MINIMISATION OF 2-COVERINGS OF GENUS 2 JACOBIANS

TOM FISHER AND MENGZHEN LIU

Abstract. An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and reduction. We give an algorithm for minimising certain pairs of quadratic forms, subject to the constraint that the first quadratic form is fixed. This has applications to 2-descent on the Jacobian of a genus 2 curve.

1. Introduction

1.1. Models for 2-coverings. We work over a field K with char(K) ≠ 2. Let C be a smooth curve of genus 2 with equation y^{2} = f(x) = f_{6}x^{6}+f_{5}x^{5}+…+f_{1}x+f_{0} where f ∈ K[x] is a polynomial of degree 6. We fix throughout the polynomial

(1)  G = z_{12}z_{34} − z_{13}z_{24} + z_{23}z_{14}.

The following two definitions are based on those in [FY, Section 2.4].

Definition 1.1. A model (for a 2-covering of the Jacobian of C) is a pair (λ, H) where λ ∈ K^{×} and H ∈ K[z_{12}, z_{13}, z_{23}, z_{14}, z_{24}, z_{34}] is a quadratic form satisfying

det(λxG − H) = −λ^{6}f_{6}^{−1}f(x)

where G and H are the matrices of second partial derivatives of G and H.

We identify the space of column vectors of length 6 and the space of 4 × 4 alternating matrices via the map

A : z = (\begin{matrix}z_{12} \\ z_{13} \\ z_{23} \\ z_{14} \\ z_{24} \\ z_{34} \end{matrix}) ↦ (\begin{matrix}0 & z_{12} & z_{13} & z_{14} \\ & 0 & z_{23} & z_{24} \\ & − & 0 & z_{34} \\ & & & 0 \end{matrix})

Date: 14th August 2024.

so that G(z) is the Pfaffian of A(z). Then each 4×4 matrix P uniquely determines a 6 × 6 matrix ∧^{2}P such that

P A(z)P^{T} = A((∧^{2}P)z)

for all column vectors z. For F ∈ K[x_{1}, …, x_{n}] and M ∈ GL_{n}(K) we write F ∘ M for the polynomial satisfying (F ∘ M)(x) = F (Mx) for all column vectors x. The Pfaffian Pf(A) of an alternating matrix A has the properties that Pf(A)^{2} = det(A) and Pf(P AP^{T}) = (det P) Pf(A). The latter tells us that G ∘ ∧^{2}P = (det P)G. It is also not hard to show that det(∧^{2}P) = (det P)^{3}.

Definition 1.2. Two models are K-equivalent if they are in the same orbit for the action of K^{×} × PGL_{4}(K) via

(c, P) : (λ, H) ↦ (cλ, \frac{c}{det P} H ∘ ∧^{2}P ) .

It may be checked using the above observations that this is a well defined (right) group action on the space of models (for a fixed choice of genus 2 curve C).

The following remark summarises part of [FY, Section 2]. It serves to motivate Definitions 1.1 and 1.2, but otherwise is not needed for what follows.

Remark 1.3. Assume for simplicity that f is irreducible and let L = K(θ) where θ is a root of f. If K is a number field then, as explained by Stoll and van Luijk [SvL], the 2-Selmer group of J = Jac C is naturally a subgroup of

H = \frac{\{(ξ, m) ∈ L^{×} × K^{×}|N_{L/K}(ξ) = m^{2}\}}{\{(rν^{2}, r^{3}N_{L/K}(ν))|r ∈ K^{×}, ν ∈ L^{×}\}}.

Starting with a 2-Selmer element (ξ, m), we define quadratic forms Q_{0}, Q_{1} …, Q_{5} ∈ K[u_{0}, u_{1}, …, u_{5}] by

ξ(u_{0} + u_{1}θ + … + u_{5}θ^{5})^{2} = Q_{0} + Q_{1}θ + … + Q_{5}θ^{5}.

Then the matrices of second partial derivatives G and H of the quadratic forms G = Q_{5}/2 and H = (f_{6}Q_{4} − f_{5}Q_{5})/(2f_{6}) satisfy

det(xG − H) = −f_{6}^{−1}m^{2}f(x).

It may further be shown (using the local conditions defining the 2-Selmer group, and a suitable local global principle) that G has a 3-dimensional isotropic subspace. We may therefore put G in the form (1) using a change of basis matrix with determinant m. We apply the same transformation to H. Our 2-Selmer group element (ξ, m) is now represented by the K-equivalence class of the model (1, H). Formulae for going the other way, that is, converting a model (λ, H) back into a pair (ξ, m), are given in [FY, Section 2.5].

We may associate to each point (x_{1} : x_{2} : x_{3} : x_{4}) ∈ ℙ^{3} a 3-dimensional isotropic subspace for G spanned by the rows of the matrix

(2)  Λ = (\begin{matrix}0 & 0 & x_{4} & 0 & −x_{3} & x_{2} \\ 0 & −x_{4} & 0 & x_{3} & 0 & −x_{1} \\ x_{4} & 0 & 0 & −x_{2} & x_{1} & 0 \\ −x_{3} & x_{2} & −x_{1} & 0 & 0 & 0 \end{matrix}) .

Let A = ΛGΛ^{T} and B = ΛHΛ^{T}. By construction A is zero. The GCD of the 3×3 minors of B is a quartic form defining a twisted Kummer surface K_{ξ,m} ⊂ ℙ^{3}. The 2-covering of J corresponding to (ξ, m) is the double cover J_{ξ,m} → K_{ξ,m} given by setting any leading 2 × 2 minor of B equal to −f_{6} times a square.

Remark 1.4. Our fixed quadratic form (1) differs by a sign choice from that in [FY]. The convention there (following [BG, SW]) was that the matrix G should have all 1’s on the antidiagonal, whereas here it is more convenient that G is the Pfaffian. We have switched the sign of the 5th column of (2) accordingly. The change is otherwise not important.

Example 1.5. Let C/ℚ be the genus 2 curve given by y^{2} = f(x) where

f(x) = −28x^{6} + 84x^{5} − 323x^{4} + 506x^{3} − 471x^{2} + 232x − 60.

One of the elements of the 2-Selmer group of Jac C is represented by the model

(λ_{1}, H_{1}) = (42336, 25128z_{12}^{2} + 24480z_{12}z_{13} + 14031z_{12}z_{23} + 15408z_{12}z_{14}
+ 13959z_{12}z_{24} + 25407z_{12}z_{34} + 2232z_{13}^{2} − 16407z_{13}z_{23} + 4464z_{13}z_{14}
− 22815z_{13}z_{24} + 1161z_{13}z_{34} + 2329z_{23}^{2} + 15282z_{23}z_{14} + 7687z_{23}z_{24}
− 19547z_{23}z_{34} − 2304z_{14}^{2} − 17838z_{14}z_{24} − 22590z_{14}z_{34} − 134z_{24}^{2}
+ 41978z_{24}z_{34} − 99584z_{34}^{2}).

Applying the transformation (c, P) with c = 1/3024 and

(3)  P = (\begin{matrix}2 & −19 & 2 & 5 \\ 4 & 4 & −31 & 38 \\ 2 & 2 & 37 & 40 \\ −7 & −7 & −14 & 7 \end{matrix})

gives the ℚ-equivalent model

(λ_{2}, H_{2}) = (14, z_{12}z_{23} + 2z_{12}z_{14} − z_{12}z_{24} + 8z_{12}z_{34} − 7z_{13}^{2} − 13z_{13}z_{23}
− 12z_{13}z_{14} − 15z_{13}z_{24} − 20z_{13}z_{34} − 5z_{23}^{2} − 2z_{23}z_{14} − 25z_{23}z_{24}
− 59z_{23}z_{34} − 4z_{14}^{2} − 14z_{14}z_{24} − 18z_{14}z_{34} + 17z_{24}^{2} − 37z_{24}z_{34} − 11z_{34}^{2}).

Remark 1.6. This is a deliberately small example, chosen to illustrate our definitions and algorithms. As discussed in Remark 1.3, converting a Selmer group element to a model involves solving a local global problem. This means that the models we first compute (analogous to (λ_{1}, H_{1}) in Example 1.5) can have very large coefficients.

1.2. Relation to previous work. The change of coordinates (3) was found by a combination of two techniques, called minimisation and reduction. Minimisation seeks to remove prime factors from a suitably defined invariant (usually the discriminant). The prototype example is using Tate’s algorithm to compute a minimal Weierstrass equation for an elliptic curve. Reduction seeks to a make a final unimodular substitution so that the coefficients are as small as possible. The prototype example is the reduction algorithm for positive definite binary quadratic forms.

Algorithms for minimising and reducing 2-, 3-, 4- and 5-coverings of elliptic curves are given by Cremona, Fisher and Stoll [CFS], and Fisher [F1], building on earlier work of Birch and Swinnerton-Dyer [BSD] for 2-coverings. Algorithms for minimising some other representations associated to genus 1 curves are given by Fisher and Radičević [FR]. A general framework for minimising hypersurfaces is described by Kollár [K], and this has been refined by Elsenhans and Stoll [ES]; in particular they give practical algorithms for plane curves (of arbitrary degree) and for cubic surfaces. Algorithms for minimising Weierstrass equations for general hyperelliptic curves are given by Q. Liu [L].

In this paper we give an algorithm for minimising 2-coverings of genus 2 Jacobians. These are represented by pairs of quadratic forms (see Definition 1.1) where the first quadratic form is fixed. We only consider minimisation and not reduction, since the latter is already treated in [FY, Remark 4.3].

Our minimisation algorithm plays a key role in the work of the first author and Jiali Yan [FY] on computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian. Indeed the method presented in loc. cit. for computing the Cassels-Tate pairing relies on being able to find rational points on the corresponding twisted Kummer surfaces (see Remark 1.3). Minimising and reducing our models for the 2-Selmer group elements simplifies the equations for these surfaces, and so makes it more likely that we will be able to find such rational points.

Earlier works on minimisation (see in particular [CFS]) considered both minimisation theorems (i.e., general bounds on the minimal discriminant) and minimisation algorithms (i.e., practical methods for finding a minimal model equivalent to a given one). For 2-coverings of hyperelliptic Jacobians, some minimisation theorems have already been proved; see the papers of Bhargava and Gross [BG, Section 8], and Shankar and Wang [SW, Section 2.4]. We will not revisit these results, as our focus is on the minimisation algorithms.

Remark 1.7. As noted in [CF, Lemma 17.1.1], [FH, Section 19.1] and [FY, Section 2.4] the quadratic form G = z_{12}z_{34} − z_{13}z_{24} + z_{23}z_{14} has two algebraic families of 3-dimensional isotropic subspaces. (One of these is specified in (2).) Moreover, the transformations considered in Definition 1.2 do not describe the full projective orthogonal group of G, but only the index 2 subgroup that preserves (rather than swaps over) these two algebraic families. Restricting attention to this index 2 subgroup (when defining equivalence) makes no difference to the minimisation problem (see Remark 3.2), but is important in the context of 2-descent, since it means we can distinguish between the pairs (ξ, m) and (ξ, −m) in Remark 1.3.

Data availability statement. Some Magma [BCP] code accompanying this article, including an implementation of our algorithm, is available from [F2].

Acknowledgements. This work originated as a summer project carried out by the second author and supervised by the first author. We thank the Research in the CMS Programme for their support.

Conflict of interest statement. On behalf of all authors, the corresponding author states that there are no conflicts of interest.

2. Statement of the algorithm

We keep the notation of Section 1.1, but now let K be a field with discrete valuation v : K^{×} → ℤ, valuation ring O_{K}, uniformiser π (i.e., v(π) = 1), and residue field k. If F is a polynomial with coefficients in K then we write v(F) for the minimum of the valuations of its coefficients.

Definition 2.1. A model (λ, H) is integral if v(H) ⩾ 0. It is minimal if v(λ) is minimal among all K-equivalent integral models.

Using the action of K^{×} (see Definition 1.2) to clear denominators it is clear that any model is K-equivalent to an integral model. By Definition 1.1 we have v(λ) ⩾ (v(f_{6}) − v(f_{i}))/(6 − i) for all i = 0, 1, …, 5. We cannot have f_{0} = … = f_{5} = 0 since C is a smooth curve of genus 2. Therefore v(λ) is bounded below, and minimal models exist.

It also follows from Definition 1.1 that if v(f_{6}) = v(disc f) = 0 then any integral model (λ, H) has v(λ) ⩾ 0. Therefore, in global applications, minimality is automatic at all but a finite set of primes, which we may determine by factoring.

Returning to the local situation, there is an evident recursive algorithm for computing minimal models if we can solve the following problem.

Minimisation problem. Given an integral quadratic form H ∈ O_{K}[z_{12}, …, z_{34}] determine whether there exists P ∈ PGL_{4}(K) such that

v (\frac{1}{det P} H ∘ ∧^{2}P ) > 0

and find such a matrix P if it exists.

Our solution to this problem (see Algorithm 2.4) is an iterative procedure that computes the required transformation as a composition of simpler transformations. These simpler transformations are either given by a matrix in GL_{4}(O_{K}), in which case we call the transformation an integral change of coordinates, or given by one of the following operations, corresponding to P = Diag(1, 1, 1, π), Diag(1, 1, π, π) or Diag(1, π, π, π).

Definition 2.2. We define the following three operations on quadratic forms H: • Operation 1. Replace H by \frac{1}{π}H(z_{12}, z_{13}, z_{23}, πz_{14}, πz_{24}, πz_{34}),

• Operation 2. Replace H by H(π^{−1}z_{12}, z_{13}, z_{23}, z_{14}, z_{24}, πz_{34}),

• Operation 3. Replace H by \frac{1}{π}H(z_{12}, z_{13}, πz_{23}, z_{14}, πz_{24}, πz_{34}),

In this section we outline our algorithm. For further details, including how to find the required integral changes of coordinates, see Section 3.

The following subalgorithm suggests some transformations that we might try applying to H. In applications W ⊂ k^{6} will be a subspace determined by the reduction of H mod π. We write e_{12}, e_{13}, e_{23}, e_{14}, e_{24}, e_{34} for the standard basis of k^{6}, and identify the dual basis with z_{12}, z_{13}, z_{23}, z_{14}, z_{24}, z_{34}.

We write \overline{G}, \overline{H} ∈ k[z_{12}, …, z_{34}] for the reductions of G, H mod π.

Algorithm 2.3. (Subalgorithm to suggest some transformations.) We take as input an integral quadratic form H ∈ O_{K}[z_{12}, …, z_{34}] and a vector space W ⊂ k^{6} that is isotropic for \overline{G}. When we make an integral change of coordinates we apply the same transformation (or rather its reduction mod π) to W. The output is either one or two transformations P ∈ PGL_{4}(K).

• If dim W = 1 then make an integral change of coordinates so that W = ⟨e_{12}⟩. Then apply Operation 2.

• If dim W = 2 then make an integral change of coordinates so that W = ⟨e_{12}, e_{13}⟩. Then apply either Operation 1 or Operation 3.

• If dim W = 3 then either make an integral change of coordinates so that W = ⟨e_{12}, e_{13}, e_{23}⟩ and apply Operation 1, or make an integral change of coordinates so that W = ⟨e_{12}, e_{13}, e_{14}⟩ and apply Operation 3.

If char(k) ≠ 2 then the rank and kernel of \overline{H} are defined as the rank and kernel of the corresponding 6 × 6 symmetric matrix. If char(k) = 2 then we assume that k is perfect, so that

\overline{H} = \frac{∂\overline{H}}{∂z_{12}} = … = \frac{∂\overline{H}}{∂z_{34}} = 0

defines a k-vector space, which we call ker \overline{H}. We then define rank \overline{H} = 6 − dim ker \overline{H}.

Algorithm 2.4. (Minimisation algorithm.) We take as input an integral quadratic form H ∈ O_{K}[z_{12}, …, z_{34}]. The output is TRUE/FALSE according as whether there exists P ∈ PGL_{4}(K) such that

v (\frac{1}{det P} H ∘ ∧^{2}P ) > 0.

Step 1. Compute r = rank \overline{H}. If r = 0 then return TRUE. Otherwise, if we have visited this step the first time and returned to it a further 4 times, then return FALSE.

Step 2. If r = 1 then try making an integral change of coordinates so that \overline{H} = z_{34}^{2}. If the reductions of G and π^{−1}H(z_{12}, …, z_{24}, 0) mod π have a common 3-dimensional isotropic subspace W ⊂ ker \overline{H}, then (since running Algorithm 2.3 on any such subspace W gives v(H) > 0) return TRUE.

Step 3. If r = 2 then try running Algorithm 2.3 on each codimension 1 subspace W ⊂ ker \overline{H} that is isotropic for \overline{G}. (There are at most 2 such codimension 1 subspaces; see Remark 3.4.) If one of the suggested transformations gives v(H) > 0 then return TRUE.

Step 4. If r ∈ \{1, 2\} and \overline{H} factors as a product of linear forms defined over k, say \overline{H} = ℓ_{1}ℓ_{2}, then for each i = 1, 2 try making an integral change of coordinates so that ℓ_{i} = z_{34} and then apply Operation 2. If at least one of these transformations gives v(H) ⩾ 0 then select one with rank \overline{H} as small as possible and go to Step 1.

Step 5. If r ∈ \{2, 3, 4, 5\} then try running Algorithm 2.3 on W = ker \overline{H} if this subspace is isotropic for \overline{G}, and otherwise on each codimension 1 subspace W ⊂ ker \overline{H} that is isotropic for \overline{G}. (There are at most 2 such codimension 1 subspaces; see Remark 3.4.) If at least one of the suggested transformations gives v(H) ⩾ 0 then select one with rank \overline{H} as small as possible and go to Step 1.

Step 6. Return FALSE.

There is no difficulty in modifying the algorithm so that when it returns TRUE the corresponding transformation P ∈ PGL_{4}(K) is also returned. In Section 3 we give further details of the implementation, in particular explaining how we make the integral changes of coordinates, and giving further details of Step 2. In Sections 4 and 5 we prove that Algorithm 2.4 is correct.

3. Remarks on implementation

In Algorithms 2.3 and 2.4 we are asked to try making various integral changes of coordinates. It is important to realise that we are restricted to considering matrices of the form ∧^{2}P for P ∈ GL_{4}(O_{K}), and not general elements of GL_{6}(O_{K}). Therefore some care is required both in determining whether a suitable transformation exists, and in finding one when it does.

Since the natural map GL_{4}(O_{K}) → GL_{4}(k) is surjective, we may concentrate on the mod π situation here. Notice however that in the global application with K = ℚ and v = v_{p} it is better to use the surjectivity of SL_{4}(ℤ) → SL_{4}(ℤ/pℤ), so that minimisation at p does not interfere with minimisation at other primes.

Let k^{4} have basis e_{1}, …, e_{4}. We identify ∧^{2}k^{4} = k^{6} via e_{i} ∧e_{j} ↦ e_{ij}. Each linear subspace W ⊂ k^{6} determines a linear subspace V_{0} ⊂ k^{4} given by

V_{0} = {v ∈ k^{4} | v ∧ w = 0 for all w ∈ W }

where ∧ is the natural map k^{4} × ∧^{2}k^{4} → ∧^{3}k^{4}. Let V_{1} be the analogue of V_{0} when W is replaced by its orthogonal complement with respect to \overline{G}.

Lemma 3.1. Let W ⊂ k^{6} be a subspace, and let P ∈ GL_{4}(k).

(i) If dim W = 1 then ∧^{2}P sends W to ⟨e_{12}⟩ if and only if P sends V_{0} to ⟨e_{1}, e_{2}⟩.

(ii) If dim W = 2 or 3 then ∧^{2}P sends W to a subspace of ⟨e_{12}, e_{13}, e_{14}⟩ if and

only if P sends V_{0} to ⟨e_{1}⟩. (iii) If dim W = 5 then ∧^{2}P sends W to ⟨e_{12}, e_{13}, e_{14}, e_{23}, e_{24}⟩ if and only if P

sends V_{1} to ⟨e_{1}, e_{2}⟩.

Proof. In (i) we have W = ⟨e_{12}⟩ if and only if V_{0} = ⟨e_{1}, e_{2}⟩, and in (ii) we have W ⊂ ⟨e_{12}, e_{13}, e_{14}⟩ if and only if V_{0} = ⟨e_{1}⟩. Since the definition of V_{0} in terms of W behaves well under all changes of coordinates this proves (i) and (ii). As noted in Section 1.1, all transformations of the form ∧^{2}P preserve \overline{G} (up to a scalar multiple). Therefore (iii) follows from (i) on replacing W by its orthogonal complement with respect to \overline{G}. □

Remark 3.2. Let G be the matrix of second partial derivatives of G, i.e., the 6×6 matrix with entries 1, −1, 1, 1, −1, 1 on the antidiagonal. A direct calculation shows that for any 4 × 4 matrix P we have

∧^{2}(adj(P)^{T}) = (det P)G(∧^{2}P)G.

Letting PGL_{4} act on the space of quadratic forms via P : H ↦ \frac{1}{det P} H ∘ ∧^{2}P, this tells us that applying P to a quadratic form H(z_{12}, z_{13}, z_{23}, z_{14}, z_{24}, z_{34}) has the same effect as applying P^{−T} to its dual quadratic form which we define to be H(z_{34}, −z_{24}, z_{14}, z_{23}, −z_{13}, z_{12}). We note that the substitution used to replace H by its dual swaps over the two families of isotropic subspaces in Remark 1.7.

We find the changes of coordinates in Algorithm 2.3 by using Lemma 3.1(i) and (ii), and the analogue of (ii) after passing to the dual as in Remark 3.2. We find the changes of coordinates in Steps 2 and 4 of Algorithm 2.4 using Lemma 3.1(iii).

Remark 3.3. In Step 2 of Algorithm 2.4 we must find if possible a 3-dimensional subspace W ⊂ ⟨e_{12}, e_{13}, e_{14}, e_{23}, e_{24}⟩ that is isotropic for both \overline{G} and \overline{H}_{1} where

H_{1}(z_{12}, …, z_{24}) = π^{−1}H(z_{12}, …, z_{24}, 0).

To be isotropic for \overline{G} we need that ⟨e_{12}⟩ ⊂ W. So such a subspace W can only exist if \overline{H}_{1}(1, 0, …, 0) = 0. We assume that this is the case and write

\overline{H}_{1}(z_{12}, …, z_{24}) = z_{12}h_{1}(z_{13}, z_{23}, z_{14}, z_{24}) + h_{2}(z_{13}, z_{23}, z_{14}, z_{24})

where h_{i} is a homogeneous polynomial of degree i. Our problem reduces to that of finding a line contained in

{z_{13}z_{24} − z_{23}z_{14} = h_{1} = h_{2} = 0} ⊂ ℙ^{3}.

The well known description of the lines on {z_{13}z_{24} − z_{23}z_{14} = 0} ⊂ ℙ^{3} suggests that we substitute (z_{13}, z_{23}, z_{14}, z_{24}) = (x_{1}y_{1}, x_{1}y_{2}, x_{2}y_{1}, x_{2}y_{2}) into h_{1} and h_{2}, take the GCD, and factor into irreducibles. The lines of interest now correspond to linear factors of the form αx_{1} + βx_{2} or γy_{1} + δy_{2}.

Remark 3.4. In Steps 3 and 5 of Algorithm 2.4, when ker \overline{H} is not itself isotropic for \overline{G}, we must find all codimension 1 subspaces of ker \overline{H} that are isotropic for \overline{G}. Since the restriction of \overline{G} to ker \overline{H} is a non-zero quadratic form, it can have at most two linear factors. There are therefore at most two codimension 1 subspaces we need to consider. In particular, the number of times that Algorithm 2.4 applies one of the operations in Definition 2.2 is uniformly bounded.

4. Weights and Admissibility

Let H ∈ O_{K}[u_{0}, …, u_{5}] be an integral quadratic form and suppose that there exists P ∈ GL_{4}(K) such that

v (\frac{1}{det P} H ∘ ∧^{2}P ) > 0.

Then P is equivalent to a matrix in Smith normal form, say

P = UDiag(π^{w_{1}}, π^{w_{2}}, π^{w_{3}}, π^{w_{4}})V

for some U, V ∈ GL_{4}(O_{K}) and w_{1}, w_{2}, w_{3}, w_{4} ∈ ℤ. We say that the weight w = (w_{1}, w_{2}, w_{3}, w_{4}) is admissible for H. It is clear that permuting the entries of w, or adding the same integer to all entries, has no effect on admissibility.

The following is based on similar definitions in [K, CFS, F1, FR, ES].

Definition 4.1. The weight w = (w_{1}, w_{2}, w_{3}, w_{4}) dominates the weight w′ = (w′_{1}, w′_{2}, w′_{3}, w′_{4}) if

(4)  \begin{matrix}max(1 + w_{1} + w_{2} + w_{3} + w_{4} − w_{i} − w_{j} − w_{k} − w_{l}, 0) \\ ⩾ max(1 + w′_{1} + w′_{2} + w′_{3} + w′_{4} − w′_{i} − w′_{j} − w′_{k} − w′_{l}, 0) \end{matrix}

for all 1 ⩽ i < j ⩽ 4 and 1 ⩽ k < l ⩽ 4.

This definition is motivated by the fact that if w dominates w′ and w is admissible for H then w′ is admissible for H. Our next lemma shows that (for the purpose of proving that Algorithm 2.4 is correct) it suffices to consider finitely many (in fact 12) weights.

Lemma 4.2. Every weight w = (0, a, b, c) ∈ ℤ^{4} with 0 ⩽ a ⩽ b ⩽ c dominates one of the following weights

(0, 0, 0, 0), (0, 0, 0, 1), (0, 1, 1, 1), (0, 0, 1, 1), (0, 0, 1, 2), (0, 1, 2, 2),
(0, 1, 1, 2), (0, 1, 1, 3), (0, 2, 2, 3), (0, 1, 2, 3), (0, 1, 2, 4), (0, 2, 3, 4).

Proof. We list the pairs (i, j) and (k, l) in Definition 4.1 in the order (1, 2), (1, 3), (2, 3), (1, 4), (2, 4), (3, 4). Taking w = (0, a, b, c), the left hand side of (4) is max(ξ, 0) where ξ runs over the entries of the following symmetric matrix.

[\begin{matrix}1 + b + c − a & 1 + c & 1 + c − a & 1 + b & 1 + b − a & 1 \\ 1 + c & 1 + a + c − b & 1 + c − b & 1 + a & 1 & 1 + a − b \\ 1 + c − a & 1 + c − b & 1 + c − a − b & 1 & 1 − a & 1 − b \\ 1 + b & 1 + a & 1 & 1 + a + b − c & 1 + b − c & 1 + a − c \\ 1 + b − a & 1 & 1 − a & 1 + b − c & 1 + b − a − c & 1 − c \\ 1 & 1 + a − b & 1 − b & 1 + a − c & 1 − c & 1 + a − b − c \end{matrix}]

We divide into 8 cases according as to which of the inequalities 0 ⩽ a ⩽ b ⩽ c are equalities. In fact we make the following more precise claims.

• If 0 = a = b = c then w = (0, 0, 0, 0).

• If 0 = a = b < c then w dominates (0, 0, 0, 1).
• If 0 = a < b = c then w dominates (0, 0, 1, 1).
• If 0 = a < b < c then w dominates (0, 0, 1, 2).
• If 0 < a = b = c then w dominates (0, 1, 1, 1).
• If 0 < a = b < c then w dominates (0, 1, 1, 3), (0, 1, 1, 2) or (0, 2, 2, 3).
• If 0 < a < b = c then w dominates (0, 1, 2, 2).
• If 0 < a < b < c then w dominates (0, 1, 2, 4), (0, 1, 2, 3) or (0, 2, 3, 4).

In each case where we list three possibilities, we further claim that these correspond to the subcases a + b < c, a + b = c and a + b > c (in that order).

Since the proofs are very similar, we give details in just one case. So suppose that 0 < a < b < c and a + b = c. Then we have a ⩾ 1, b ⩾ 2, c ⩾ 3, b − a ⩾ 1, c − a ⩾ 2 and c − b ⩾ 1. Listing the pairs (i, j) and (k, l) in the same order as before, the left hand side of (4) is at least

[\begin{matrix}5 & 4 & 3 & 3 & 2 & 1 \\ 4 & 3 & 2 & 2 & 1 & 0 \\ 3 & 2 & 1 & 1 & 0 & 0 \\ 3 & 2 & 1 & 1 & 0 & 0 \\ 2 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

with equality if (a, b, c) = (1, 2, 3). Therefore w dominates (0, 1, 2, 3). □

Our next remark further reduces the number of weights we must consider.

Remark 4.3. It is clear from Remark 3.2 that if w ∈ ℤ^{4} is admissible for H then −w is admissible for the dual of H. We say that the weights w and −w (or any weights equivalent to these, in the sense of permuting the entries, or adding the same integer to all entries) are dual. The list of 12 weights in Lemma 4.2 consists of 4 dual pairs (0, a, b, c) and (0, c − b, c − a, c) with a + b ≠ c, and 4 self-dual weights (0, a, b, a + b).

5. Completion of the proof

In this section we complete the proof that Algorithm 2.4 is correct.

We first note that if H and H′ are related by an integral change of coordinates, and the algorithm works for H then it works for H′. This is because before applying Operations 1, 2 or 3 we always make an integral change of coordinates that, by Lemma 3.1, is unique up to an element of GL_{4}(O_{K}) whose reduction mod π preserves a suitable subspace of k^{4}. The following elementary lemma then shows that the transformed quadratic forms are again related by an integral change of coordinates.

Lemma 5.1. Let α = Diag(I_{r}, πI_{4−r}) and P ∈ GL_{4}(O_{K}). Then P ∈ α GL_{4}(O_{K})α^{−1} if and only if the reduction of P mod π preserves the subspace ⟨e_{1}, …, e_{r}⟩.

Proof. This is [CFS, Lemma 4.1]. □

Let H ∈ O_{K}[z_{12}, …, z_{34}] be a quadratic form. If there exists P ∈ PGL_{4}(K) such that

(5)  v (\frac{1}{det P} H ∘ ∧^{2}P ) > 0,

then, as explained in Section 4, one of the 12 weights in Lemma 4.2 is admissible for H. Since the analysis for dual weights (see Remark 4.3) is essentially identical, we only need to consider one weight from each dual pair. It therefore suffices to consider the 8 weights listed in the table below.

In the case of weight (w_{1}, …, w_{4}) we may suppose, by an integral change of coordinates, that (5) holds with P = Diag(π^{w_{1}}, …, π^{w_{4}}). This implies certain lower bounds on the valuations of the coefficients of H. To specify these (in a way that is valid even when char(k) = 2), we relabel the variables z_{12}, z_{13}, z_{23}, z_{14}, z_{24}, z_{34} as z_{1}, …, z_{6} and write H = ∑_{i⩽j} H_{ij}z_{i}z_{j}. We also put H_{ji} = H_{ij}. Then the lower bounds on the v(H_{ij}) are as recorded in the table.

Case 1: (0, 0, 0, 0) Case 2: (0, 0, 0, 1) Case 3: (0, 0, 1, 1) Case 4: (0, 1, 1, 2)

[\begin{matrix}1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}] [\begin{matrix}2 & 2 & 2 & 1 & 1 & 1 \\ 2 & 2 & 2 & 1 & 1 & 1 \\ 2 & 2 & 2 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \end{matrix}] [\begin{matrix}3 & 2 & 2 & 2 & 2 & 1 \\ 2 & 1 & 1 & 1 & 1 & 0 \\ 2 & 1 & 1 & 1 & 1 & 0 \\ 2 & 1 & 1 & 1 & 1 & 0 \\ 2 & 1 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix}3 & 3 & 2 & 2 & 1 & 1 \\ 3 & 3 & 2 & 2 & 1 & 1 \\ 2 & 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \end{matrix}]

r = 0  r = 1, 2, 3  r = 1, 2  r = 1, 2, 3, 4
Case 5: (0, 0, 1, 2) Case 6: (0, 1, 1, 3) Case 7: (0, 1, 2, 3) Case 8: (0, 1, 2, 4)

[\begin{matrix}4 & 3 & 3 & 2 & 2 & 1 \\ 3 & 2 & 2 & 1 & 1 & 0 \\ 3 & 2 & 2 & 1 & 1 & 0 \\ 2 & 1 & 1 & 0 & 0 & 0 \\ 2 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix}4 & 4 & 3 & 2 & 1 & 1 \\ 4 & 4 & 3 & 2 & 1 & 1 \\ 3 & 3 & 2 & 1 & 0 & 0 \\ 2 & 2 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix}5 & 4 & 3 & 3 & 2 & 1 \\ 4 & 3 & 2 & 2 & 1 & 0 \\ 3 & 2 & 1 & 1 & 0 & 0 \\ 3 & 2 & 1 & 1 & 0 & 0 \\ 2 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix}6 & 5 & 4 & 3 & 2 & 1 \\ 5 & 4 & 3 & 2 & 1 & 0 \\ 4 & 3 & 2 & 1 & 0 & 0 \\ 3 & 2 & 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

r = 3, 4  r = 3, 4  r = 3, 4  r = 5

In our analysis of each case, we will assume we are not in an earlier case. The possibilities for r = rank \overline{H} will be justified below, but are recorded in the table for convenience. We complete the proof that Algorithm 2.4 is correct by going through the 8 cases. In fact we show that if the cases are grouped as

Case 1 Case 2 Case 4 Case 5 Case 7
Case 3  Case 6 Case 8

then at each iteration of the algorithm we move at least one column to the left. Therefore, if after visiting Step 1 the first time and returning to it a further 4 times we still do not have v(H) > 0 then the algorithm is correct to return FALSE.

Case 1: w = (0, 0, 0, 0). In this case we already have v(H) > 0, so r = 0 and we are done by Step 1.

Case 2: w = (0, 0, 0, 1). We see from the table that ⟨e_{12}, e_{13}, e_{23}⟩ ⊂ ker \overline{H} and so r ⩽ 3. We cannot have r = 0, otherwise we would be in Case 1. If r = 1 then we are done by Step 2. If r = 2 then we are done by Step 3. If r = 3 then Step 5 directly applies Operation 1. (By “directly” we mean that there is no preliminary integral change of coordinates.) Since this gives v(H) > 0 we are in Case 1 on the next iteration.

Case 3: w = (0, 0, 1, 1). We see from the table that \overline{H} = ℓz_{34} for some linear form ℓ. One of the transformations considered in Step 4 is to directly apply Operation 2. Since this gives v(H) > 0 we are in Case 1 on the next iteration.

Case 4: w = (0, 1, 1, 2). We see from the table that ⟨e_{12}, e_{13}⟩ ⊂ ker \overline{H} and so r ⩽ 4. If r = 4 then Step 5 directly applies Operation 1 or Operation 3. Then on the next iteration either (0, 0, 0, 1) or (0, 1, 1, 1) is admissible, which means we are in Case 2 or its dual. If r ⩽ 3 then by applying a block diagonal element of GL_{4}(O_{K}) with blocks of sizes 1, 2 and 1, we may suppose that H_{35} ≡ H_{45} ≡ 0 (mod π). If r = 3 then ker \overline{H} = ⟨e_{12}, e_{13}, ae_{23} + be_{14}⟩ for some a, b ∈ k. If a = 0 or b = 0 then ker \overline{H} is isotropic for \overline{G}. Otherwise ⟨e_{12}, e_{13}⟩ is the unique codimension 1 isotropic subspace. Either way, Step 5 directly applies Operation 1 or Operation 3, and we are done as before.

We now suppose that r ⩽ 2 and divide into the following cases.

• Suppose that H_{36} ≢ 0 (mod π) and H_{46} ≢ 0 (mod π). Since r ⩽ 2 we have \overline{H} = ℓz_{34} for some linear form ℓ. Since there is no integral change of coordinates taking ℓ to z_{34} the only possible outcome of Step 4 is to directly apply Operation 2. This brings us to Case 3.

• Suppose that H_{36} ≡ 0 (mod π) and H_{46} ≢ 0 (mod π). Then v(H_{33}) = 1, otherwise we would be in Case 2. We again have \overline{H} = ℓz_{34} for some linear form ℓ. Although there does now exist an integral change of coordinates taking ℓ to z_{34}, following this up with Operation 2 does not preserve that v(H) ⩾ 0. So again the only possible outcome of Step 4 is to directly apply Operation 2. This brings us to Case 3.

• Suppose that H_{36} ≢ 0 (mod π) and H_{46} ≡ 0 (mod π). This is essentially the same as the previous case by duality.

• Suppose that H_{36} ≡ H_{46} ≡ 0 (mod π). Then \overline{H} is a quadratic form in z_{24} and z_{34} only. If this factors over k then either of the transformations in Step 4 brings us to Case 3. Otherwise we proceed to Step 5 which directly applies Operation 1 or Operation 3. As before, this brings us to Case 2 or its dual.

Case 5: w = (0, 0, 1, 2). Applying a block diagonal element of GL_{4}(O_{K}) with blocks of sizes 2, 1 and 1, we may suppose that H_{26} ≡ 0 (mod π). Then H_{36} ≢ 0 (mod π) (otherwise we would be in Case 2) and H_{44}, H_{45}, H_{55} cannot all vanish mod π (otherwise we would be in Case 3). Therefore ⟨e_{12}, e_{13}⟩ ⊂ ker \overline{H} and r = 3 or 4. The only 3-dimensional isotropic subspaces for \overline{G} that contain ⟨e_{12}, e_{13}⟩ are ⟨e_{12}, e_{13}, e_{23}⟩ and ⟨e_{12}, e_{13}, e_{14}⟩. Therefore one of the transformations considered in Step 5 is to directly apply Operation 1 or Operation 3 (the latter only being a possibility if H_{44} ≡ 0 (mod π)). It follows that at the next iteration we have r ⩽ 2, and so are in Case 4 or earlier.

Case 6: w = (0, 1, 1, 3). Applying a block diagonal element of GL_{4}(O_{K}) with blocks of sizes 1, 2 and 1, we may suppose that H_{15} ≡ 0 (mod π^{2}). We have H_{44} ≢ 0 (mod π) (otherwise we would be in Case 4) and H_{35} ≢ 0 (mod π) (otherwise we would be in Case 5). Therefore ⟨e_{12}, e_{13}⟩ ⊂ ker \overline{H} and r = 3 or 4. Exactly as in Case 5 we find that at the next iteration we have r ⩽ 2, and so are in Case 4 or earlier.

Case 7: w = (0, 1, 2, 3). We have H_{26} ≢ 0 (mod π) (otherwise we would be in Case 4), and H_{35}, H_{45}, H_{55} cannot all vanish mod π (otherwise we would be in Case 3). Therefore r = 3 or 4, and ⟨e_{12}⟩ ⊂ ker \overline{H} ⊂ ⟨e_{12}, e_{13}, e_{23}, e_{14}⟩.

If r = 4 then ker \overline{H} = ⟨e_{12}, ae_{13} + be_{23} + ce_{14}⟩ for some a, b, c ∈ k. If b, c ≠ 0 then ⟨e_{12}⟩ is the unique codimension 1 subspace of ker \overline{H} that is isotropic for \overline{G}. Therefore, Step 5 directly applies Operation 2, which brings us to Case 4. If b = 0 then c ≠ 0, and by applying a block diagonal element of GL_{4}(O_{K}) with blocks of sizes 1, 1 and 2, we may suppose that a = 0. Then the 3-dimensional isotropic subspaces for \overline{G} containing ker \overline{H} = ⟨e_{12}, e_{14}⟩ are ⟨e_{12}, e_{13}, e_{14}⟩ and ⟨e_{12}, e_{14}, e_{24}⟩. Step 5 applies either Diag(1, π, π, π) or Diag(1, 1, π, 1) bringing us to Case 5 or Case 6. The case c = 0 is similar by duality.

If r = 3 then ker \overline{H} = ⟨e_{12}, e_{23} +ae_{13}, e_{14} +be_{13}⟩ for some a, b ∈ k. By applying a block diagonal element of GL_{4}(O_{K}) with blocks of sizes 2 and 2, we may suppose that a = b = 0. Then H_{35} ≡ H_{36} ≡ H_{45} ≡ H_{46} ≡ 0 (mod π) and H_{55} ≢ 0 (mod π). The codimension 1 subspaces of ker \overline{H} = ⟨e_{12}, e_{23}, e_{14}⟩ that are isotropic for \overline{G} are ⟨e_{12}, e_{23}⟩ and ⟨e_{12}, e_{14}⟩. The 3-dimensional isotropic subspaces for \overline{G} containing one of these spaces are

⟨e_{12}, e_{13}, e_{23}⟩, ⟨e_{12}, e_{13}, e_{14}⟩, ⟨e_{12}, e_{23}, e_{24}⟩, ⟨e_{12}, e_{14}, e_{24}⟩.

The first two of these correspond to directly applying Operation 1 or Operation 3, which brings us to Case 5 or its dual. The last two correspond to transformations which fail to preserve that v(H) ⩾ 0, and so cannot be selected by Step 5.

Case 8: w = (0, 1, 2, 4). We have H_{35} ≢ 0 (mod π) (otherwise we would be in Case 5), H_{26} ≢ 0 (mod π) (otherwise we would be in Case 6), and H_{44} ≢ 0 (mod π) (otherwise we would be in Case 7). Therefore r = 5 and ker \overline{H} = ⟨e_{12}⟩. Step 5 directly applies Operation 2 which brings us to Case 6.

Example 5.2. We give three examples where Algorithm 2.4 takes the maximum of 4 iterations to give v(H) > 0. The first two examples start in Case 7, with rank \overline{H} = 3 or 4, and the final one starts in Case 8. In the first two examples there are two choices on the first iteration. We made an arbitrary choice in each case, but in fact with the other choices the algorithm would still have taken 4 iterations.

Let K = ℚ and v = v_{p} for any choice of prime number p. An arrow labelled (w_{1}, …, w_{4}) indicates that we replace H by \frac{1}{det P} H ∘ ∧^{2}P where P = Diag(p^{w_{1}}, …, p^{w_{4}}).

p^{5}z_{12}^{2} + z_{13}z_{34} + pz_{23}^{2} + pz_{14}^{2} + z_{24}^{2} ⟶^{(0,0,0,1)} p^{4}z_{12}^{2} + z_{13}z_{34} + z_{23}^{2} + p^{2}z_{14}^{2} + pz_{24}^{2}

⟶^{(0,0,1,0)} p^{3}z_{12}^{2} + pz_{13}z_{34} + pz_{23}^{2} + pz_{14}^{2} + z_{24}^{2}

⟶^{(0,1,0,1)} p^{3}z_{12}^{2} + z_{13}z_{34} + pz_{23}^{2} + pz_{14}^{2} + p^{2}z_{24}^{2}

⟶^{(0,0,1,1)} p(z_{12}^{2} + z_{13}z_{34} + z_{23}^{2} + z_{14}^{2} + pz_{24}^{2}).

p^{5}z_{12}^{2} + z_{13}z_{34} + pz_{23}^{2} + z_{14}z_{24} ⟶^{(0,0,0,1)} p^{4}z_{12}^{2} + z_{13}z_{34} + z_{23}^{2} + pz_{14}z_{24}

⟶^{(0,0,1,0)} p^{3}z_{12}^{2} + pz_{13}z_{34} + pz_{23}^{2} + z_{14}z_{24}

⟶^{(0,1,0,1)} p^{3}z_{12}^{2} + z_{13}z_{34} + pz_{23}^{2} + pz_{14}z_{24}

⟶^{(0,0,1,1)} p(z_{12}^{2} + z_{13}z_{34} + z_{23}^{2} + z_{14}z_{24}).

p^{6}z_{12}^{2} + z_{13}z_{34} + z_{23}z_{24} + z_{14}^{2} ⟶^{(0,0,1,1)} p^{4}z_{12}^{2} + pz_{13}z_{34} + z_{23}z_{24} + z_{14}^{2}

⟶^{(0,0,0,1)} p^{3}z_{12}^{2} + pz_{13}z_{34} + z_{23}z_{24} + pz_{14}^{2}

⟶^{(0,1,0,1)} p^{3}z_{12}^{2} + z_{13}z_{34} + pz_{23}z_{24} + pz_{14}^{2}

⟶^{(0,0,1,1)} p(z_{12}^{2} + z_{13}z_{34} + z_{23}z_{24} + z_{14}^{2}).

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

Email address: T.A.Fisher@dpmms.cam.ac.uk

Harvard University, Department of Mathematics, Science Center Room 325, 1 Oxford Street, Cambridge, MA 02138, USA

Email address: aliu@math.harvard.edu