining

Ξ : PGL(V_{m}) × PGL(V_{n}) → PGL(V_{mn})
(α, β) ↦ π ∘ (α ⊠ β) ∘ π^{−1}.

In general, if u^{2}n ≡ −1 (mod m) or v^{2}m ≡ −1 (mod n) then we replace V_{m}^{(a)} or V_{n}^{(b)} by its dual in Proposition 3.10, and the proof carries through as before. □

Next we prove our refined version of the theorem in the case m and n are consecutive integers.

Proof of Theorem 2.4: The analogue of Proposition 3.10 gives a k-isomorphism of H_{n(n+1)}-modules

π : V_{n} ⊗ V_{n+1}^{∗} ≅ V_{n(n+1)}.

Hence there is a base diagram

(5)  ϕ_{n,n+1} : E → ℙ(V_{n} ⊗ V_{n+1}^{∗})

with associated character

χ_{n,n+1}(S + T) = χ_{n}(S) ⊠ χ_{n+1}(T)^{∗}.

Picking bases for V_{n} and V_{n+1}, we identify ℙ(V_{n}) = ℙ^{n−1}, ℙ(V_{n+1}) = ℙ^{n} and ℙ(V_{n} ⊗ V_{n+1}^{∗}) = ℙ(Mat_{n,n+1}). Then χ_{n,n+1} is given by

χ_{n,n+1}(S + T) : A ↦ χ_{n}(S)Aχ_{n+1}(±T)

where the sign ± is immaterial by the case a = −1 of Lemma 3.5. This proves the first statement of Theorem 2.4.

The base diagram (5) is given by a matrix 𝔄 ∈ Mat_{n,n+1}(k(E)) whose entries are a basis for L(n(n + 1).O). Let r be the rank of this matrix. Then the r × r minors define a morphism

Φ : E → ℙ(∧^{r}V_{n+1}^{∗})

with the property that Φ ∘ τ_{S} = Φ for all S ∈ E[n](\overline{k}). Hence Φ factors through [n] : E → E. Therefore n^{2} | deg(Φ^{∗}H) where H is the hyperplane section on ℙ(∧^{r}V_{n+1}^{∗}). Since deg(Φ^{∗}H) = rn(n + 1) it follows that r = n. Thus there is a commutative diagram

\xymatrix{E \ar[r]^{ϕ_{n,n+1}} \ar[d]_{[n]} & ℙ(V_{n} ⊗ V_{n+1}^{∗}) \ar@{-->}[d]^{μ} \\ E \ar[r]^{γ} & ℙ(V_{n+1})}

where the pull back of the hyperplane section by γ has degree n + 1. We must show that γ = ϕ_{n+1}.

It is easy to see that γ shares with ϕ_{n+1} the property

γ(T + P) = χ_{n+1}(T)γ(P)

for all T ∈ E[n + 1](\overline{k}) and P ∈ E(\overline{k}). Since V_{n+1} is an irreducible representation of H_{n+1} the image of γ spans ℙ(V_{n+1}). So γ is an embedding by a complete linear system.

By Lemma 3.9 the subspace of L(n^{2}(n + 1).O) fixed by H_{n} has dimension n + 1. Since the n × n minors of 𝔄 are linearly independent they are a basis for this space. We show in Proposition 3.11 that if g ∈ \overline{k}(E)^{×} with

div(g) = (n + 1)[n]^{∗}O − n^{2}(n + 1).O

then g is fixed by H_{n}. Hence we may assume that the first n × n minor of 𝔄, viewed as a homogeneous form of degree n in the co-ordinate ring of ℙ(V_{n} ⊗ V_{n+1}^{∗}), meets E with divisor (n + 1)[n]^{∗}O. Then (n + 1).O is the pull back of a hyperplane section by γ, and hence γ is a base diagram of level n + 1.

Since γ and ϕ_{n+1} are base diagrams of level n+1 they can only differ by an element of PGL(V_{n+1}). But they also have the same character χ_{n+1}. Since the image of χ_{n+1} is its own centraliser in PGL(V_{n+1}), it follows that γ = ϕ_{n+1} as required. □

Let σ : Div(E) → E be the summation map. An unexpected difficulty in the proof of Theorem 2.5 is showing that, if D is a hyperplane section for the image of γ, then σ(D) = O. (Comparing pull backs via μ ∘ ϕ_{n,n+1} and γ ∘ [n] only gives that σ(D) is an n-torsion point.) We appeal to the case N = n + 1 of the following proposition.

Proposition 3.11. Let g ∈ \overline{k}(E)^{×} with div(g) = [n]^{∗}O − n^{2}.O. Then g^{N} is fixed by the natural action of H_{n} on L(n^{2}N.O).

For the proof we need two lemmas.

Lemma 3.12. Let T ∈ E(\overline{k}) be a point of exact order n. Let f ∈ \overline{k}(E)^{×} with div(f) = n.T − n.O. If S ∈ E(\overline{k}) with 2S = T and nS ≠ O then

∏_{i=0}^{n−1} f(S + iT) = f(S)^{n}.

Proof: The rational function P ↦ f(P)f(T − P) has trivial divisor and is therefore constant. Hence for any integer i,

f(S + iT)f(S − iT) = f(S)^{2}.

We are immediately done in the case n is odd. In the case n is even it remains to show that f(S) = f(S + T_{2}) where T_{2} = \frac{n}{2}T ∈ E[2](\overline{k}). Let x ∈ \overline{k}(E)^{×} with div(x) = 2.T_{2} − 2.O. Then comparing divisors gives

∏_{i=0}^{(n−2)/2} τ_{−iT}^{∗} f = cx^{n/2}

for some constant c ∈ \overline{k}^{×}. Evaluating each side at ±S we deduce

\frac{f(S + T_{2})}{f(S)} = (\frac{x(−S)}{x(S)})^{n/2} = 1.

□

Lemma 3.13. Let T ∈ E(\overline{k}) be a point of exact order n. Let f, g ∈ \overline{k}(E)^{×} with div(f) = n.T − n.O and div(g) = [n]^{∗}O − n^{2}.O. Then (i) The pair (f, −T) belongs to H_{n} if and only if ∏_{i=0}^{n−1} τ_{iT}^{∗} f = (−1)^{n−1}. (ii) If f satisfies (i) then \frac{g}{τ_{−T}^{∗} g} = f^{n}.

Proof: (i) By Lemma 3.7 the pair (f, −T) belongs to H_{n} if and only if (f, −T)^{n} = (−1)^{n−1}. By the group law (3) this is equivalent to the stated condition. (ii) Each side has divisor n^{2}.T −n^{2}.O. So it suffices to check equality at S ∈ E(\overline{k}) with 2S = T and nS ≠ O. Since g has a pole of order n^{2} − 1 at O, we deduce [−1]^{∗}g = (−1)^{n−1}g. So the left hand side evaluated at S is (−1)^{n−1}. By (i) and Lemma 3.12 we also get (−1)^{n−1} on the right hand side. □

Proof of Proposition 3.11: If (f, −T) ∈ H_{n} then

(f, −T)g^{N} = τ_{T}^{∗}(g^{N}/f^{nN}) = g^{N}

where for the first equality we use (4), and for the second equality we use Lemma 3.13. □

4. Invariant Theory

We recall some classical invariant theory of binary quartics and ternary cubics, as surveyed in [1]. We then add to this theory by introducing what we call “covariant columns”. These are used in §5 to give formulae for ϕ_{2,3} and ϕ_{3,4}. In this section we give a complete classification of the covariant columns. This is more that we need in §5, but serves to explain where our formulae come from.

In this section k will be a field with char (k) ≠ 2, 3.

4.1. Binary quartics. We study the invariants and covariants of the binary quartic

U(x_{1}, x_{2}) = ax_{1}^{4} + bx_{1}^{3}x_{2} + cx_{1}^{2}x_{2}^{2} + dx_{1}x_{2}^{3} + ex_{2}^{4}.

For a polynomial F ∈ k[x_{1}, x_{2}] and matrix g ∈ GL_{2}(k) we write

(F ∘ g)(x_{1}, x_{2}) = F (g_{11}x_{1} + g_{12}x_{2}, g_{21}x_{1} + g_{22}x_{2})

Thus (F ∘ g)(x) = F (gx) where x is the column vector (x_{1}, x_{2})^{T}.

Definition 4.1. A covariant F = F (U; x) of order m, degree d and weight p, is a homogeneous polynomial of degree m in x_{1}, x_{2}, whose coefficients are homogeneous polynomials of degree d in the coefficients of the binary quartic U, such that

F (U ∘ g; x) = (det g)^{p}F (U; gx)

for all g ∈ GL_{2}(\overline{k}).

By considering g a scalar matrix, it is clear that the order m, degree d and weight p of a covariant are related by 4d = 2p + m. It is well known that the ring of invariants (an invariant is a covariant of order 0) is generated by c_{4} and c_{6} where

\begin{aligned}c_{4} & = & 2^{4}(12ae − 3bd + c^{2}) \\ c_{6} & = & 2^{5}(72ace − 27ad^{2} − 27b^{2}e + 9bcd − 2c^{3}). \end{aligned}

Moreover, the ring of covariants is generated by c_{4}, c_{6}, U, H and J, where

H(x_{1}, x_{2}) = \frac{1}{3} det (\frac{∂^{2}U}{∂x_{i}∂x_{j}} )_{i,j=1,2}

and

(6)  J(x_{1}, x_{2}) = \frac{1}{12} \frac{∂(U,H)}{∂(x_{1},x_{2})} ,

subject only to the relation

(7)  27J^{2} = −H^{3} + 3c_{4}HU^{2} − 2c_{6}U^{3}.

Since c_{4}, c_{6}, U, H and J have weights 4, 6, 0, 2 and 3 we deduce Lemma 4.2. Every covariant of odd weight is divisible by J.

We now define what we call a “covariant column”.

Definition 4.3. A covariant column v = v(U; x) of order m, degree d and weight p, is a column vector v = (v_{1}, v_{2})^{T} of degree m homogeneous polynomials in x_{1}, x_{2}, whose coefficients are homogeneous polynomials of degree d in the coefficients of the binary quartic U, such that

v(U ∘ g; x) = (det g)^{p}g^{−1}v(U; gx)

for all g ∈ GL_{2}(\overline{k}).

By considering g a scalar matrix, it is clear that the order m, degree d and weight p of a covariant column are related by 4d = 2p+m−1. The column vector x itself is a covariant column of order 1, degree 0 and weight 0. The proof of the following lemma is entirely straightforward, and so will be omitted.

Lemma 4.4. (i) If F is a covariant of order m, degree d and weight p then ∂F = (−\frac{∂F}{∂x_{2}} , \frac{∂F}{∂x_{1}} )^{T} is a covariant column of order m − 1, degree d and weight p + 1, (ii) If v_{1} and v_{2} are covariant columns of orders m_{1}, m_{2}, degrees d_{1}, d_{2} and weights p_{1}, p_{2}, then the determinant [v_{1}, v_{2}] is a covariant of order m_{1} + m_{2}, degree d_{1} + d_{2} and weight p_{1} + p_{2} − 1.

For F a covariant of order m we have [x, ∂F] = mF.

Theorem 4.5. (i) The covariant columns of even weight form a free k[c_{4}, c_{6}, U, H]-module of rank 2 with basis x, ∂J. (ii) The covariant columns of odd weight form a free k[c_{4}, c_{6}, U, H]- module of rank 2 with basis ∂U, ∂H.

Proof: (i) Since [x, ∂J] = 6J ≠ 0, we can write any covariant column as v = F_{1}x + F_{2}∂J for some rational functions F_{1} and F_{2}. Then [v, ∂J] = 6JF_{1} is a covariant of odd weight. It follows by Lemma 4.2 that F_{1} is a covariant. The same argument shows that F_{2} is a covariant. (ii) Since we can rewrite (6) as [∂U, ∂H] = 12J, the proof carries over exactly as in case (i). □

Corollary 4.6. The covariant columns form a module over the ring of covariants, generated by x, ∂U, ∂H, ∂J, subject only to the relations

\begin{aligned}3Jx & = & H∂U − U∂H \\ 18J∂J & = & 2(c_{4}UH − c_{6}U^{2})∂U + (c_{4}U^{2} − H^{2})∂H \\ 9J∂U & = & (c_{4}U^{2} − H^{2})x + 6U∂J \\ 9J∂H & = & 2(c_{6}U^{2} − c_{4}UH)x + 6H∂J \end{aligned}

Proof: It only remains to describe the action of multiplication by J. The first relation is obtained by applying the proof of Theorem 4.5(ii) to v = Jx, and the second is obtained by differentiating the syzygy (7). We take linear combinations, and use the syzygy once more to obtain the final two relations. □

4.2. Ternary cubics. We study the invariants and covariants of the ternary cubic

U(x_{1}, x_{2}, x_{3}) = ax_{1}^{3} + bx_{2}^{3} + cx_{3}^{3} + … + mx_{1}x_{2}x_{3}.

For a polynomial F ∈ k[x_{1}, x_{2}, x_{3}] and matrix g ∈ GL_{3}(k) we write (F ∘ g)(x) = F (gx) where x is the column vector (x_{1}, x_{2}, x_{3})^{T}. The definition of a covariant is exactly analogous to that in the case of a binary quartic. By considering g a scalar matrix, it is clear that the order m, degree d and weight p of a covariant are related by 3d = 3p+m.

The Hessian is a covariant of order 3, degree 3 and weight 2 given by

H(x_{1}, x_{2}, x_{3}) = −\frac{1}{2} det (\frac{∂U}{∂x_{i}∂x_{j}} )_{i,j=1,2,3} .

There are invariants c_{4} and c_{6} such that

H(λU + μH) = 3(c_{4}λ^{2}μ + 2c_{6}λμ^{2} + c_{4}^{2}μ^{3})U + (λ^{3} − 3c_{4}λμ^{2} − 2c_{6}μ^{3})H,

and it is well known that these generate the ring of invariants.

If Q_{1}(x) and Q_{2}(x) are ternary quadrics, with corresponding 3 × 3 symmetric matrices A_{1} and A_{2}, i.e.

Q_{1}(x) = \frac{1}{2}x^{T} A_{1}x and Q_{2}(x) = \frac{1}{2}x^{T} A_{2}x,

then we write {Q_{1}, Q_{2}} for the 3 × 3 symmetric matrix satisfying

adj(A_{1} + tA_{2}) = adj(A_{1}) + t{Q_{1}, Q_{2}} + t^{2} adj(A_{2}).

We can then define a symmetric matrix of quadrics

M = ∑_{i,j=1}^{3} \{\frac{∂U}{∂x_{i}} , \frac{∂H}{∂x_{j}} \} x_{i}x_{j}

and covariants

(8)  \begin{aligned}Θ(x_{1}, x_{2}, x_{3}) & = & ∑_{r,s=1}^{3} M_{rs} \frac{∂U}{∂x_{r}} \frac{∂H}{∂x_{s}} \\ J(x_{1}, x_{2}, x_{3}) & = & \frac{1}{3} \frac{∂(U,H,Θ)}{∂(x_{1},x_{2},x_{3})}. \end{aligned}

Again it is well known that the ring of covariants is generated by c_{4}, c_{6}, U, H, Θ and J subject only to a relation which reduces mod U to

(9)  J^{2} ≡ Θ^{3} − 27c_{4}ΘH^{4} − 54c_{6}H^{6} (mod U).

Since c_{4}, c_{6}, U, H, Θ and J have weights 4, 6, 0, 2, 6 and 9 we deduce Lemma 4.7. Every covariant of odd weight is divisible by J.

Our definition of a covariant column is exactly analogous to that in the case of a binary quartic. However we now also need to work with “contravariant columns”.

Definition 4.8. A covariant column, respectively contravariant column, v = v(U; x) of order m, degree d and weight p, is a column vector v = (v_{1}, v_{2}, v_{3})^{T} of degree m homogeneous polynomials in x_{1}, x_{2}, x_{3}, whose coefficients are homogeneous polynomials of degree d in the coefficients of the ternary cubic U, such that

v(U ∘ g; x) = (det g)^{p}g^{−1}v(U; gx),

respectively

v(U ∘ g; x) = (det g)^{p}g^{T} v(U; gx),

for all g ∈ GL_{3}(\overline{k}).

By considering g a scalar matrix, it is clear that the order m, degree d and weight p of a covariant column, respectively contravariant column, are related by 3d = 3p + m − 1, respectively 3d = 3p + m + 1. The column vector x itself is a covariant column of order 1, degree 0 and weight 0. The proof of the following lemma is entirely straightforward, and so will be omitted.

Lemma 4.9. (i) If F is a covariant of order m, degree d and weight p then ∇F = (\frac{∂F}{∂x_{1}} , \frac{∂F}{∂x_{2}} , \frac{∂F}{∂x_{3}} )^{T} is a contravariant column of order m − 1, degree d and weight p. (ii) Let v_{1} and v_{2} be covariant or contravariant columns of orders m_{1}, m_{2}, degrees d_{1}, d_{2} and weights p_{1}, p_{2}. Then

(1) If v_{1} is a covariant column, and v_{2} a contravariant column then the dot product v_{1} · v_{2} is a covariant of order m_{1} + m_{2}, degree d_{1} + d_{2} and weight p_{1} + p_{2}.

(2) If v_{1} and v_{2} are covariant columns then the cross product v_{1}×v_{2} is a contravariant column of order m_{1} + m_{2}, degree d_{1} + d_{2} and weight p_{1} + p_{2} − 1.

(3) If v_{1} and v_{2} are contravariant columns then the cross product v_{1} × v_{2} is a covariant column of order m_{1} + m_{2}, degree d_{1} + d_{2} and weight p_{1} + p_{2} + 1.

(iii) If v is a contravariant column of order m, degree d and weight p, then Mv is covariant column of order m + 2, degree d + 4 and weight p + 4.

For F a covariant of order m we have x.∇F = mF. The determinant of the three vectors v_{1}, v_{2}, v_{3} will be denoted

[v_{1}, v_{2}, v_{3}] = (v_{1} × v_{2}) · v_{3} = v_{1} · (v_{2} × v_{3}).

We define contravariant columns u = ∇U, h = ∇H, t = ∇Θ and covariant columns e = Mu and f = Mh.

Theorem 4.10. (i) The covariant columns, respectively contravariant columns, of even weight form a free k[c_{4}, c_{6}, U, H, Θ]-module of rank 3 with basis x, e, f, respectively u, h, t. (ii) The covariant columns, respectively contravariant columns, of odd weight form a free k[c_{4}, c_{6}, U, H, Θ]-module of rank 3 with basis u × h, u × t, h × t, respectively x × e, x × f, e × f.

Proof: (i) Since [x, e, f] = −2J ≠ 0, we can write any covariant column as v = F_{1}x + F_{2}e + F_{3}f for some rational functions F_{1}, F_{2}, F_{3}. Then [v, e, f] = −2JF_{1} is a covariant of odd weight. It follows by Lemma 4.7 that F_{1} is a covariant, and likewise for F_{2} and F_{3}. The case of a contravariant column is similar, since we can rewrite (8) as [u, h, t] = 3J. (ii) Since [u × h, u × t, h × t] = [u, h, t]^{2} = 9J^{2} ≠ 0, we can write any covariant column as v = F_{1}(u × h) + F_{2}(u × t) + F_{3}(h × t) for some rational functions F_{1}, F_{2}, F_{3}. Then v · t = 3JF_{1} is a covariant of odd weight. It follows by Lemma 4.7 that F_{1} is a covariant, and likewise for F_{2} and F_{3}. The case of a contravariant column is similar. □

Corollary 4.11. (i) The covariant columns are generated as a module over the ring of covariants by x, e, f, u × h, u × t and h × t. (ii) The contravariant columns are generated as a module over the ring of covariants by u, h, t, x × e, x × f and e × f.

As in the case of binary quartics, there is no difficulty in finding the relations (describing the effect of multiplication by J). Since these relations are somewhat messy, we instead record the identities

x · u = 3U, x · h = 3H, x · t = 6Θ, e · h = f · u = Θ

and

e · u = 3(H^{2} − 3c_{4}U^{2})
f · h = 3(3c_{4}H^{2} − 8c_{6}UH + 3c_{4}^{2}U^{2})
e · t = 12(3c_{4}H^{3} − c_{4}UΘ − 12c_{6}UH^{2} + 9c_{4}^{2}U^{2}H)
f · t = 12(c_{4}HΘ − 3c_{6}H^{3} − 3c_{6}UΘ + 9c_{4}^{2}UH^{2} − 3c_{4}c_{6}U^{2}H − 9c_{4}^{3}U^{3})

from which the relations may be recovered by following the proof of Theorem 4.10.

5. Covariant matrices

We use the invariant theory of binary quartics and ternary cubics to given an alternative proof of Theorem 2.5 in the cases n = 2, 3.

5.1. The case n = 2. A binary quartic U ∈ k[x_{1}, x_{2}] is non-singular if its discriminant Δ = (c_{4}^{3} − c_{6}^{2})/1728 is non-zero. We write ℙ(1, 1, 2) for the weighted projective space where the co-ordinates x_{1}, x_{2}, y are assigned degrees 1, 1, 2.

Proposition 5.1. Let U ∈ k[x_{1}, x_{2}] be a non-singular binary quartic with invariants c_{4}, c_{6} ∈ k and covariants H, J ∈ k[x_{1}, x_{2}]. Then

(i) The equation y^{2} = U(x_{1}, x_{2}) defines a smooth curve of genus one C_{2} ⊂ ℙ(1, 1, 2). (ii) The Jacobian E of C_{2} has Weierstrass equation

Y^{2}Z = X^{3} − 27c_{4}XZ^{2} − 54c_{6}Z^{3}.

(iii) The 2-covering map π : C_{2} → E is given by

(Z : X : Y) = (yU(x_{1}, x_{2}) : −3yH(x_{1}, x_{2}) : 27J(x_{1}, x_{2})).

(iv) Let ϕ_{3} : E → ℙ^{2} be the natural inclusion. Then the Brauer-Severi diagrams ϕ_{2} : C_{2} → ℙ^{1} ; (x_{1} : x_{2} : y) ↦ (x_{1} : x_{2}) and ϕ_{2,3} : C_{2} → ℙ(Mat_{2,3}) ; (x_{1} : x_{2} : y) ↦ A_{2,3}, where

A_{2,3} = (\begin{matrix}−9\frac{∂H}{∂x_{2}} & −3\frac{∂U}{∂x_{2}} & x_{1}y \\ 9\frac{∂H}{∂x_{1}} & 3\frac{∂U}{∂x_{1}} & x_{2}y \end{matrix}) ,

satisfy the conclusions of Theorem 2.5.

Proof: Statements (i)–(iii) are well known: see [1], [23]. (iv) Let D be the hyperplane section for ϕ_{2} : C → ℙ^{1}. Then L(3D) has basis x_{1}^{3}, x_{1}^{2}x_{2}, x_{1}x_{2}^{2}, x_{2}^{3}, x_{1}y, x_{2}y. We write the entries of A_{2,3} as linear combinations of these basis elements, and arrange the coefficients in a 6 × 6 matrix. The determinant of this matrix is 2^{2}3^{8}Δ. Hence ϕ_{2,3} : C_{2} → ℙ(Mat_{2,3}) is an embedding by the complete linear system |3D|.

In the notation of §4.1 we have A_{2,3} = (9∂H, 3∂U, yx). The 2 × 2 minors of this matrix are

(\begin{matrix}−3y[x, ∂U] \\ 9y[x, ∂H] \\ −27[∂U, ∂H] \end{matrix}) = 12 (\begin{matrix}−yU \\ 3yH \\ −27J \end{matrix})

So the final statement of Theorem 2.5 is immediate from (iii).

It remains to show that S ∈ E[2](\overline{k}), respectively T ∈ E[3](\overline{k}), acts on the image of ϕ_{2,3} as left multiplication by χ_{2}(S), respectively right multiplication by χ_{3}(T).

The statement for S ∈ E[2](\overline{k}) follows formally from the covariance of the columns of A_{2,3}. Indeed, writing A_{2,3} = (a_{1}, a_{2}, ya_{3}), where the a_{i} are covariant columns, Definition 4.3 gives

a_{i}(U ∘ g; x) = g^{−1}a_{i}(U, gx)

for all g ∈ SL_{2}(\overline{k}). So if g ∈ SL_{2}(\overline{k}) is a lift of χ_{2}(S) then U ∘ g = U and

g ϕ_{2,3}(x : y) = ϕ_{2,3}(g x : y)

as required.

We deduce the statement for T ∈ E[3](\overline{k}) from the parts of Theorem 2.5 already established. To this end, let M_{S} and M_{T} be endomorphisms of Mat_{2,3}(\overline{k}) lifting χ_{2,3}(S) and χ_{2,3}(T) respectively. We have shown that M_{S} is left multiplication by a 2×2 matrix. Accordingly we view Mat_{2,3}(\overline{k}) as an H_{2}-module via left multiplication. (In this proof we write H_{n} as a shorthand for H_{n}(\overline{k}).) Since 2 and 3 are coprime, the commutator of M_{S} and M_{T} is trivial. Hence M_{T} is an endomorphism of Mat_{2,3}(\overline{k}) as an H_{2}-module. Since the standard representation V_{2} of H_{2} is irreducible, it follows by Schur’s lemma that M_{T} belongs to

End_{H_{2}}(Mat_{2,3}(\overline{k})) ≅ End_{H_{2}}(V_{2} ⊕ V_{2} ⊕ V_{2}) ≅ Mat_{3}(\overline{k}).

Thus M_{T} is right multiplication by a 3 × 3 matrix. Let ψ(T) be the image of this matrix in PGL_{3}(\overline{k}). It remains to show that the characters χ_{3} : E[3] → PGL_{3} and ψ : E[3] → PGL_{3} are equal. Recalling that ϕ_{3} ∘ π = μ ∘ ϕ_{2,3}, we take P ∈ C_{2}(\overline{k}) and compute

ϕ_{3}(π(P + T)) = μ(ϕ_{2,3}(P + T))
⟹  ϕ_{3}(π(P) + 2T) = μ(ϕ_{2,3}(P)ψ(T))
⟹ χ_{3}(T)^{−1}ϕ_{3}(π(P)) = ψ(T)^{−1}μ(ϕ_{2,3}(P)).

Hence χ_{3} = ψ as required. □

5.2. The case n = 3. A ternary cubic U ∈ k[x_{1}, x_{2}, x_{3}] is non-singular if its discriminant Δ = (c_{4}^{3} − c_{6}^{2})/1728 is non-zero. The covariant columns e, f and contravariant columns u, h were defined in §4.2.

Proposition 5.2. Let U ∈ k[x_{1}, x_{2}, x_{3}] be a non-singular ternary cubic with invariants c_{4}, c_{6} ∈ k and covariants H, Θ, J ∈ k[x_{1}, x_{2}, x_{3}]. Then (i) The equation U(x_{1}, x_{2}, x_{3}) = 0 defines a smooth curve of genus one C_{3} ⊂ ℙ^{2}. (ii) The Jacobian E of C_{3} has Weierstrass equation

Y^{2}Z = X^{3} − 27c_{4}XZ^{2} − 54c_{6}Z^{3}.

(iii) The 3-covering map π : C_{3} → E is given by

(Z : X : Y) = (H^{3} : ΘH : J).

(iv) Let ϕ_{4} : E → ℙ^{3} ; (Z : X : Y) ↦ (Z^{2} : XZ : Y Z : X^{2}). Then the Brauer-Severi diagrams ϕ_{3} : C_{3} → ℙ^{2} and ϕ_{3,4} : C_{3} → ℙ(Mat_{3,4}) ; (x_{1} : x_{2} : x_{3}) ↦ A_{3,4}, where

A_{3,4} = (−3f + 9c_{4}Hx e \frac{2}{3}(u × h) −\frac{1}{3}Hx) ,

satisfy the conclusions of Theorem 2.5.

Proof: Statements (i)–(iii) are well known: see [1]. (iv) By Theorem 4.10 the covariant columns of order 4 form a free k[c_{4}, c_{6}]-module with basis

(10)  Ux, Hx, e, f, u × h.

The entries of these columns give us 15 ternary quartics. We arrange the coefficients of these quartics in a 15 × 15 matrix, and find that the determinant is 2^{42}3^{12}Δ^{5}. (The calculation is made easier if we first put U in Hesse normal form:

U(x_{1}, x_{2}, x_{3}) = a(x_{1}^{3} + x_{2}^{3} + x_{3}^{3}) + bx_{1}x_{2}x_{3}.)

Let D the hyperplane section for ϕ_{3} : C_{3} → ℙ^{2}. Since the only ternary quartics vanishing on C_{3} are the entries of Ux, the above calculation shows that ϕ_{3,4} : C_{3} → ℙ(Mat_{3,4}) is an embedding by the complete linear system |4D|.

A direct calculation (carried out for U in Hesse normal form) shows that the 3 × 3 minors of A_{3,4} are

μ_{1} = 2H^{4} − 6c_{4}U^{2}H^{2} − \frac{2}{3}UHΘ
μ_{2} = 2ΘH^{2} − 18c_{4}^{2}U^{3}H − 18c_{4}UH^{3} + 48c_{6}U^{2}H^{2}
μ_{3} = 2JH
μ_{4} = 2Θ^{2} + 162c_{4}^{3}U^{4} − 54c_{4}^{2}U^{2}H^{2} − 432c_{4}c_{6}U^{3}H
− 18c_{4}UHΘ + 144c_{6}UH^{3}.

The final statement of Theorem 2.5 follows since by (iii) the composition ϕ_{4} ∘ π is given by

(x_{1} : x_{2} : x_{3}) ↦ (H^{4} : ΘH^{2} : JH : Θ^{2}).

The remainder of the proof now carries through exactly as in the case n = 2. □

6. Computations

Let E be an elliptic curve over ℚ. We use 6-descent and 12-descent to assist in the search for generators of E(ℚ) of large height. The method is of greatest interest when E(ℚ) has rank at least 2, or E has large conductor, i.e. in those cases where we cannot use Heegner points. In this section we give some details of our implementation in the computer algebra system Magma [15]. Further remarks accompany the numerical examples in §7.

6.1. The method in outline. We begin by using the existing functions in Magma to compute n-coverings for n = 2, 3, 4.

• The Magma function TwoDescent, takes as input a Weierstrass equation for E, and returns a list of 2^{s} − 1 binary quartics representing the non-zero elements of the 2-Selmer group S^{(2)}(E/ℚ) ≅ (ℤ/2ℤ)^{s}.

• The Magma function ThreeDescent, written by Stoll, Donnelly and the author, takes as input a Weierstrass equation for E, and returns a list of (3^{t} − 1)/2 ternary cubics representing the non-zero elements of the 3-Selmer group S^{(3)}(E/ℚ) ≅ (ℤ/3ℤ)^{t}.

• The Magma function FourDescent, written by Womack and Watkins, takes as input a binary quartic representing a non-zero element α ∈ S^{(2)}(E/ℚ), and returns a list of pairs of quadrics in four variables, representing the elements of the 4-Selmer group in the fibre of S^{(4)}(E/ℚ) → S^{(2)}(E/ℚ) above α.

Each element of the n-Selmer group is now represented by (equations for) a Brauer-Severi diagram [C_{n} → ℙ^{n−1}]. The Selmer group elements may equally be viewed as n-coverings, where the covering maps π : C_{n} → E are computed using the classical formulae surveyed in [1]. (For n = 2, 3 we recalled these formulae in Propositions 5.1 and 5.2.) Replacing the covering map π by [−1] ∘ π corresponds to taking the inverse in the Selmer group. So in the cases n = 3, 4 each ternary cubic, respectively pair of quadrics, represents both a Selmer group element and its inverse.

Definition 6.1. Let [C_{n} → ℙ^{n−1}] be a Brauer-Severi diagram with hyperplane section D. A point P ∈ C_{n}(\overline{k}) is a flex if n.P ∼ D.

The flex points of a 2-covering are the roots of the binary quartic. In the cases n = 3, 4 the flex points (also known as points of inflection, or hyperosculating points) are the intersections with H = 0, respectively J = 0, where H is the Hessian of a ternary cubic, and J is the covariant defined in [1, §3.3].

We recall that if [C_{n} → ℙ^{n−1}] is a Brauer-Severi diagram, then the morphism C_{n} → ℙ^{n−1} is that determined by a complete linear system of degree n. So if n ≥ 3 then C_{n} → ℙ^{n−1} is an embedding. We identify C_{n} with its image, which is called a genus one normal curve of degree n. It is well known that if n ≥ 4 then the homogeneous ideal I(C_{n}) is generated by a vector space of quadrics of dimension n(n − 3)/2.

The details of 6-descent are as follows. We start with a 2-covering C_{2} = {y^{2} = U_{2}(x_{1}, x_{2})} and a 3-covering C_{3} = {U_{3}(x_{1}, x_{2}, x_{3}) = 0}, each defined over ℚ. Since these are coverings of the same elliptic curve E, we may assume that U_{2} and U_{3} have the same invariants c_{4} and c_{6}. Then E has Weierstrass equation

y^{2} = x^{3} − 27c_{4}x − 54c_{6}.

We compute a flex point on C_{3} with co-ordinates in a number field, L say. Typically [L : ℚ] = 9. Then Algorithm 6.3 finds a matrix g ∈ GL_{3}(L) with

(U_{3} ∘ g)(z, x, y) = λ(y^{2}z − x^{3} + 27c_{4}xz^{2} + 54c_{6}z^{3})

for some λ ∈ L^{×}. (In the notation of §2 we have g = B_{3}^{−1}.)

Next we let ϕ_{2,3} : C_{2} → ℙ(Mat_{2,3}) be the embedding defined in Proposition 5.1(iv). The image is a genus one normal curve of degree 6. We use linear algebra to compute a basis Q_{1}, …, Q_{9} for the space of quadrics vanishing on this curve. Writing these as polynomials in variables X_{ij} for 1 ≤ i ≤ 2 and 1 ≤ j ≤ 3, we make the substitution

(\begin{matrix}X_{11} & X_{12} & X_{13} \\ X_{21} & X_{22} & X_{23} \end{matrix}) = (\begin{matrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \end{matrix}) g.

The new quadrics have coefficients in L, but the vector space they span has a basis with coefficients in ℚ. We compute an LLL-reduced basis for the intersection of this space with ℤ[x_{11}, …, x_{23}]. These are now the equations for a 6-covering C_{6} ⊂ ℙ(Mat_{2,3}). Moreover, by Theorem 2.5, the covering map C_{6} → C_{3} is defined by the 2 × 2 minors, i.e.

(\begin{matrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \end{matrix}) ↦ (x_{12}x_{23} −x_{22}x_{13} : x_{13}x_{21} −x_{23}x_{11} : x_{11}x_{22} −x_{21}x_{12}).

The details of 12-descent are similar. We start with a 3-covering C_{3} = {U_{3}(x_{1}, x_{2}, x_{3}) = 0} and a 4-covering C_{4} = {Q_{1} = Q_{2} = 0}, each defined over ℚ, and with the same invariants c_{4} and c_{6}. We compute a flex point on C_{4} with co-ordinates in a number field, L say. Typically [L : ℚ] = 16. Then Algorithm 6.4 finds a matrix g ∈ GL_{4}(L) with

⟨Q_{1} ∘ g, Q_{2} ∘ g⟩ = ⟨x_{1}x_{4} − x_{2}^{2}, x_{2}x_{4} − x_{3}^{2} − 27c_{4}x_{1}x_{2} − 54c_{6}x_{1}^{2}⟩.

Next we let ϕ_{3,4} : C_{3} → ℙ(Mat_{3,4}) be the embedding defined in Proposition 5.2(iv). The image is a genus one normal curve of degree 12. We use linear algebra to compute a basis Q_{1}, …, Q_{54} for the space of quadrics vanishing on this curve. As in the case of 6descent, we then twist by g ∈ GL_{4}(L) to obtain equations for a 12covering C_{12} ⊂ ℙ(Mat_{3,4}). Moreover, by Theorem 2.5, the covering map C_{12} → C_{4} is defined by the 3 × 3 minors.

Unlike the case of 6-descent, we can combine a 3-covering and a 4covering to give a 12-covering in two essentially different ways. This is because each of C_{3} and C_{4} represents both a Selmer group element and its inverse. It is important that we compute both 12-coverings, since in the case they are soluble, their rational points will cover ±P + 12E(ℚ) and ±5P + 12E(ℚ) respectively. In practice the second 12-covering is obtained by switching the sign in the third column of the matrix defining ϕ_{3,4}.

It remains to search for rational points on C_{6} and C_{12}. We use the p-adic point searching method due independently to Elkies and Heath-Brown, as implemented by Watkins in the Magma function PointSearch. Descriptions may be found in [22] and [24, §2.9]. (Elkies’ original paper [10] only considers real approximations.) The method first chooses an auxiliary prime p, whose size depends on the height bound set for the search. The points on the reduction of C mod p are then enumerated, and for each such point P_{0} a lattice method variant of Hensel’s lemma is used to search for rational points on C with reduction P_{0}. A variant of the method uses two primes. The method works particularly well for curves of high codimension as considered here.

Finally, our search for points is significantly improved if we “minimise” our equations for C_{6} and C_{12} before running PointSearch. We give details in §6.3.

6.2. Computing flex matrices. Let [C → ℙ^{n−1}] be a Brauer-Severi diagram. To compute a flex matrix for C, as defined in §2, we first find a flex point P on C. We then follow an inductive procedure, based on the idea of projecting away from P. This method is a by-product of the standard procedures for putting an elliptic curve in Weierstrass form, as described in [3, §8]. We therefore simply list the algorithms used. Notice that we do not use the general Riemann-Roch machinery implemented in Magma, as this would be unnecessarily slow in our applications.

Algorithm 6.2. Let U ∈ k[x_{1}, x_{2}] be a non-singular binary quartic with invariants c_{4} and c_{6}. Given (α : β) ∈ ℙ^{1}(k) with U(α, β) = 0 we compute g ∈ GL_{2}(k) with last column (α, β)^{T} satisfying

(U ∘ g)(z, x) = \frac{1}{36}(det g)^{2}(x^{3}z − 27c_{4}xz^{3} − 54c_{6}z^{4}).

(1) Choose any g_{1} ∈ GL_{2}(k) with last column (α, β)^{T}.

(2) Compute (U ∘ g_{1})(z, x) = (det g_{1})^{2}(bx^{3}z + cx^{2}z^{2} + …) and put

g_{2} = (\begin{matrix}36b & 0 \\ −12c & 1 \end{matrix}).

(3) Return g_{1}g_{2}.

Algorithm 6.3. Let U ∈ k[x_{1}, x_{2}, x_{3}] be a non-singular ternary cubic with invariants c_{4} and c_{6}. Given (α : β : γ) ∈ ℙ^{2}(k) a flex point on the curve {U = 0} we compute g ∈ GL_{3}(k) with last column (α, β, γ)^{T} satisfying

(U ∘ g)(z, x, y) = \frac{1}{6}(det g)(y^{2}z − x^{3} + 27c_{4}xz^{2} + 54c_{6}z^{3}).

(1) Choose any g_{1} ∈ GL_{3}(k) with last column (α, β, γ)^{T}.

(2) Write (U ∘g_{1})(z, x, y) = (det g_{1})(f_{1}(z, x)y^{2}+f_{2}(z, x)y+f_{3}(z, x)) and let α, β ∈ k with f_{1}(z, x) = βz − αx. Then run Algorithm 6.2 on \frac{1}{4}f_{2}^{2} − f_{1}f_{3} to obtain g ∈ GL_{2}(k).

(3) Compute p, q ∈ k with

\begin{aligned}(f_{1} ∘ g)(z, x) & = & (det g)z \\ (f_{2} ∘ g)(z, x) & = & (det g)(pz + qx)z \end{aligned}

and put g_{2} = ( \begin{matrix}6g & {0 \atop 0} \\ −3p −3q & 1 \end{matrix} ).

(4) Return g_{1}g_{2}.

In the case n = 4 the invariants are again described in [1]. We label them c_{4} and c_{6}, with scalings as specified in [11].

Algorithm 6.4. Let Q_{1}, Q_{2} ∈ k[x_{1}, x_{2}, x_{3}, x_{4}] be a pair of quadrics, with invariants c_{4} and c_{6}. We suppose that {Q_{1} = Q_{2} = 0} is a smooth curve of genus one. Given (α : β : γ : δ) ∈ ℙ^{3}(k), a flex point on this curve, we compute g ∈ GL_{4}(k) with last column (α, β, γ, δ)^{T} satisfying

⟨Q_{1} ∘ g, Q_{2} ∘ g⟩ = ⟨x_{1}x_{4} − x_{2}^{2}, x_{2}x_{4} − x_{3}^{2} − 27c_{4}x_{1}x_{2} − 54c_{6}x_{1}^{2}⟩.

(1) Choose any g_{1} ∈ GL_{4}(k) with last column (α, β, γ, δ)^{T}.

(2) Write

\begin{aligned}Q_{1} ∘ g_{1} & = & ℓ_{1}(x_{1}, x_{2}, x_{3})x_{4} + q_{1}(x_{1}, x_{2}, x_{3}) \\ Q_{2} ∘ g_{1} & = & ℓ_{2}(x_{1}, x_{2}, x_{3})x_{4} + q_{2}(x_{1}, x_{2}, x_{3}) \end{aligned}

and let

α = |\begin{matrix}ℓ_{12} & ℓ_{13} \\ ℓ_{22} & ℓ_{23} \end{matrix}| , β = |\begin{matrix}ℓ_{13} & ℓ_{11} \\ ℓ_{23} & ℓ_{21} \end{matrix}| , γ = |\begin{matrix}ℓ_{11} & ℓ_{12} \\ ℓ_{21} & ℓ_{22} \end{matrix}| ,

where ℓ_{i} = ∑ ℓ_{ij}x_{j}. Then run Algorithm 6.3 on (det g_{1})^{−1}(ℓ_{2}q_{1}− ℓ_{1}q_{2}) to obtain g ∈ GL_{3}(k).

(3) Replace Q_{1} and Q_{2} by linear combinations (and update the ℓ_{i} and q_{i} of Step 2) so that ℓ_{i} ∘ g = x_{i} for i = 1, 2.

(4) Compute a, b, c ∈ k with

\begin{aligned}q_{1} ∘ g & = & \frac{1}{6}(x_{1}(ax_{1} + bx_{2} + cx_{3}) − x_{2}^{2}) \\ q_{2} ∘ g & = & \frac{1}{6}(x_{2}(ax_{1} + bx_{2} + cx_{3}) − x_{3}^{2} − 27c_{4}x_{1}x_{2} − 54c_{6}x_{1}^{2}) \end{aligned}

and put g_{2} = ( \begin{matrix}6g & 0 0 0 \\ −a −b −c & 1 \end{matrix} ).

(5) Return g_{1}g_{2}. 6.3. Minimisation. If an n-covering π : C → E is to be useful in the search for rational points on E, not only must we find explicit equations for C ⊂ ℙ^{n−1}, but we must also find a change of co-ordinates on ℙ^{n−1} so that these equations have reasonably small coefficients. The task naturally falls into two parts which, following terminology introduced by Cremona, we call minimisation and reduction.

Minimisation is the task of removing as many prime factors as possible from a suitably defined discriminant. The most familiar example is that of minimising a Weierstrass equation. By reduction we mean the use of unimodular transformations to further decrease the size of the coefficients. The basic example is reduction of binary quadratic forms, or more generally lattice reduction. Thus minimisation is concerned with the finite places, and reduction with the infinite places. The need to perform reduction is our main reason for working over the rationals (instead of a more general number field).

The minimisation and reduction of 2-coverings has been studied in [2], [4], and [8]. The generalisations to 3-coverings and 4-coverings are described in [6] and [24]. These algorithms have been implemented in Magma, and are called by the functions TwoDescent, ThreeDescent and FourDescent. Hence in §6.1 we start with an n-covering and an (n + 1)-covering both of which are already minimised and reduced. So it would not be unreasonable to hope that the n(n + 1)-covering computed from them will automatically be minimised and reduced. Numerical examples suggest that this is true for reduction, but not for minimisation.

The following is a description of our current ad hoc approach to the minimisation of n-coverings for n > 5. Although this method works reasonably well in practice, there remains considerable room for both theoretical and practical improvements.

Let C ⊂ ℙ^{n−1} be a genus one normal curve, defined over ℚ, and of degree n ≥ 4. We recall that the homogeneous ideal I(C) in ℚ[x_{1}, …, x_{n}] is generated by a vector space of quadrics of dimension N = n(n−3)/2. Then for A ∈ GL_{N}(ℚ) and B ∈ GL_{n}(ℚ) we define

[A, B](q_{1}, …, q_{N}) = (∑_{i=1}^{N} a_{1i}q_{i} ∘ B^{T} , …, ∑_{i=1}^{N} a_{Ni}q_{i} ∘ B^{T}).

Definition 6.5. (i) An integral model (q_{1}, …, q_{N}) for C is a tuple of quadrics in ℤ[x_{1}, …, x_{n}] generating I(C).

(ii) An integral model (q_{1}, …, q_{N}) is minimal at a prime p if whenever A ∈ GL_{N}(ℚ) and B ∈ GL_{n}(ℚ) with [A, B](q_{1}, …, q_{N}) integral, then

ord_{p}(det A) + (n − 3) ord_{p}(det B) ≥ 0.

(iii) An integral model is minimal if it is minimal at all primes p.

It is not even clear from our definition that minimal models exist, although in the cases n = 4, 5 this can be proved using the invariants defined in [1] and [11]. The condition in Definition 6.5(ii) is motivated by considering what happens when A and B are scalar matrices.

We attempt to minimise at p as follows. Let I_{p} be the ideal in 𝔽_{p}[x_{1}, …, x_{n}] generated by the reductions of q_{1}, …, q_{N} mod p. We compute the 𝔽_{p}-vector space V_{p} of linear forms in the radical of I_{p}. By a unimodular change of co-ordinates we may suppose that V_{p} = ⟨x_{1}, …, x_{d}⟩ for some 0 ≤ d < n. Then we put

B = (\begin{matrix}pI_{d} & 0 \\ 0 & I_{n−d} \end{matrix})

and compute the index p^{m} of the lattice spanned by the q_{i} ∘ B in its saturation in ℤ[x_{1}, …, x_{n}]. We call the integer m − (n − 3)d the gain. If the gain is positive then we switch to the new quadrics and start over again. Otherwise we stick with the old quadrics. Sometimes it is worth trying other choices for the vector space V_{p}, for example the space of linear forms in one of the minimal primes containing I_{p}. There is no guarantee that these methods will produce a p-minimal model (and in general they do not).

The end result of our attempts at minimisation is a change of coordinates on ℙ^{n−1}. We run the LLL algorithm on the rows (or columns depending on conventions) of the change of basis matrix, before applying it to the original quadrics. This is to ensure that we do not throw away the fact our quadrics are already (close to being) reduced.

7. Numerical Examples

7.1. An example of 6-descent. The following problem falls into the class of problems discussed on pages 480-481 of [9].

Given an integer N > 2, decide whether there are nonzero integers x and y such that both x^{2} + Nxy + y^{2} and x^{2} − Nxy + y^{2} are squares.

Elementary manipulations show that the problem is equivalent to deciding whether the elliptic curve

E_{N} : y^{2} = x(x + (N + 2)^{2})(x + (N − 2)^{2})

has positive rank^{2}. MacLeod and Rathbun [14] have undertaken to find a solution for x and y (where one exists) for all N < 1000. The one case to elude them (as of November 2006) was N = 809, for which the rank is 1 and the generator is predicted^{3} to have height 617.88. On the 2-isogenous curve

E′_{N} : y^{2} = x^{3} + 2(N^{2} + 12N + 4)x^{2} + (N − 2)^{4}x

the predicted height is half this value, yet still beyond the range that can be found using 4-descent. The conductor of E = E′_{809} is sufficiently large that a Heegner point calculation ran into difficulties (and for this reason the curve was reported to Magma as a bug).

We find a point of infinite order on E using 6-descent. The existing Magma functions for 2-descent and 3-descent give us a 2-covering

C_{2} = {y^{2} = 138546x_{1}^{4} + 225978x_{1}^{3}x_{2} + 435649x_{1}^{2}x_{2}^{2} + 3884x_{1}x_{2}^{3} + 183499x_{2}^{4}}

and a 3-covering

C_{3} = \left\{{54x^{3} − 84y^{3} − 258z^{3} + 144x^{2}y + 87x^{2}z − 350xy^{2} \atop + 71y^{2}z − 1656xz^{2} − 986yz^{2} − 388xyz = 0}\right\} .

To compute this 3-covering we had to find the class group and units for a number field of degree 8. This is by far the most time consuming part of the 6-descent calculation, taking a couple of hours, as compared to at most a couple of minutes for each of the other steps.

Following the method described in §6.1 we compute 9 quadrics defining a 6-covering C_{6} ⊂ ℙ(Mat_{2,3}). These are quadrics in 6 variables, labelled x_{ij} for 1 ≤ i ≤ 2 and 1 ≤ j ≤ 3. The coefficients are reasonably small integers, the largest in absolute value being 142.

Minimising (at the primes 2, 3, 809 and 811), as described in §6.3, suggests making the substitution

(\begin{matrix}x_{11} \\ x_{12} \\ x_{13} \\ x_{21} \\ x_{22} \\ x_{23} \end{matrix}) = (\begin{matrix}−70 & 455 & 293 & 700 & −63 & 437 \\ 104 & 417 & −363 & 290 & 745 & −579 \\ −268 & −89 & 205 & −60 & 1223 & 817 \\ −320 & −335 & −839 & 386 & 147 & −311 \\ −318 & 411 & −123 & −696 & −405 & 561 \\ 284 & 59 & −523 & 226 & −15 & 1973 \end{matrix}) (\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \end{matrix}) .

This decreases the size of the coefficients. More importantly, but as we will only see in hindsight, it also reduces the (naive) height of the point we are looking for. The new quadrics are

2The trivial solutions correspond to a subgroup T ⊂ E_{N}(ℚ) with T ≅ ℤ/2ℤ × ℤ/4ℤ. Since ±(N^{2} −4) ∉ (ℚ^{∗})^{2}, the image of T under the 2-descent map E_{N}(ℚ) → ℚ^{∗}/(ℚ^{∗})^{2} × ℚ^{∗}/(ℚ^{∗})^{2} has order 4. It follows that E_{N}(ℚ)_{tors} = T.

3This estimate comes from the Birch–Swinnerton-Dyer conjecture, assuming that the Tate-Shafarevich group is trivial.

q_{1} = x_{1}x_{4} + 2x_{2}x_{3} + 2x_{2}x_{4} − 2x_{2}x_{5} − 2x_{2}x_{6} + 2x_{3}^{2} − 2x_{3}x_{4} − 6x_{3}x_{6} − x_{4}x_{5} + 4x_{4}x_{6}
−2x_{5}^{2} + 5x_{5}x_{6} + x_{6}^{2}
q_{2} = x_{1}x_{2} + x_{1}x_{3} − 2x_{1}x_{4} + 3x_{1}x_{5} − 4x_{1}x_{6} − x_{2}^{2} + 2x_{2}x_{5} − 7x_{2}x_{6} + x_{3}^{2} − x_{3}x_{4} + 4x_{3}x_{5}
−2x_{3}x_{6} + 5x_{4}^{2} − 3x_{4}x_{5} − 8x_{4}x_{6} + x_{5}^{2} + 2x_{5}x_{6} − 6x_{6}^{2}
q_{3} = x_{1}^{2} + x_{1}x_{3} + 3x_{1}x_{4} + 5x_{1}x_{5} − 2x_{1}x_{6} − 3x_{2}^{2} + 5x_{2}x_{3} + x_{2}x_{4} − 4x_{2}x_{5} − x_{2}x_{6} + 2x_{3}^{2}
+4x_{3}x_{4} + 7x_{3}x_{5} − 5x_{3}x_{6} + 3x_{4}^{2} − 5x_{4}x_{6} − 2x_{5}x_{6} − x_{6}^{2}
q_{4} = x_{1}^{2} − 3x_{1}x_{2} + 3x_{1}x_{4} + x_{1}x_{5} + 5x_{1}x_{6} + 7x_{2}x_{3} + 3x_{2}x_{4} − 5x_{2}x_{5} + 3x_{2}x_{6} + 7x_{3}^{2}
−2x_{3}x_{4} − x_{3}x_{5} + 7x_{3}x_{6} − 5x_{4}^{2} + x_{4}x_{5} − x_{5}^{2} − x_{5}x_{6}
q_{5} = 2x_{1}x_{2} − x_{1}x_{3} + x_{1}x_{5} − 7x_{1}x_{6} − x_{2}^{2} − 6x_{2}x_{3} − 2x_{2}x_{4} + x_{2}x_{5} − 8x_{2}x_{6} − 2x_{3}^{2} − 3x_{3}x_{4}
−5x_{3}x_{5} + 2x_{4}^{2} + 3x_{4}x_{6} + x_{5}^{2} − 7x_{5}x_{6} − x_{6}^{2}
q_{6} = x_{1}^{2} + 3x_{1}x_{2} + 5x_{1}x_{3} − 3x_{1}x_{4} − 2x_{1}x_{5} + 2x_{1}x_{6} + 2x_{2}^{2} + 4x_{2}x_{3} + x_{2}x_{5} + 9x_{2}x_{6}
−x_{3}^{2} + 2x_{3}x_{4} + 4x_{3}x_{5} + 9x_{4}^{2} + 6x_{4}x_{5} + 5x_{4}x_{6} − x_{5}^{2} + 4x_{6}^{2}
q_{7} = x_{1}^{2} + 2x_{1}x_{2} + 3x_{1}x_{3} − 3x_{1}x_{4} + x_{1}x_{6} − 6x_{2}^{2} + 3x_{2}x_{3} + 9x_{2}x_{4} + x_{2}x_{5} + 2x_{2}x_{6} + 3x_{3}^{2}
+x_{3}x_{4} + 7x_{3}x_{5} − x_{3}x_{6} − 6x_{4}^{2} − 2x_{4}x_{5} − 6x_{4}x_{6} + 3x_{5}^{2} − x_{5}x_{6} − x_{6}^{2}
q_{8} = x_{1}^{2} − 2x_{1}x_{2} + x_{1}x_{3} − 2x_{1}x_{4} + 3x_{1}x_{5} + 2x_{1}x_{6} − x_{2}x_{3} − x_{2}x_{4} + 3x_{2}x_{5} − 8x_{2}x_{6}
−x_{3}^{2} + 7x_{3}x_{4} + 8x_{3}x_{5} + 12x_{3}x_{6} + 4x_{4}^{2} − 2x_{4}x_{5} + 2x_{4}x_{6} + 6x_{5}x_{6} − 5x_{6}^{2}
q_{9} = x_{1}x_{2} − 6x_{1}x_{3} + 6x_{1}x_{4} − 2x_{1}x_{5} + 6x_{1}x_{6} − x_{2}^{2} + x_{2}x_{3} − 5x_{2}x_{4} − 2x_{2}x_{5} − 2x_{2}x_{6}
−7x_{3}^{2} − 3x_{3}x_{4} + 5x_{3}x_{5} + x_{3}x_{6} − 3x_{4}^{2} + 10x_{4}x_{5} + 5x_{4}x_{6} − 4x_{5}^{2} + 5x_{5}x_{6} + 2x_{6}^{2}.

The PointSearch function (see §6.1 for references) finds a solution

\begin{aligned}(x_{1} : …: x_{6}) & = & (7439932626 : −837815413 : −525136075 : \\ & & 2262805710 : −3465232629 : −1122238333). \end{aligned}

Mapping back to C_{6} this point becomes

(\begin{matrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \end{matrix}) = (\begin{matrix}42664889066 & −110100049465 & −1219600972219 \\ −157741863480 & −574453039469 & 114558712088 \end{matrix})

The 2 × 2 minors are the co-ordinates of a point in C_{3}(ℚ) which then maps down to a point P ∈ E(ℚ) of canonical height 308.94. It is routine to check (using the bounds given in [7]) that P is a generator for the Mordell-Weil group modulo torsion.

The final values of x and y for which both x^{2} + 809xy + y^{2} and x^{2} − 809xy + y^{2} are squares may be found on the website [14]. They have 534 and 537 decimal digits respectively.

7.2. An example of 12-descent. As described in the introduction, Mark Watkins provided me with a list of 35 elliptic curves over ℚ of analytic rank 2 for which only one generator of the Mordell-Weil group was known. The Birch–Swinnerton-Dyer conjecture gives an estimate for the height of the supposed second generator. The curves were ordered by this estimated height. For the first 30 curves on the list, the estimated height was in the range 220 up to 370. The last 5 were as follows. We list the conductor N_{E}, the coefficients a_{1}, …, a_{6} of a minimal Weierstrass equation, and the canonical heights of the generators. (The last column was computed in hindsight.)

\begin{matrix}N_{E} & [a_{1}, a_{2}, a_{3}, a_{4}, a_{6}] & ĥ(P_{1}) & ĥ(P_{2}) \\ 8423178259 & [0, −1, 1, −6286122, −6064183289] & 17.2636 & 442.070 \\ 4817824003 & [0, −1, 1, −91969194, −339447383999] & 15.4617 & 445.878 \\ 4353186907 & [1, 1, 0, −14176508, −20550712585] & 14.4505 & 488.336 \\ 5242805459 & [1, 1, 0, −5078887, −4407675042] & 2.9643 & 527.301 \\ 7800899941 & [0, 0, 1, −237882589, −1412186639384] & 5.3208 & 642.626 \end{matrix}

Each curve on the list is the only curve in its isogeny class, and so in particular has trivial torsion subgroup. According to Magma we have S^{(3)}(E/ℚ) ≅ (ℤ/3ℤ)^{2} and S^{(4)}(E/ℚ) ≅ (ℤ/4ℤ)^{2}. Magma also returns 4 ternary cubics and 6 pairs of quadrics, representing the inverse pairs of elements of order n in S^{(n)}(E/ℚ) for n = 3, 4. Following the method described in §6.1 we compute 48 different 12-coverings, each corresponding to an inverse pair of elements of order 12 in S^{(12)}(E/ℚ) ≅ (ℤ/12ℤ)^{2}. Since E(ℚ) has one generator of small height, we only needed to search on a few of these before a second independent generator was found.

We give brief details for the last curve on the list. In this case the first generator is

P_{1} = (−2003564/15^{2}, −1691/15^{3}).

To compute a second independent generator we start with a 3-covering

C_{3} = \left\{{13x^{3} − 87y^{3} − 142z^{3} + 17x^{2}y + 28x^{2}z + 77xy^{2} \atop − 187y^{2}z − 35xz^{2} − 30yz^{2} − 118xyz = 0}\right\}

and a 4-covering

C_{4} = \{ \begin{matrix}x_{1}^{2} + 3x_{1}x_{2} + 13x_{1}x_{4} − 2x_{2}^{2} − 2x_{2}x_{3} \\ + x_{2}x_{4} − 6x_{3}^{2} − 9x_{3}x_{4} + 7x_{4}^{2} & = & 0 \\ 3x_{1}^{2} − 6x_{1}x_{2} + 5x_{1}x_{3} − 14x_{1}x_{4} + 8x_{2}^{2} \\ − 7x_{2}x_{3} − 2x_{2}x_{4} + 5x_{3}^{2} + 6x_{4}^{2} & = & 0 \end{matrix} \} .

Following the method described in §6.1 we compute 54 quadrics in 12 variables defining a 12-covering C_{12} ⊂ ℙ(Mat_{3,4}). The coefficients are integers of absolute value at most 40. After minimising at the unique bad prime of E, the largest absolute value was 7. On this modified curve, PointSearch (see §6.1 for references) found a solution

\begin{matrix}(−38935814 : 66676907 : 35419393 : −17989378 : 14587909 : −9597188 : \\ −41856515 : −6994528 : −103052506 : 12269644 : 11697462 : 25846956) \end{matrix}

Mapping back to C_{12} this point becomes

(\begin{matrix}−585852746652 & −134738830676 & 992806781984 & −476555121265 \\ −5994121237 & 8026743882 & −211970353911 & 286395682995 \\ 303306392932 & −167866472332 & −273061778593 & 215669566507 \end{matrix}) .

The 3 × 3 minors are the co-ordinates of a point in C_{4}(ℚ) which then maps down to a point P = (r/t^{2}, s/t^{3}) ∈ E(ℚ) of canonical height 651.86 where

t = 19114217356093463705777747876066898415631548291608697\

40922807612824612875940389382477232533975065261036903\ 1136244375962645684728831244647511

r = 93385419996781156236208893304670769704360761931620474\

91160376652094516256058095438975234936485365750728672\ 93638862617394747880602761519393543195699455909538302\ 59168129312401737073248837456279406678810951156628252\ 40211217008647003170248465787238475381689553329226658\ 862657964535534165

s = 21189601910515224224247520792578674272370041362778083\

56705954773720391166818153294963600750782215820469113\ 74353930791392149260850703573807892173379962268109766\ 98439592570904852474980215470887488235939468315716611\ 87491555874815362407229178054307290009804071221273367\ 65774805454336495291566121830488793684956520543942634\ 32595140366259647660234205784539280961702449802725098\ 961125300545865563681315860704624955352014647220765212

A second generator of slightly smaller height is P_{2} = P + P_{1} with ĥ(P_{2}) = 642.63. (In hindsight we could find P_{2} directly by starting with different C_{3} and C_{4}.) According to Magma the regulator of the subgroup generated by P_{1} and P_{2} is 3415.49, the non-zero value confirming that these points are independent. Again it is routine to check (using the bounds given in [7]) that P_{1} and P_{2} generate the Mordell-Weil group.

References

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The Magma home page is at http://magma.maths.usyd.edu.au/magma/

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[22] M. Watkins, Searching for points p-adically, notes available from

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[24] T. Womack, Explicit descent on elliptic curves, PhD thesis, University of Nottingham, 2003.

[25] Ju. G. Zarhin, Noncommutative cohomology and Mumford groups, Math. Notes 15 (1974), 241–244.

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

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