Some improvements to 4-descent on an elliptic curve

Tom Fisher

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

                        T.A.Fisher@dpmms.cam.ac.uk
                   http://www.dpmms.cam.ac.uk/˜ taf1000

Abstract. The theory of 4-descent on elliptic curves has been developed in the PhD theses of Siksek [18], Womack [21] and Stamminger [20]. Prompted by our use of 4-descent in the search for generators of large height on elliptic curves of rank at least 2, we explain how to cut down the number of class group and unit group calculations required, by using the group law on the 4-Selmer group.

1 Introduction

Let E be an elliptic curve over a number field K. A 2-descent (see e.g. [2], [4], [19]) furnishes us with a list of quartics g(X) ∈ K[X] representing the everywhere locally soluble 2-coverings of E, and hence the elements of the 2-Selmer group S^{(2)}(E/K). If we are unable to resolve the existence of K-rational points on the curves Y^{2} = g(X), then it may be necessary to perform a 4-descent. Cassels [3] has constructed a pairing on S^{(2)}(E/K) whose kernel is the image of [2]_{∗} in the exact sequence

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

We have checked [11] that this pairing agrees with the usual Cassels-Tate pairing on �(E/K)[2]. An improved method for computing the pairing has recently been found by Steve Donnelly [7].

Computing this pairing is sufficient to determine the structure of S^{(4)}(E/K) as an abelian group, but if our aim is to find generators of E(K) of large height, then we also need to find equations for the 4-coverings parametrised by this group. For this we use the theory of 4-descent, as developed in [14], [21] and [20]. Each quartic g(X) has an associated flex algebra^{1} F = K[X]/(g(X)), which is usually a degree 4 field extension of K. The existing methods of 4-descent (as implemented in Magma [13] by Tom Womack, and improved by Mark Watkins) require us to compute the class group and units for the flex field of every quartic in the image of [2]_{∗}. In this article we explain how to cut down the number of class

1 We keep the terminology of [6, Paper 1]. Were we to use a term specific to 2-descent then “ramification algebra” would seem more appropriate.

group and unit group calculations, by using the group law on S^{(4)}(E/K). This is a non-trivial task since by properties of the obstruction map [6], [15], we expect to have to solve an explicit form of the local-to-global principle for the Brauer group Br(K). We also give a test for equivalence of 4-coverings (generalising the tests for 2-coverings and 3-coverings given in [4], [5] and [8]).

Even when the calculation of class groups and unit groups does finish, the output may be unmanageably large. We get round this by using a method described in §2, to find good representatives for elements of K^{×}/(K^{×})^{n}. This technique is not specific to descent calculations on elliptic curves.

2 Selmer groups of number fields

Let K be a number field of degree [K : ℚ] = d and let S be a finite set of primes of K. The n-Selmer group

K(S, n) = {x(K^{×})^{n} ∈ K^{×}/(K^{×})^{n} : ord_{𝔭}(x) ≡ 0 (mod n) for all 𝔭 ∉ S}

plays an important role in the construction of number fields via Kummer theory, and in the theory of descent on elliptic curves.

The height of an algebraic integer x in K is H(x) = ∏_{i=1}^{d} max(|σ_{i}(x)|, 1) where σ_{1}, …, σ_{d} are the distinct embeddings of K into ℂ. We write r_{1} (resp. r_{2}) for the number of real (resp. complex) places, and Δ_{K} for the absolute discriminant. The Minkowski bound is

m_{K} = (\frac{4}{π} )^{r_{2}} \frac{d!}{d^{d}} \sqrt{|Δ_{K}|} .

Theorem 2.1. Let n ≥ 1 be an integer. Let α ∈ K^{×} with (α) = 𝔟𝔠^{n} and 𝔟 an integral ideal. Then there exists β ∈ 𝔟 with αβ^{−1} ∈ (K^{×})^{n} and

H(β) ≤ max(m_{K}^{n}N𝔟, exp(nd)) .

The proof uses two lemmas.

Lemma 2.2. If a_{1}, …, a_{d} are positive real numbers with ∑_{i=1}^{d} a_{i} ≤ dc^{1/d} then

∏_{i=1}^{d} max(a_{i}, 1) ≤ max(c, exp(d)) .

Proof. We may assume that a_{i} ≥ 1 for 1 ≤ i ≤ r and a_{i} < 1 for r + 1 ≤ i ≤ d. By the inequality of the arithmetic and geometric means we obtain

∏_{i=1}^{d} max(a_{i}, 1) = ∏_{i=1}^{r} a_{i} ≤ f(r/d)

where f(x) = x^{−dx}c^{x}. If log(c) ≥ d then f′(x) ≥ 0 for all 0 < x ≤ 1. Thus f(r/d) ≤ f(1) = c. On the other hand if log(c) ≤ d we obtain

log f(x) ≤ dx(1 − log x) ≤ d .
⊓ ⊔

We extend the embeddings σ_{i} : K → ℂ to maps defined on K ⊗_{ℚ} ℝ.

Lemma 2.3. Let Λ be a lattice in K ⊗_{ℚ} ℝ of covolume V. Then there exists non-zero ξ ∈ Λ with

∑_{i=1}^{d} |σ_{i}(ξ)| ≤ ((\frac{4}{π} )^{r_{2}} d! V )^{1/d} .

Proof. This is a standard application of Minkowski’s convex body theorem. ⊓ ⊔

The usual application of Lemma 2.3 is to show that every fractional ideal 𝔟 in K contains an element β with |N_{K/ℚ}(β)| ≤ m_{K}N𝔟.

Proof of Theorem 2.1. Let | · | be the map on K ⊗_{ℚ} ℝ ≅ ℝ^{r_{1}} ⊕ ℂ^{r_{2}} given componentwise by x ↦ |x|. We apply Lemma 2.3 to the lattice Λ = |α|^{1/n}𝔠^{−1} and let β = \frac{α}{|α|}ξ^{n}. The covolume of Λ is

|N_{K/ℚ}(α)|^{1/n}(N𝔠)^{−1}\sqrt{|Δ_{K}|} = (N𝔟)^{1/n}\sqrt{|Δ_{K}|} .

Thus β satisfies

∑_{i=1}^{d} |σ_{i}(β)|^{1/n} ≤ d (m_{K}(N𝔟)^{1/n})^{1/d} .

Since β ∈ 𝔟 is an algebraic integer, we deduce by Lemma 2.2 that

H(β)^{1/n} ≤ max(m_{K}(N𝔟)^{1/n}, exp(d))

as required. ⊓ ⊔

Theorem 2.1 shows that every element of K(S, n) is represented by an element of K of height at most

max (m_{K}^{n}( ∏_{𝔭∈S} N𝔭)^{n−1}, exp(nd)) .  (2)

Since there are only finitely many elements of K of height less than a given bound, this gives a new proof that K(S, n) is finite. More importantly for us, replacing Minkowski’s convex body theorem by the LLL algorithm, we obtain an algorithm for computing small representatives of Selmer group elements from large ones. This is particularly useful when using Magma’s function pSelmerGroup (so n = p a prime here) which returns a list of “small” elements of K^{×}, and a list of exponents to which they must be multiplied to give generators for K(S, p). In many examples of interest to us, multiplying out directly in K^{×} gives elements of unfeasibly large height. Using our algorithm (after every few multiplications) eliminates this problem. Moreover, the process can be arranged so that the only factorisations required are of the original list of “small” elements.

In principle one could also compute K(S, n) by searching up to the bound (2), but of course this would be absurdly slow in practice.

3 Background on quadric intersections

Let QI(K) be the space of “quadric intersections” i.e. pairs of homogeneous polynomials of degree 2 in K[x_{1}, x_{2}, x_{3}, x_{4}]. Given (A, B) ∈ QI(K) we identify A and B with their matrices of second partial derivatives, and compute

g(X) = det(AX + B) = aX^{4} + bX^{3} + cX^{2} + dX + e .

The invariants of the quartic g(X) are I = 12ae − 3bd + c^{2} and J = 72ace − 27ad^{2} − 27b^{2}e + 9bcd − 2c^{3}, and the invariants of (A, B) are c_{4} = I and c_{6} = \frac{1}{2}J. It is well known (see [1]) that if Δ = (c_{4}^{3} − c_{6}^{2})/1728 is non-zero then the curves C_{2} = {Y^{2} = g(X)} and C_{4} = {A = B = 0} ⊂ ℙ^{3} are smooth curves of genus one with Jacobian

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

Moreover C_{4} is a 2-covering of C_{2} (see [1], [14]) the composite C_{4} → C_{2} →^{X} ℙ^{1} being given by −T_{1}/T_{2} where T_{1} and T_{2} are the quadrics determined by

adj((adjA)X + (adjB)) = a^{2}AX^{3} + aT_{1}X^{2} + eT_{2}X + e^{2}B.

Following [5], we say that quartics g_{1}, g_{2} ∈ K[X] are K-equivalent if their homogenisations satisfy g_{1} = μ^{2}g_{2} ∘ M for some μ ∈ K^{×} and M ∈ GL_{2}(K). Quadric intersections (A, B), (A′, B′) ∈ QI(K) are K-equivalent if

(A′, B′) = (m_{11}A ∘ N + m_{12}B ∘ N, m_{21}A ∘ N + m_{22}B ∘ N)

for some (M, N) ∈ G_{4}(K) := GL_{2}(K) × GL_{4}(K). It is routine to check that the quartics associated to equivalent quadric intersections are themselves equivalent.

In the course of a 4-descent, a 2-covering C_{4} of C_{2} is computed as follows. Let C_{2} have equation Y^{2} = g(X) and flex algebra F = K[θ] = K[X]/(g(X)). Suppose we are given ξ ∈ F^{×} with N_{F/K}(ξ) ≡ a mod (K^{×})^{2} where a is the leading coefficient of g. (The existence of such a ξ is clearly necessary for the existence of K-rational points on C_{2}.) We consider the equation

X − θ = ξ(x_{1} + x_{2}θ + x_{3}θ^{2} + x_{4}θ^{3})^{2} .

A quadric intersection, defining a 2-covering C_{4} of C_{2}, is obtained by expanding in powers of θ and taking the coefficients of θ^{2} and θ^{3}. The answer only depends (up to K-equivalence) on the class of ξ in F^{×}/K^{×}(F^{×})^{2}. Using the method of §2 to find a good representative for this class, can significantly decrease the time subsequently taken to find a good choice of co-ordinates on ℙ^{3}, that is, to minimise and reduce the quadric intersection (using the algorithms in [21]).

4 Galois cohomology

We keep the notation and conventions of [6, Paper I]. Let π : C → E be the 2-covering corresponding to ξ ∈ H^{1}(K, E[2]). The flex algebra of ξ is F = Map_{K}(Φ, \overline{K}) where Φ is the fibre of π above 0_{E}. We note that C is a torsor under E, and Φ is a torsor under E[2]. Let ⟨ξ⟩ be the subgroup of H^{1}(K, E[2]) generated by ξ, and let ∪ be the map H^{1}(K, E[2]) × H^{1}(K, E[2]) → Br(K)[2] induced by cup product and the Weil pairing e_{2} : E[2]×E[2] → μ_{2}. The following theorem is a variant of a standard result (see for example [17], [20]).

Theorem 4.1. There is a canonical isomorphism

ker (\frac{H^{1}(K, E[2])}{⟨ξ⟩} ⟶^{∪ξ} Br(K)) ≅ ker (F^{×}/K^{×}(F^{×})^{2} ⟶^{N_{F/K}} K^{×}/(K^{×})^{2}) .

Proof. Let \overline{F} = F ⊗_{K} \overline{K}. We may identify \overline{F} = Map(Φ, \overline{K}) and μ_{2}(\overline{F}) = Map(Φ, μ_{2}). These are identifications as Galois modules, the action of Galois being given by σ(f) = (P ↦ σ(f(σ^{−1}P))). An easy generalisation of Hilbert’s theorem 90 shows that H^{1}(K, \overline{F}^{×}) = 0 and hence H^{1}(K, μ_{2}(\overline{F})) = F^{×}/(F^{×})^{2}. We define N : Map(Φ, μ_{2}) → μ_{2} by N(f) = ∏_{P ∈Φ} f(P). The constant maps give an inclusion μ_{2} → Map(Φ, μ_{2}) with quotient X (say). We thus have short exact sequences of Galois modules

0 ⟶ μ_{2} ⟶ Map(Φ, μ_{2}) ⟶^{q} X ⟶ 0

and

0 ⟶ E[2] ⟶^{w} X ⟶^{N} μ_{2} ⟶ 0

where w(T) is the class of P ↦ e_{2}(P − P_{0}, T), for any fixed choice of P_{0} ∈ Φ. Taking the long exact sequences of Galois cohomology we obtain a diagram

\xymatrix{ &  & K^{×}/(K^{×})^{2} \ar[d] \\  &  & F^{×}/(F^{×})^{2} \ar[d]_{q_{∗}} \ar[dr]^{N_{F/K}} \\ μ_{2} \ar[r]^{ξ} & H^{1}(K, E[2]) \ar[dr]_{∪ξ} \ar[r]^{w_{∗}} & H^{1}(K, X) \ar[d]^{Δ} \ar[r]^{N_{∗}} & K^{×}/(K^{×})^{2} \\  &  & Br(K)[2] .}

Once we have shown that the diagram commutes, the theorem follows by a routine diagram chase.

We check that the lower left triangle commutes. Let η ∈ Z^{1}(K, E[2]) be a cocycle. Then w_{∗}(η)_{σ} is the map P ↦ e_{2}(P − P_{0}, η_{σ}). Applying the connecting map Δ gives a ∈ Z^{2}(K, μ_{2}) with

a_{στ} = e_{2}(P − σ(P_{0}), σ(η_{τ})) e_{2}(P − P_{0}, η_{σ}) e_{2}(P − P_{0}, η_{στ})^{−1}
= e_{2}(P_{0} − σ(P_{0}), σ(η_{τ})) e_{2}(P − P_{0}, σ(η_{τ}) + η_{σ} − η_{στ})
= e_{2}(ξ_{σ}, σ(η_{τ})) .

This is the cup product of ξ and η. The commutativity of the upper right triangle is clear. ⊓ ⊔

The case ξ = 0 of Theorem 4.1 is well-known. In this case F is the étale algebra K × L of E[2] where E : Y^{2} = f(X) and L = K[X]/(f(X)).

Corollary 4.2. There is a canonical isomorphism

H^{1}(K, E[2]) ≅ ker (L^{×}/(L^{×})^{2} ⟶^{N_{L/K}} K^{×}/(K^{×})^{2}) .

The following theorem, due to Steve Donnelly, gives an explicit description of the isomorphism of Theorem 4.1 (in one direction). We make the identification of Corollary 4.2 so that now ξ is represented by some α ∈ L^{×}. Let LF be the tensor product L ⊗_{K} F and let L[\sqrt{α}] be the algebra L[X]/(X^{2} − α). By the formulae in [4, §3] there is a natural inclusion L[\sqrt{α}] ⊂ LF. (If Gal(F/K) ≅ S_{4} then L is the resolvent cubic field, LF is the usual composite of fields, and we are quoting that α is a square in LF.) Let τ be the non-trivial automorphism of L[\sqrt{α}] that fixes L.

Theorem 4.3. Let δ ∈ F^{×} with N_{F/K}(δ) = k^{2} for some k ∈ K. Suppose we are given ν ∈ L[\sqrt{α}]^{×} with N_{LF/L[\sqrt{α}]}(δ)/k = τ(ν)/ν. Then

β := N_{LF/L[\sqrt{α}]}(δ)ν^{2} = kN_{L[\sqrt{α}]/L}(ν) ∈ L^{×}  (4)

represents an element of ker (L^{×}/(L^{×})^{2} ⟶^{N_{L/K}} K^{×}/(K^{×})^{2}) mapping to δ under the isomorphisms of Theorem 4.1 and Corollary 4.2.

Proof. We identify

LF = L ⊗_{K} F = Map_{K}((E[2] \ \{0\}) × Φ, \overline{K}) .

Then N_{LF/L[\sqrt{α}]}(δ) is the map (T, P) ↦ δ(P)δ(T + P). So fixing a base point P_{0} ∈ Φ we can rewrite the first equality in (4) as

β(P − P_{0}) = δ(P)δ(P_{0})ν(P − P_{0}, P)^{2}  (5)

for all P ∈ Φ with P ≠ P_{0}.

The image of β in H^{1}(K, X) is represented by a cocycle (ψ_{σ}) where

ψ_{σ}(P) = \{ \begin{matrix}\frac{σ\sqrt{β}}{\sqrt{β}} (P − P_{0}) & if P ≠ P_{0} \\ 1 & if P = P_{0} . \end{matrix}

It follows by (5) that

ψ_{σ}(P) = \frac{σ\sqrt{δ}}{\sqrt{δ}} (P)\frac{σ\sqrt{δ}}{\sqrt{δ}} (P_{0})

for all P ∈ Φ. (The case P = P_{0} is just 1 = (±1)^{2}.) By the definition of X we may ignore the term involving P_{0}, and so (ψ_{σ}) also represents the image of δ in H^{1}(K, X). ⊓ ⊔ Remark 4.4. If ε = N_{LF/L[\sqrt{α}]}(δ)/k then N_{L[\sqrt{α}]/L}(ε) = 1. So by Hilbert’s theorem 90 there exists ν ∈ L[\sqrt{α}]^{×} with ε = τ(ν)/ν. The construction of Theorem 4.3 therefore gives a well-defined map

ker (F^{×}/K^{×}(F^{×})^{2} ⟶^{N_{F/K}} K^{×}/(K^{×})^{2}) → L^{×}/{1, α}(L^{×})^{2} .

The ambiguity up to multiplication by α is predicted by Theorem 4.1, and in this construction comes from the arbitrary choice of sign for k.

5 Testing equivalence of 4-coverings

Let g(X) ∈ K[X] be a (non-singular) quartic with flex algebra F = K[θ] = K[X]/(g(X)). We put QI(K)^{det=g} = {(A, B) ∈ QI(K) : det(AX+B) = g(X)}. If (A, B) ∈ QI(K)^{det=g} then keeping the notation of §3 we define

Q = θ^{−1}eA + T_{1} + θT_{2} + θ^{2}aB  (6)

with suitable modifications if ae = 0. (For example if e = 0 then the “θ = 0 component” of Q is −dA + T_{1}.) Then Q is a rank 1 quadratic form, i.e. Q = ξℓ^{2} for some ξ ∈ F^{×} and ℓ ∈ F [x_{1}, x_{2}, x_{3}, x_{4}] a linear form. This defines a map

λ : QI(K)^{det=g} ⟶ F^{×}/(F^{×})^{2} ; (A, B) ↦ ξ

inverse to the construction of §3.

Lemma 5.1. Quadric intersections in QI(K)^{det=g} define isomorphic coverings of C_{2} = {Y^{2} = g(X)} if and only if they are related by a transformation (μI_{2}, N) ∈ G_{4}(K) with μ^{2} det(N) = 1.

Proof. If π : C_{4} → C_{2} is the 2-covering defined by (A, B) ∈ QI(K)^{det=g} and P_{0} ∈ C_{2} is a ramification point of C_{2} → ℙ^{1} then the divisor π^{∗}(P_{0}) is a hyperplane section of C_{4} (in fact cut out by the linear form ℓ). So if a pair of quadric intersections determine isomorphic 2-coverings of C_{2}, then they must be K-equivalent. Moreover, the equivalence (M, N) ∈ G_{4}(K) is of the form described since, by definition of a 2-covering, the induced self-equivalence of g must be trivial as an automorphism of C_{2}. ⊓ ⊔

If (A_{0}, B_{0}) ∈ QI(K)^{det=g} defines C_{4} ⊂ ℙ^{3} then the 2-coverings of C_{2} are parametrised as twists of C_{4} → C_{2} by H^{1}(K, E[2]). This defines a map

ϕ_{0} : \frac{QI(K)^{det=g}}{\{(μI_{2}, N) ∈ G_{4}(K) : μ^{2} det N = 1\}} ⟶ H^{1}(K, E[2]) .

We find that quotienting out by the transformations with μ^{2} det N = −1 corresponds to quotienting out by ⟨ξ_{2}⟩ where ξ_{2} ∈ H^{1}(K, E[2]) is the class of g.

Theorem 5.2. The following diagram is commutative.

[figure: \frac{QI(K)^{det=g}}{\{(μI_{2},N)∈G_{4}(K) : μ^{2} det N=±1\}}; ϕ_{0}; λ; F^{×}/K^{×}(F^{×})^{2}; ·λ(A_{0},B_{0}); F^{×}/K^{×}(F^{×})^{2}; q_{∗}; \frac{H^{1}(K,E[2])}{⟨ξ_{2}⟩}; w_{∗}; H^{1}(K, X)]

Proof. This is a variant of [20, Theorem 6.1.4]. Let Q_{0} = ξ_{0}ℓ_{0}^{2} and Q_{1} = ξ_{1}ℓ_{1}^{2} be the rank 1 quadratic forms determined by (A_{0}, B_{0}) and (A, B). If (μI_{2}, N) ∈ G_{4}(\overline{K}) relates (A, B) and (A_{0}, B_{0}) then by properties of the Weil pairing

w_{∗}(ϕ_{0}(A, B)) = (σ ↦ \frac{ℓ_{0} ∘ σ(N)N^{−1}}{ℓ_{0}} = \frac{σ(ℓ_{0} ∘ N)}{ℓ_{0} ∘ N}) .

Since Q_{1} = μ Q_{0} ∘ N, this works out as q_{∗}(ξ_{0}ξ_{1}). ⊓ ⊔

The maps ϕ_{0} and w_{∗} of Theorem 5.2 are injective. It follows that λ is injective. So to test whether a pair of quadric intersections (A_{1}, B_{1}), (A_{2}, B_{2}) ∈ QI(K) are equivalent we proceed as follows. We have implemented this test in the case K = ℚ and contributed it to Magma [13]. Step 1. Let g_{i}(X) = det(A_{i}X + B_{i}) for i = 1, 2. We test whether g_{1} and g_{2} are equivalent, using one of the tests in [4], [5]. We are now reduced to the case g_{1} = g_{2}. (If there is more than one equivalence between g_{1} and g_{2} then we must repeat the remaining steps for each of these.) Step 2. Compute ξ_{i} = λ(A_{i}, B_{i}) for i = 1, 2 by evaluating the quadratic form (6) at points in ℙ^{3}(K). It helps with Step 3 if we use several points in ℙ^{3}(K) to give several representatives for the class of ξ_{i} in F^{×}/(F^{×})^{2}. (Spurious prime factors can then be removed from consideration by computing gcd’s.) Step 3. Let S be a finite set of primes of K, including all primes that ramify in F. We enlarge S so that ξ_{1}, ξ_{2} ∈ F (S′, 2) where S′ is the set of primes of F above S. Step 4. The quadric intersections are equivalent if and only if ξ_{1}ξ_{2}^{−1} is in the image of the natural map K(S, 2) → F (S′, 2). We cut down the subgroup of K(S, 2) to be considered by reducing modulo some random primes, and then loop over all possibilities.

In the case that (A_{1}, B_{1}) and (A_{2}, B_{2}) are equivalent, we can reduce to the case Q_{1} = ξ_{1}ℓ_{1}^{2} and Q_{2} = ξ_{2}ℓ_{2}^{2} with ξ_{1}ξ_{2}^{−1} ∈ K. Then solving ℓ_{1} ∘ N = ℓ_{2} for N ∈ Mat_{4}(K), gives the change of co-ordinates relating the two quadric intersections. This transformation is also returned by our Magma function.

6 Adding 2-Selmer and 4-Selmer elements

In §8 we describe a general method for adding 4-Selmer group elements. This involves solving an explicit form of the local-to-global principle for Br(K). But in the special case where we add 2-Selmer and 4-Selmer elements, no such problem need be solved. This is essentially because (by a theorem of Zarhin [22] relating the cup product in Theorem 4.1 to the obstruction map in [6, Paper I], [15]) we have already solved all the conics we need when doing the original 2-descent. To make this explicit we have found the following partial description of the isomorphism of Theorem 4.1.

Let g(X) ∈ K[X] be a quartic with invariants I and J. Let L = K[φ] where φ is a root of f(X) = X^{3} − 3IX + J. We assume that the discriminant Δ_{0} = 27(4I^{3} − J^{2}) is non-zero. Formulae in [4], [5] allow us to represent g by α = a_{0} + a_{1}φ ∈ L^{×} with a_{0}, a_{1} ∈ K and N_{L/K}(α) ∈ (K^{×})^{2}. We assume α ∉ (L^{×})^{2}. As in §4 we put F = K[X]/(g(X)) and LF = L ⊗_{K} F.

Theorem 6.1. If β, γ ∈ L^{×} are linear in φ with N_{L/K}(β), N_{L/K}(γ) ∈ (K^{×})^{2} and αβγ ∈ (L^{×})^{2}, then the isomorphisms of Theorem 4.1 and Corollary 4.2 map each of β and γ to the class of

δ := Tr_{LF/F} (\frac{\sqrt{α}}{f′(φ)}) Tr_{LF/F} (\frac{\sqrt{βγ}}{f′(φ)}) ∈ F^{×} .

Proof. Let φ_{1} = φ, φ_{2}, φ_{3} be the K-conjugates of φ, and likewise for α, β, γ, m where αβγ = m^{2}. Using that α, β, γ are linear in φ we compute

N_{LF/L[\sqrt{α}]}(δ) = \frac{(\sqrt{α_{2}} − \sqrt{α_{3}})^{2}(\sqrt{β_{2}γ_{3}} − \sqrt{β_{3}γ_{2}})^{2}}{Δ_{0} (φ_{2} − φ_{3})^{2}} .

The hypotheses of Theorem 4.3 are therefore satisfied with

k = \frac{(α_{2} − α_{3})(β_{2}γ_{3} − β_{3}γ_{2})}{Δ_{0} (φ_{2} − φ_{3})^{2}} = \frac{a_{1}(b_{1}c_{0} − b_{0}c_{1})}{Δ_{0}} ∈ K^{×}

and (swapping β and γ if necessary to avoid dividing by zero)

ν^{−1} = \frac{(\sqrt{α_{2}} − \sqrt{α_{3}})(\sqrt{β_{2}γ_{3}} − \sqrt{β_{3}γ_{2}})}{\sqrt{α_{2}β_{2}γ_{3}} + \sqrt{α_{3}β_{3}γ_{2}}} = 1 − \frac{m_{2}β_{3} + m_{3}β_{2}}{m_{2}γ_{3} + m_{3}γ_{2}} \sqrt{\frac{γ_{2}γ_{3}}{β_{2}β_{3}}} .

We are done since

(\sqrt{α_{2}β_{2}γ_{3}} + \sqrt{α_{3}β_{3}γ_{2}})^{2} = \frac{(m_{2}γ_{3} + m_{3}γ_{2})^{2}}{γ_{2}γ_{3}} ≡ γ mod (L^{×})^{2} .

⊓ ⊔

We give an example in the case K = ℚ. The quartics

\begin{aligned}g_{1}(X) & = & −675X^{4} − 7970X^{3} − 18923X^{2} + 27176X − 7848 \\ g_{2}(X) & = & −5483X^{4} + 10470X^{3} + 8869X^{2} − 13240X − 8768 \\ g_{3}(X) & = & −3728X^{4} − 8536X^{3} + 9037X^{2} + 15940X − 13000 \end{aligned}

have invariants I = 1071426889 and J = 70141299507574. Moreover they sum to zero in S^{(2)}(E/ℚ) where E : −3Y^{2} = f(X) = X^{3} −3IX +J. Let L = ℚ(φ) = ℚ[X]/(f(X)) and F_{1} = ℚ(θ) = ℚ[X]/(g_{1}(X)). We use the existing FourDescent routine in Magma to compute 2-coverings D_{i} of C_{i} = {Y^{2} = g_{i}(X)} for i = 2, 3 and then add these using the method of §8 to give a 2-covering D_{1} of C_{1} = {Y^{2} = g_{1}(X)}. By a formula in [4] the quartics g_{1}, g_{2}, g_{3} are represented by

\begin{aligned}α & = & −900φ + 29459500 \\ β & = & (−21932φ + 717892516)/3 \\ γ & = & (−14912φ + 488109376)/3 \end{aligned}

in L^{×}/(L^{×})^{2}. Theorem 6.1 and the map λ in §5 convert C_{2} and D_{1} to

δ = 26565975θ^{3} + 327644415θ^{2} + 917786936θ − 582546987

and ξ_{1} = 4725θ^{3}+59165θ^{2}+168496θ−106600 in F_{1}^{×}/ℚ^{×}(F_{1}^{×})^{2}. We then multiply δ and ξ_{1} in F_{1}^{×} and recover a new 2-covering D′_{1} of C_{1} by the method of §3. By Theorem 5.2 this new 4-covering of E represents the sum of ι_{∗}(C_{2}) and D_{1} in S^{(4)}(E/ℚ) where ι_{∗} is the map in (1). Notice that at no stage of the computation of D_{1} and D′_{1} did we need to find the class group and units of F_{1}, although it is only for much larger examples that this saving becomes worthwhile.

7 Computing the action of the Jacobian

In this section we generalise the formulae of [8, §7] from 3-coverings to 4coverings. The main new ingredient is a certain generalisation of the Hessian, introduced in [9]. This is an SL_{2}(K) × SL_{4}(K)-equivariant polynomial map H : QI(K) → QI(K). In the notation of §3 it is given by

H : (A, B) ↦ (6T_{2} − cA − 3bB, 6T_{1} − cB − 3dA) .  (7)

The analogue of the Hesse pencil of plane cubics, is the “Hesse family” of quadric intersections

U(a, b) = (a(x_{1}^{2} + x_{3}^{2}) − 2bx_{2}x_{4}, a(x_{2}^{2} + x_{4}^{2}) − 2bx_{1}x_{3})

with invariants

\begin{aligned}c_{4}(a, b) & = & 2^{8}(a^{8} + 14a^{4}b^{4} + b^{8}) \\ c_{6}(a, b) & = & −2^{12}(a^{12} − 33a^{8}b^{4} − 33a^{4}b^{8} + b^{12}) \\ Δ(a, b) & = & 2^{20}a^{4}b^{4}(a^{4} − b^{4})^{4} \end{aligned}

and Hessian U(a′, b′) where a′ = −2^{4}a(a^{4} − 5b^{4}) and b′ = 2^{4}b(5a^{4} − b^{4}).

If U ∈ QI(K) is a non-singular quadric intersection with Jacobian E, then the pencil of quadric intersections spanned by U and its Hessian is a twist of the Hesse family. So there are exactly six singular fibres, and each singular fibre is a “square” (really a quadrilateral spanning ℙ^{3}). Each square is uniquely the intersection of a pair of rank 2 quadrics and the union of these quadrics is the set of fixed planes for the action of M_{T} on ℙ^{3} for some T ∈ E[4] \ E[2]. So there is a Galois equivariant bijection between the syzygetic squares and the cyclic subgroups of E[4] of order 4. (Our terminology generalises that in [12, §II.7].)

Lemma 7.1. Let U be a non-singular quadric intersection with invariants c_{4}, c_{6} and Hessian H. Let T = (x_{T} , y_{T}) be a point of order 4 on the Jacobian (3). Then the syzygetic square corresponding to ±T is defined by S = \frac{1}{3}x_{T} U + H, and this quadric intersection satisfies H(S) = ν_{T}^{2} S where

ν_{T} = (x_{T}^{4} − 54c_{4}x_{T}^{2} − 216c_{6}x_{T} − 243c_{4}^{2})/(18y_{T}) .

Proof. We may assume that U belongs to the Hesse family and that T = (2^{4}3(a^{4} − 5b^{4}), 2^{7}3^{3}i(a^{4} − b^{4})b^{2}). The lemma follows by direct calculation. ⊓ ⊔

Let C ⊂ ℙ^{3} be a genus one normal curve of degree 4, defined over K, and with Jacobian E. Let L/K be any field extension. Given T ∈ E(L) a point of order 4, we aim to construct M_{T} ∈ GL_{4}(L) describing the action of T on C. We start with a quadric intersection U defining C. Then we compute the syzygetic square S = \frac{1}{3}x_{T} U + H as described in Lemma 7.1. Making a change of co-ordinates (defined over K) we may assume

– The point (1 : 0 : 0 : 0) does not lie on either of the rank 2 quadrics whose intersection is the syzygetic square.

– The line {x_{3} = x_{4} = 0} does not meet either diagonal of the square.

Let A and B be the rank 2 quadrics in the pencil spanned by S, scaled so that the coefficient of x_{1}^{2} is 1 in each case. These quadrics are defined over a field L′ with [L′ : L] ≤ 2, and are easily found by factoring the determinant of a generic quadric in the pencil. We factor A and B over \overline{K} as

\begin{aligned}A & = & (x_{1} + α_{1}x_{2} + β_{1}x_{3} + γ_{1}x_{4})(x_{1} + α_{3}x_{2} + β_{3}x_{3} + γ_{3}x_{4}) \\ B & = & (x_{1} + α_{2}x_{2} + β_{2}x_{3} + γ_{2}x_{4})(x_{1} + α_{4}x_{2} + β_{4}x_{3} + γ_{4}x_{4}) . \end{aligned}

Then we put

P = ( \begin{matrix}1 & α_{1} & β_{1} & γ_{1} \\ 1 & α_{2} & β_{2} & γ_{2} \\ 1 & α_{3} & β_{3} & γ_{3} \\ 1 & α_{4} & β_{4} & γ_{4} \end{matrix} )

and ξ = α_{1} − iα_{2} − α_{3} + iα_{4} where i = \sqrt{−1}.

Theorem 7.2. If ξ ≠ 0 then the matrix

M_{T} = ξP^{−1} ( \begin{matrix}1 \\ & i \\ & & −1 \\ & & & −i \end{matrix} ) P

belongs to GL_{4}(L) and describes the action of T (or −T) on C.

Proof. The image of this matrix in PGL_{4} has order 4, and acts on ℙ^{3} with fixed planes defined by the linear factors of A and B. So the second statement is clear. Theorem 7.3 shows that M_{T} has entries in L. (It may also be checked directly that each entry is fixed by Gal(L′(i)/L).) ⊓ ⊔

Any polynomial in the α_{i}, β_{i}, γ_{i} invariant under the action of C_{2} × C_{2} that swaps the subscripts 1 ↔ 3 and 2 ↔ 4 may be rewritten as a polynomial in the coefficients of A and B. We write A = ∑_{i≤j} a_{ij}x_{i}x_{j} and B = ∑_{i≤j} b_{ij}x_{i}x_{j}. Then by computer algebra we find an expression for κ = (α_{1} − α_{3})(α_{2} − α_{4}) det(P) as a polynomial in the a_{ij} and b_{ij}, and likewise for the entries of

M_{1} = (α_{2} − α_{4})adj(P)Diag(1, 0, −1, 0)P

and

M_{2} = (α_{1} − α_{3})adj(P)Diag(0, 1, 0, −1)P .

Let S = (λ_{1}A + μ_{1}B, λ_{2}A + μ_{2}B) with λ_{i}, μ_{i} ∈ L′. Then κ ∈ L, whereas if A and B are not defined over L then Gal(L′/L) interchanges λ_{1} ↔ λ_{2}, μ_{1} ↔ μ_{2} and M_{1} ↔ M_{2}.

Theorem 7.3. The matrix M_{T} of Theorem 7.2 is given by

M_{T} = \frac{a_{12}^{2} − 4a_{22}}{κ} M_{1} + \frac{b_{12}^{2} − 4b_{22}}{κ} M_{2} ± \frac{λ_{1}μ_{2} − λ_{2}μ_{1}}{ν_{T}} (M_{1} − M_{2})

where ν_{T} = (x_{T}^{4} − 54c_{4}x_{T}^{2} − 216c_{6}x_{T} − 243c_{4}^{2})/(18y_{T}).

Proof. By our choice of co-ordinates we have α_{1} ≠ α_{3} and α_{2} ≠ α_{4}. So κ ∈ L is non-zero. We compute

κM_{T} = ξ(α_{1} − α_{3})(α_{2} − α_{4})adj(P)Diag(1, i, −1, −i)P
= ξ(α_{1} − α_{3})M_{1} + iξ(α_{2} − α_{4})M_{2}
= (α_{1} − α_{3})^{2}M_{1} + (α_{2} − α_{4})^{2}M_{2} − iκ(det P)^{−1}(M_{1} − M_{2}) .

Since H(x_{1}x_{3}, x_{2}x_{4}) = (−x_{1}x_{3}, −x_{2}x_{4}) we have

H(S) = −(λ_{1}μ_{2} − λ_{2}μ_{1})^{2} det(P)^{2}S .

By Lemma 7.1 we deduce ν_{T} = ±i(λ_{1}μ_{2} − λ_{2}μ_{1}) det(P), and substituting this into the above expression for κM_{T} completes the proof of the theorem. ⊓ ⊔

By our choice of co-ordinates it is impossible that both ξ = α_{1}−iα_{2}−α_{3}+iα_{4} and ξ′ = α_{1} + iα_{2} − α_{3} − iα_{4} vanish. So if our formula for M_{T} gives the zero matrix, we can instead use the formula for M_{−T} and take the inverse.

8 Adding 4-Selmer group elements

Finally we outline how the theory in [6] can be used to add elements of S^{(4)}(E/K). (Of course, the method in §6 should be used in preference whenever it applies.)

Let C ⊂ ℙ^{3} be a 4-covering of E. We embed E in ℙ^{3} via (x, y) ↦ (1 : x : y : x^{2}). In [10, §6.2] we gave a practical algorithm for computing B ∈ GL_{4}(\overline{K}) describing a change of co-ordinates on ℙ^{3} taking C to E.

Now let R be the étale algebra of E[4]. Applying the formulae of §7 over each constituent field of R, we compute M, M′ ∈ GL_{4}(R) describing the actions of E[4] on E ⊂ ℙ^{3} and C ⊂ ℙ^{3} respectively. We scale these matrices by using the method of §2 to find good representatives for their determinants in R^{×}/(R^{×})^{4}. These matrices now determine γ ∈ \overline{R}^{×} = Map(E[4], \overline{K}^{×}) by the rule

BM′_{T} B^{−1} = γ(T)M_{T}

for all T ∈ E[4]. It is shown in [6, Paper I] that we may identify H^{1}(K, E[4]) with a certain subquotient of (R ⊗ R)^{×}. Our 4-covering corresponds to ρ ∈ (R ⊗ R)^{×} given by the rule

ρ(S, T) = \frac{γ(S)γ(T)}{γ(S + T)}  (8)

for all S, T ∈ E[4]. So if 4-coverings C_{1} and C_{2} determine γ_{1}, γ_{2} ∈ \overline{R}^{×}, then their sum (by the group law of H^{1}(K, E[4])) corresponds to the product γ_{1}γ_{2}.

It remains to explain how, if C is everywhere locally soluble, we can recover equations for C ⊂ ℙ^{3} from γ ∈ \overline{R}^{×}. Let ε ∈ (R⊗R)^{×} be the element determined by ε(S, T)I_{4} = M_{S}M_{T} M_{S+T}^{−1} for all S, T ∈ E[4], and let ρ be given by (8). We view R ⊗ R as an R-algebra via the comultiplication R → R ⊗ R and write Tr : R ⊗ R → R for the corresponding trace map. In [6, Paper I] we defined the obstruction algebra A_{ρ} = (R, +, ∗_{ερ}) to be the K-vector space R equipped with a new multiplication z_{1} ∗_{ερ} z_{2} = Tr(ερ.(z_{1} ⊗ z_{2})).

In our situation, we already have a trivialisation of A_{ρ} over \overline{K}, namely the isomorphism of \overline{K}-algebras A_{ρ} ⊗_{K} \overline{K} ≅ Mat_{4}(\overline{K}) given by

z ↦ ∑_{T ∈E[4]} z(T)γ(T)M_{T} .

So picking a basis r_{1}, …, r_{16} for R gives matrices M_{1}, …, M_{16} ∈ Mat_{4}(\overline{K}). We then compute structure constants c_{ijk} ∈ K for the obstruction algebra A_{ρ} by the rule M_{i}M_{j} = ∑_{k=1}^{16} c_{ijk}M_{k}.

Our only implementation so far is in the case K = ℚ. In practice we fix an embedding \overline{ℚ} ⊂ ℂ, and so γ is represented by a 16-tuple of complex numbers (to some precision). In [6, Paper III] we will explain how to choose a basis for R so that the structure constants c_{ijk} are (reasonably small) integers. This makes it easy to recognise them from their floating point approximations.

Since C is everywhere locally soluble, it is guaranteed by class field theory that there is an isomorphism of K-algebras A_{ρ} ≅ Mat_{4}(K). We must find such an isomorphism explicitly, and for this we use the method of Pílniková [16], who reduces the problem to that of solving conics over (at most quadratic) extensions of K. Finally any one of the three methods in [6, Paper I, §5] may be used to recover equations for C. In practice we use the Hesse pencil method, which by virtue of the Hessian (7) has a natural generalisation from 3-descent to 4-descent.

Acknowledgements

I would like to thank John Cremona, Michael Stoll and Denis Simon for many useful discussions in connection with this work, and Steve Donnelly for providing me with the construction described in Theorem 4.3 and Remark 4.4. All computer calculations in support of this work were performed using Magma [13].

References

1. S.Y. An, S.Y. Kim, D.C. Marshall, S.H. Marshall, W.G. McCallum and A.R. Perlis, Jacobians of genus one curves, J. Number Theory 90 (2001), no. 2, 304–315.

2. J.W.S. Cassels, Lectures on elliptic curves, LMS Student Texts, 24, CUP, Cambridge, 1991.

3. J.W.S. Cassels, Second descents for elliptic curves, J. reine angew. Math. 494 (1998), 101–127.

4. J.E. Cremona, Classical invariants and 2-descent on elliptic curves, J. Symbolic Comput. 31 (2001), no. 1-2, 71–87.

5. J.E. Cremona and T.A. Fisher, On the equivalence of binary quartics, submitted.

6. J.E. Cremona, T.A. Fisher, C. O’Neil, D. Simon and M. Stoll, Explicit n-descent on elliptic curves, I Algebra, J. reine angew. Math. 615 (2008), II Geometry, submitted, III Algorithms, in preparation.

7. S. Donnelly, Computing the Cassels-Tate pairing, in preparation.

8. T.A. Fisher, Testing equivalence of ternary cubics, Algorithmic number theory, 333– 345, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, 2006.

9. T.A. Fisher, The Hessian of a genus one curve, preprint.

10. T.A. Fisher, Finding rational points on elliptic curves using 6-descent and 12descent, submitted.

11. T.A. Fisher, E.F. Schaefer and M. Stoll, The yoga of the Cassels-Tate pairing, submitted.

12. D. Hilbert, Theory of algebraic invariants, CUP, Cambridge, 1993.

13. MAGMA is described in W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. The

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

14. J.R. Merriman, S. Siksek and N.P. Smart, Explicit 4-descents on an elliptic curve, Acta Arith. 77 (1996), no. 4, 385–404.

15. C. O’Neil, The period-index obstruction for elliptic curves, J. Number Theory 95 (2002), no. 2, 329–339.

16. J. Pílniková, Trivializing a central simple algebra of degree 4 over the rational numbers, J. Symbolic Comput. 42 (2007), no. 6, 579–586.

17. B. Poonen and E.F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. reine angew. Math. 488 (1997), 141–188.

18. S. Siksek, Descent on curves of genus 1, PhD thesis, University of Exeter, 1995.

19. D. Simon, Computing the rank of elliptic curves over number fields, LMS J. Comput. Math. 5 (2002), 7–17 (electronic).

20. S. Stamminger, Explicit 8-descent on elliptic curves, PhD thesis, International University Bremen, 2005.

21. T. Womack, Explicit descent on elliptic curves, PhD thesis, University of Nottingham, 2003.

22. Yu. G. Zarhin, Noncommutative cohomology and Mumford groups, Math. Notes 15 (1974), 241–244.