EXPLICIT n-DESCENT ON ELLIPTIC CURVES II. GEOMETRY

J.E. CREMONA, T.A. FISHER, C. O’NEIL, D. SIMON, AND M. STOLL

Abstract. This is the second in a series of papers in which we study the n-Selmer group of an elliptic curve. In this paper, we show how to realize elements of the n-Selmer group explicitly as curves of degree n embedded in ℙ^{n−1}. The main tool we use is a comparison between an easily obtained embedding into ℙ^{n^{2}−1} and another map into ℙ^{n^{2}−1} that factors through the Segre embedding ℙ^{n−1}×ℙ^{n−1} → ℙ^{n^{2}−1}. The comparison relies on an explicit version of the local-to-global principle for the n-torsion of the Brauer group of the base field.

1. Introduction

This paper is the second in a series of papers discussing ‘Explicit ndescent on elliptic curves.’ Let E be an elliptic curve over a number field K, and let n ≥ 3. The elements of the n-Selmer group of E may be viewed as isomorphism classes of n-coverings C → E. For us, to do an ‘explicit n-descent’ means to represent each such isomorphism class by giving equations for C as a curve of degree n in ℙ^{n−1}.

It is well known that computing the n-Selmer group as an abstract group gives partial information about both the Mordell-Weil group E(K) and the Tate-Shafarevich group � (K, E). There are likewise several motivations for computing the n-Selmer group in the explicit sense described above. Firstly, it is known that a point in E(K) of xcoordinate height h lifts to a point of height approximately h/(2n) on one of the Selmer n-coverings (see Theorem B.3.2 in [6]). Thus explicit n-descent enables us to find generators for the Mordell-Weil group more easily, and hence sometimes to show that the n-torsion of the Tate-Shafarevich group is trivial. Secondly, if we have already computed the Mordell-Weil group, for instance using descent at some other n′, then we can use explicit n-descent to exhibit concrete examples of nontrivial n-torsion elements of � (K, E). Our work is also likely to form a useful starting point for performing higher descents, and for computing the Cassels-Tate pairing.

Date: 17th November 2006.

The first paper in this series [3] gives more background on the theory of descent. In particular we explain a number of different interpretations of the elements of the group H^{1}(K, E[n]), which contains the Selmer group as a subgroup. One of these interpretations, and the one which is relevant to this paper, is as ‘Brauer-Severi diagrams’ [C → S]; such a diagram consists of a morphism that is a twist of the embedding E → ℙ^{n−1} associated to the complete linear system |n(O)|, where O is the origin of E. In particular C is a torsor under E and S is a Brauer-Severi variety of dimension n − 1. If C has points everywhere locally, then so does S, hence (by Global Class Field Theory) S is isomorphic to ℙ^{n−1}. Our goal in this paper is to explain how one can obtain equations for the image of C in S ≅ ℙ^{n−1} in this case.

In brief, we compare two different embeddings of C into ℙ^{n^{2}−1}. The first comes from the fact that, even without assuming C has points everywhere locally, the cohomology map H^{1}(K, E[n]) → H^{1}(K, E[n^{2}]) sends the Brauer-Severi diagram [C → S] to a diagram [C → S′] with S′ ≅ ℙ^{n^{2}−1}. The second map is more abstract: starting with the diagram C → S we form a dual map C → S^{∨} and thus C → S × S^{∨}. We then compose with the (generalised) Segre embedding to obtain

C → S × S^{∨} → ℙ^{n^{2}−1}.

Equations for the image of the first map to ℙ^{n^{2}−1} are given in Section 3. By a suitable change of coordinates on ℙ^{n^{2}−1} followed by projection to a hyperplane, we obtain equations for the image of the second map. Finally, we pull back to C → S and, when we start with a Selmer group element, obtain equations for C → ℙ^{n−1}. For obvious reasons we refer to this as the Segre embedding method.

We will not be concerned in this paper with the details of implementation; these will be discussed in the third paper of the series [4]. However, we would like to mention that the Segre embedding method, as well as two more methods discussed in [3], have been implemented for n = 3 and K = ℚ and are available as part of the MAGMA computer algebra system [7] (version 2.13 or later).

All three methods rely for their practical implementation on a ‘Black Box’ that computes, for a given central simple K-algebra A of dimension n^{2} that is known to be isomorphic to the matrix algebra Mat_{n}(K), an explicit isomorphism A → Mat_{n}(K). Algorithms for this when K = ℚ will be described in [4].

EXPLICIT n-DESCENT ON ELLIPTIC CURVES 3

2. Background and overview

Unless stated otherwise, K will denote a number field, with absolute Galois group G_{K}, and E will be an elliptic curve over K with origin O. Let n be a positive integer. Recall the definition of the n-Selmer group of E: the short exact sequence of K-group schemes

0 ⟶ E[n] ⟶ E ⟶^{[n]} E ⟶ 0

gives rise to the following commutative diagram with exact rows.

\xymatrix{E(K) \ar[r]^{δ} \ar[d] & H^{1}(K, E[n]) \ar[r] \ar[d] \ar[dr]^{α} & H^{1}(K, E) \ar[d] \\ ∏_{v} E(K_{v}) \ar[r]^{δ} & ∏_{v} H^{1}(K_{v}, E[n]) \ar[r] & ∏_{v} H^{1}(K_{v}, E)}

Here, v runs through all places of K. The n-Selmer group Sel^{(n)}(K, E) is then defined to be the kernel of α. The following notation and facts can be found in [3]. We assume that n ≥ 3. Then there is an embedding f : E → ℙ^{n−1} associated to the complete linear system |n(O)|. In fact, if n = 2 one would work with a double cover, in which case our algorithm still works with minor changes. Indeed it reduces to the classical number field method for 2-descent (see for example [2], Lecture 15). We can view an element of H^{1}(K, E[n]) as a twist of the diagram f : E → ℙ^{n−1}, i.e., as a diagram of the form [C → S], where C is a torsor under E and S is a Brauer-Severi variety of dimension n−1. We call such a diagram a Brauer-Severi diagram. In this interpretation, a diagram [C → S] corresponds to an element of the n-Selmer group if and only if C has points everywhere locally. The period-index obstruction map, defined in [9], is a quadratic map

Ob_{n} : H^{1}(K, E[n]) ⟶ Br(K)[n].

It sends the diagram [C → S] to the class of S in Br(K)[n]. We say an element ξ of H^{1}(K, E[n]) has trivial obstruction if Ob_{n}(ξ) = 0, equivalently the corresponding diagram [C → S] has S ≅ ℙ^{n−1}. Alternatively we can view an element ξ of H^{1}(K, E[n]) as a pair (C, [D]) where [D] is a G_{K}-invariant divisor class on C. The class [D] is represented by a rational divisor if and only if the element ξ has trivial obstruction. Our algorithm applies not only to elements of the Selmer group, but more generally to any element in H^{1}(K, E[n]) with trivial obstruction.

The natural map H^{1}(K, E[n]) → H^{1}(K, E[n^{2}]) brings the pair (C, [D]) to the pair (C, [nD]). Moreover, composing the above with Ob_{n^{2}} gives the zero map: there is a commutative diagram

\xymatrix{H^{1}(K, E[n]) \ar[r]^{^{Ob^{n}}} \ar[d] & Br(K)[n] \ar[d]^{·n} \\ H^{1}(K, E[n^{2}]) \ar[r]^{_{Ob_{n^{2}}}} & Br(K)[n^{2}]}

In other words, the divisor class [nD] is always represented by a rational divisor of degree n^{2}, or equivalently, the above map takes any diagram [C → S] to a diagram [C → ℙ^{n^{2}−1}] with trivial obstruction. It is relatively easy to find equations for the image of this map. We do this in Section 3.

Our basic question then is, starting with an element of H^{1}(K, E[n]) with trivial obstruction, how do we reverse the map

[C → ℙ^{n−1}] ⟼ [C → ℙ^{n^{2}−1}] ?

The diagram [C → S] naturally extends to give a map

λ_{C} : C ⟶ S × S^{∨} ⟶ ℙ(A)

where A is the obstruction algebra, the central simple K-algebra associated to the Brauer group element Ob_{n}(C → S) = [S] (see [10, p. 160]). In this paper we describe an algorithm for writing down both structure constants for A and equations for C as a curve of degree n^{2} in ℙ(A) ≅ ℙ^{n^{2}−1}. In fact we specify the equations in Section 3 and the structure constants in Section 4.

In the case of trivial obstruction, we know that there exists an isomorphism of K-algebras A ≅ Mat_{n}(K), which is called a trivialisation of A. Using this isomorphism, we may obtain equations for C in ℙ^{n−1} as a curve of degree n by projecting onto a column.

The algorithm is split into parts, each of which (except the first) correspond to a piece of the ‘master diagram’ found in Theorem 8.1 below, which we reproduce here.

\xymatrix{C \ar[r]^{f′_{C}×f′_{C}^{∨}} \ar[d]^{g_{C}} & ℙ^{n−1} × (ℙ^{n−1})^{∨} \ar[r]^{Segre} & ℙ(Mat_{n}) \\ ℙ(R) \ar[r]_{φ_{ρ}}^{∼} & ℙ(A_{ρ}) \ar[r]_{τ_{ρ}}^{∼} & ℙ(Mat_{n}) \ar@{-->}[u]^{proj}}

The first step is to realize H^{1}(K, E[n]) as a subgroup of a concrete group. Let R be the affine algebra of the group scheme E[n]. Addition in E[n] translates into the comultiplication homomorphism Δ : R → R ⊗_{K} R. We use this to define a group homomorphism

∂ : R^{×} ⟶ (R ⊗_{K} R)^{×}, α ⟼ \frac{α ⊗ α}{Δ(α)}.

Then there is a natural embedding of H^{1}(K, E[n]) into (R⊗_{K}R)^{×}/∂R^{×}. See Section 3 below and [3, Section 3]. We can then compute the Selmer group as a subgroup of (R ⊗_{K} R)^{×}/∂R^{×}, see [4] for details. Therefore, in the following we can assume that our element of H^{1}(K, E[n]) is represented by some ρ ∈ (R ⊗_{K} R)^{×}.

In the second step, starting with ρ ∈ (R ⊗_{K} R)^{×} coming from an element of H^{1}(K, E[n]), we construct an embedding g_{C} : C → ℙ(R), where ℙ(R) is the projective space associated to the K-vector space R. We need to choose a K-basis of R in order to write down explicit equations for the image of g_{C}. This step is explained in Section 3.

In the third step, we define a new multiplication on R, which depends on ρ, that turns it into a central simple K-algebra A_{ρ}. This is the obstruction algebra for the element of H^{1}(K, E[n]) represented by ρ. In other words, we create an explicit isomorphism of K-vector spaces φ_{ρ} : R → A_{ρ}. This will be explained in Section 4.

The fourth step makes use of the fact that S ≅ ℙ^{n−1}, or equivalently that A_{ρ} ≅ Mat_{n}(K), when ρ represents an element of H^{1}(K, E[n]) which has trivial obstruction. In Section 5, we give an explicit trivialisation of the algebra A_{1}, the central simple algebra coming from the trivial element of H^{1}(K, E[n]), which serves as a normalisation. For the general case a trivialisation map τ_{ρ} : A_{ρ} → Mat_{n}(K) will come from our ‘Black Box’, to be described in more detail in the third paper in this series [4].

In the fifth step, we project the image of τ_{ρ} ∘φ_{ρ} ∘g_{C} into the hyperplane of trace zero matrices and show that our total map factors through the Segre embedding:

C ⟶ ℙ^{n−1} × (ℙ^{n−1})^{∨} ⟶^{Segre} ℙ(Mat_{n}).

This step makes up Section 6 for the case ρ = 1, i.e. [C → ℙ^{n−1}] ≅ [E → ℙ^{n−1}]. The general case is described in Sections 7 and 8.

The final step of the algorithm is to make use of the Segre factorisation. We pull back under the Segre embedding and project to the first factor, which gives us equations for C → ℙ^{n−1}. This is explained in Section 8.

3. Finding Equations for C in ℙ^{n^{2}−1}

We continue to take K a number field, but in fact our algorithm works over any field whose characteristic is prime to n (although below, we assume for simplicity that the characteristic is neither 2 nor 3). We denote by [n] the multiplication-by-n map on E. We fix a Weierstrass equation

E : y^{2} = x^{3} + ax + b

and define for T_{1}, T_{2} ∈ E[n](K̄) a rational function r_{(T_{1},T_{2})} in K̄(E)^{×},

r_{(T_{1},T_{2})}(P) = \{ \begin{matrix}1 & if T_{1} = O or T_{2} = O \\ x(P) − x(T_{1}) & if T_{1} + T_{2} = O and T_{1} ≠ O \\ \frac{y(P)+y(T_{1}+T_{2})}{x(P)−x(T_{1}+T_{2})} − λ(T_{1}, T_{2}) & otherwise, \end{matrix}

where λ(T_{1}, T_{2}) denotes the slope of the line joining T_{1} and T_{2}, respectively of the tangent line at T_{1} = T_{2} if the points are equal.

For T ∈ E[n](K̄), there exists a rational function G_{T} ∈ K̄(E) with divisor

div(G_{T}) = ∑_{nS=T}(S) − ∑_{nS=O}(S) = [n]^{∗}(T) − [n]^{∗}(O).

(See [11, Section III.8].)

Proposition 3.1. We can scale the {G_{T} } such that

(i) The map T ↦ G_{T} is G_{K}-equivariant,

(ii) For each T_{1}, T_{2} ∈ E[n](K̄) and P ∈ E(K̄)\E[n^{2}](K̄) we have

r_{(T_{1},T_{2})}(nP) = \frac{G_{T_{1}}(P)G_{T_{2}}(P)}{G_{T_{1}+T_{2}}(P)}.

(iii) G_{O} = 1, and for T ≠ O the residue of G_{T} at O with respect to the local parameter x/y is \frac{1}{n}.

Proof. Define the scalings of the G_{T} so that condition (iii) holds. That means that with respect to the local parameter t = x/y, when T ≠ O, we can write G_{T} (t) = \frac{1}{n}t^{−1} + …, where ‘… ’ signifies ‘higher order terms.’ This choice of scaling makes the map T ↦ G_{T} visibly G_{K}- equivariant.

When T_{1} = O or T_{2} = O, condition (ii) holds trivially. In any case, a calculation of divisors gives

div(r_{(T_{1},T_{2})}) = (T_{1}) + (T_{2}) − (O) − (T_{1} + T_{2})

for all choices of T_{1} and T_{2}. So the divisor of r_{(T_{1},T_{2})} ∘ [n] is

div(r_{(T_{1},T_{2})} ∘ [n]) = ∑_{nS=T_{1}}(S) + ∑_{nS=T_{2}}(S) − ∑_{nS=O}(S) − ∑_{nS=T_{1}+T_{2}}(S).

This is exactly the divisor of P ↦ G_{T_{1}}(P)G_{T_{2}}(P)/G_{T_{1}+T_{2}}(P). Therefore condition (ii) holds up to a scalar. In the following, we consider the case T_{1} + T_{2} ≠ O. We write x(t) = t^{−2} + … and y(t) = t^{−3} + . . .. Then locally at O,

r_{(T_{1},T_{2})}(t) = \frac{y(t) + y(T_{1} + T_{2})}{x(t) − x(T_{1} + T_{2})} − λ(T_{1}, T_{2})

= \frac{t^{−3} + …}{t^{−2} + …} − λ(T_{1}, T_{2})

= t^{−1} + …

Next, from [11, Prop. IV.2.3], we have [n](t) = nt + …, hence

r_{(T_{1},T_{2})} ∘ [n](t) = (nt)^{−1} + ⋯ = \frac{1}{n}t^{−1} + …

Comparing this with

\frac{G_{T_{1}}(t)G_{T_{2}}(t)}{G_{T_{1}+T_{2}}(t)} = \frac{(\frac{1}{n}t^{−1} + …)(\frac{1}{n}t^{−1} + …)}{\frac{1}{n}t^{−1} + …} = \frac{1}{n}t^{−1} + …

shows that the scalar is 1. The case T_{1} + T_{2} = O is similar. □

In preparation for defining the embedding of C in ℙ^{n^{2}−1}, we recall some facts from [3]. Let R be the affine algebra of E[n], i.e.,

R = Map_{K}(E[n](K̄), K̄).

It is isomorphic to a product of (finite) field extensions of K, one for each G_{K}-orbit in E[n](K̄). We also work with the algebra

R̄ = R ⊗_{K} K̄ = Map(E[n](K̄), K̄).

The Weil pairing e_{n} : E[n] × E[n] → μ_{n} determines an injection

w : E[n](K̄) ↪ R̄^{×} = Map(E[n](K̄), K̄^{×})

via w(S)(T) = e_{n}(S, T).

As in Section 2, we define ∂ : R̄^{×} → (R̄ ⊗K̄ R̄)^{×} via

(1)  ∂α = \frac{α ⊗ α}{Δ(α)},  i.e., (∂α)(T_{1}, T_{2}) = \frac{α(T_{1})α(T_{2})}{α(T_{1} + T_{2})};

then there is an exact sequence (cf. [3, Section 3])

(2)  0 ⟶ E[n](K̄) ⟶^{w} R̄^{×} ⟶^{∂} (R̄ ⊗K̄ R̄)^{×}.

For V a vector space over K, we write ℙ(V) = Proj(K[V]), where K[V] = ⊕_{d≥0} Sym^{d}(V^{∗}) is the ring of polynomial functions on V.

We define R = Res_{R/K}(𝔸^{1}), or equivalently, R = Spec(K[R]). For any K-scheme X, we have

R(X) = 𝔸^{1}(Spec(R) ×_{Spec(K)} X).

In particular, R(L) = R ⊗_{K} L for any field extension L/K.

We also define R^{×} = Res_{R/K}(𝔾_{m}) and S^{×} = Res_{R⊗_{K}R/K}(𝔾_{m}). These schemes inherit a multiplication from 𝔾_{m}. The groups of K-rational points are R^{×}(K) = R^{×} and S^{×}(K) = (R ⊗_{K} R)^{×}. We may identify R^{×} with an open subscheme of R.

With this notation, the exact sequence of G_{K}-modules (2), becomes an exact sequence of K-group schemes:

(3)  0 ⟶ E[n] ⟶^{w} R^{×} ⟶^{∂} S^{×}.

Part (i) of Proposition 3.1 allows us to package the functions G_{T} to form a scheme map g_{E} : E → ℙ(R) sending P ∈ E(K̄) to the class of the map T ↦ G_{T} (P). Away from the subscheme E[n^{2}], the map g_{E} can be lifted to a map g_{E}^{0} to R^{×}. Then we have a commutative diagram:

\xymatrix{E \ E[n^{2}] \ar@{^{(}->}[d] \ar[r]^{g_{E}^{0}} & R^{×} \ar[d] \\ E \ar[r]^{g_{E}} & ℙ(R)}

Next, we use the G_{K}-equivariance of (T_{1}, T_{2}) ↦ r_{(T_{1},T_{2})} to package the functions r_{(T_{1},T_{2})} to form a scheme map r : E \ E[n] → S^{×} sending P ∈ E(K̄) \ E[n](K̄) to the map r(P) : (T_{1}, T_{2}) ↦ r_{(T_{1},T_{2})}(P).

Proposition 3.2. The following diagram commutes.

\xymatrix{E \ E[n^{2}] \ar[r]^{[n]} \ar[d]_{g_{E}^{0}} & E \ E[n] \ar[d]^{r} \\ R^{×} \ar[r]^{∂} & S^{×}}

Proof. We take a geometric point P ∈ E(K̄) \ E[n^{2}](K̄) and observe that

∂(g_{E}^{0}(P))(T_{1}, T_{2}) = \frac{G_{T_{1}}(P)G_{T_{2}}(P)}{G_{T_{1}+T_{2}}(P)}.

By Proposition 3.1(ii), this equals r_{(T_{1},T_{2})}(nP). □

For T ∈ E[n](K̄) we denote by z_{T} the coordinate function on R×_{Spec(K)} Spec(K(T)) given by evaluating at T, so z_{T} (α) = α(T).

Proposition 3.3. Given a Weierstrass equation for E, we can explicitly compute a set of n^{2}(n^{2} −3)/2 linearly independent quadrics over K which define the image of

g_{E} : E ⟶ ℙ(R) ≅ ℙ^{n^{2}−1}.

If E[n](K̄) = E[n](K), then the z_{T} are coordinate functions on R, and the defining quadrics can be split into two groups as follows. For all T_{1}, T_{2} ∈ E[n](K̄) \ {O}, we have

(x(T_{1}) − x(T_{2}))z_{O}^{2} + z_{T_{1}}z_{−T_{1}} − z_{T_{2}}z_{−T_{2}},

and for all T_{11}, T_{12}, T_{21}, T_{22} ∈ E[n](K̄) \ {O} such that

T_{11} + T_{12} = T_{21} + T_{22} = T ≠ O,

we have

(λ(T_{21}, T_{22}) − λ(T_{11}, T_{12}))z_{O}z_{T} − z_{T_{11}}z_{T_{12}} + z_{T_{21}}z_{T_{22}}.

Proof. We first note that the G_{T} are linearly independent. This follows from the fact they are eigenfunctions for distinct characters with respect to the action of E[n] by translation. (We are using the definition of the Weil pairing in [11, Section III.8].) Since there are n^{2} functions G_{T}, they form a basis for the Riemann-Roch space of the divisor [n]^{∗}(O). Hence g_{E} embeds E into ℙ(R) ≅ ℙ^{n^{2}−1} as an elliptic normal curve of degree n^{2}.

Now let P ∈ E(K̄) \ E[n^{2}](K̄), and let z = g_{E}^{0}(P) ∈ R^{×}(K̄) be projective coordinates for g_{E}(P). By Proposition 3.2, we then have r(nP) = ∂z, or equivalently, r(nP)Δ(z) = z ⊗ z. We wish to eliminate P from this equation. Since z_{O}(g_{E}^{0}(P)) = 1, we can make the equation homogeneous by multiplying the left hand side with z(O). This gives

r(nP) z(O)Δ(z) = z ⊗ z.

Writing everything out in terms of a K-basis of R, we obtain n^{4} quadrics, some of whose coefficients involve rational functions of nP. We can eliminate these rational functions by linear algebra over K, to obtain a set of quadrics in K[R] — this is what we do in practice. In order to determine the dimension of the space spanned by them and the geometry of the object defined by them, we can work over K̄. In fact, it is sufficient to work over L = K(E[n]).

Over L, the coordinate functions z_{T} are defined. In terms of these, our system of equations is

r_{(T_{1},T_{2})}(nP) z_{O}z_{T_{1}+T_{2}} = z_{T_{1}}z_{T_{2}},

parametrised by (T_{1}, T_{2}) ∈ E[n](K̄) × E[n](K̄).

If T_{1} = O or T_{2} = O, this reduces to a tautology.

If T_{1} = T ≠ O and T_{2} = −T, then we get

(x(nP) − x(T))z_{O}^{2} = z_{T} z_{−T} .

We can eliminate x(nP) by taking differences. Taking into account the symmetry T ↔ −T, this gives us d_{1} independent quadrics, where

d_{1} = #(\frac{E[n](K̄) \ \{O\}}{\{±1\}}) − 1 = \begin{cases}(n^{2} − 3)/2 & if n is odd \\ n^{2}/2 & if n is even. \end{cases}

If T_{1} + T_{2} = T ≠ O and T_{1}, T_{2} ≠ O, we obtain

(\frac{y(nP) + y(T)}{x(nP) − x(T)} − λ(T_{1}, T_{2}))z_{O}z_{T} = z_{T_{1}}z_{T_{2}}.

Fixing T, we can again eliminate the dependence on P by taking differences. Taking into account the symmetry (T_{1}, T_{2}) ↔ (T_{2}, T_{1}), this provides us with d_{2} independent quadrics, where

d_{2} = #(\frac{\{(T_{1}, T_{2}) : T_{1}, T_{2}, T_{1} + T_{2} ≠ O\}}{(T_{1}, T_{2}) ∼ (T_{2}, T_{1})}) − #(E[n](K̄) \ \{O\})

= \begin{cases}(n^{2} − 1)(n^{2} − 3)/2 & if n is odd \\ n^{2}(n^{2} − 4)/2 & if n is even. \end{cases}

Together, we obtain d_{1} + d_{2} = n^{2}(n^{2} − 3)/2 independent quadrics in either case.

We have found an n^{2}(n^{2} − 3)/2-dimensional space of quadrics vanishing on the image of g_{E}. In general (see for example the Corollary to Theorem 8 in [8]) the homogeneous ideal of an elliptic normal curve of degree m ≥ 4 is generated by a vector space of quadrics of dimension m(m − 3)/2. Therefore, our quadrics define the image of E in ℙ(R) under g_{E}. □

By Section 3 of [3], we can identify H^{1}(K, E[n]) with a subgroup of (R ⊗_{K} R)^{×}/∂R^{×}. Specifically, we define an injective map

H^{1}(K, E[n]) ⟶ (R ⊗_{K} R)^{×}/∂R^{×}

by sending ξ ∈ H^{1}(K, E[n]) to ρ ∂R^{×} where ρ = ∂γ for some γ ∈ R̄^{×} such that w(ξ_{σ}) = σ(γ)/γ for all σ ∈ G_{K}.

Definition 3.4. We let H denote the subgroup of (R ⊗_{K} R)^{×} that maps to the image of H^{1}(K, E[n]) in (R ⊗_{K} R)^{×}/∂R^{×}.

Starting with a representative ρ ∈ H, we fix a choice of γ as above. Then γ determines a cocycle class ξ ∈ H^{1}(K, E[n]). We let π : C → E be the twist of the trivial n-covering [n] : E → E by ξ. In other words, there is a genus 1 curve C defined over K and an isomorphism ϕ : C → E defined over K̄ with σ(ϕ) ∘ ϕ^{−1} = τ_{ξ_{σ}} (translation by ξ_{σ} ∈ E[n](K̄)) for all σ ∈ G_{K}. The covering map is then π = [n] ∘ ϕ. It is easy to check that π is defined over K.

Proposition 3.5. Given ρ and C as above, there are rational functions G_{T,C} ∈ K̄(C), indexed by T ∈ E[n](K̄), such that

(i) The divisor of G_{T,C} is

div(G_{T,C}) = ∑_{π(S)=T}(S) − ∑_{π(S)=O}(S) = π^{∗}(T) − π^{∗}(O).

(ii) The map T ↦ G_{T,C} is G_{K}-equivariant.

(iii) The functions G_{T,C} are scaled so that

r_{(T_{1},T_{2})}(π(P)) = ρ(T_{1}, T_{2})\frac{G_{T_{1},C}(P)G_{T_{2},C}(P)}{G_{T_{1}+T_{2},C}(P)}.

(iv) We can package these G_{T,C} to form morphisms of schemes

g_{C} : C → ℙ(R)  and  g_{C}^{0} : C \ π^{∗}E[n] → R^{×}.

(v) The following diagram, with vertical maps defined over K̄,

commutes:

\xymatrix{C \ar[r]^{g_{C}} \ar[d]_{ϕ} & ℙ(R) \ar[d]^{·γ} \\ E \ar[r]^{g_{E}} & ℙ(R)}

Proof. We take γ ∈ R̄^{×} as above and define

(4)  G_{T,C}(P) = γ(T)^{−1}G_{T} (ϕ(P))

for P ∈ C(K̄). Since π = [n] ∘ ϕ, statement (i) is immediate from the corresponding statement for the G_{T}. Next we check Galois equivariance. Since σ(ϕ) ∘ ϕ^{−1} = τ_{ξ_{σ}} we have

σ(G_{T} (ϕP)) = G_{σT} (ϕ(σP) + ξ_{σ}) = e_{n}(ξ_{σ}, σT)G_{σT} (ϕ(σP))

where the second equality is the definition of the Weil pairing in [11, Section III.8]. On the other hand, since σ(γ)/γ = w(ξ_{σ}), we have

σ(γ(T)) = w(ξ_{σ})(σT)γ(σT) = e_{n}(ξ_{σ}, σT)γ(σT).

We deduce that

σ(G_{T,C})(σP) = σ(G_{T,C}(P)) = γ(σT)^{−1}G_{σT} (ϕ(σP)) = G_{σT,C}(σP).

This proves (ii). For (iii) we compute

r_{(T_{1},T_{2})}(π(P)) = r_{(T_{1},T_{2})}(n · ϕ(P))

= \frac{G_{T_{1}}(ϕ(P))G_{T_{2}}(ϕ(P))}{G_{T_{1}+T_{2}}(ϕ(P))} = ρ(T_{1}, T_{2}) \frac{G_{T_{1},C}(P)G_{T_{2},C}(P)}{G_{T_{1}+T_{2},C}(P)}.

Here the second equality comes from Proposition 3.1. The third equality follows from (4) and ∂γ = ρ.

Statement (iv) is a formal consequence of (ii), and (v) then follows from (4). □

The proofs of the following propositions are very similar to those of Propositions 3.2 and 3.3, and so will be omitted.

Proposition 3.6. The following diagram commutes.

\xymatrix{C \ π^{∗}E[n] \ar[rr]^{π} \ar[d]_{g_{C}^{0}} &  & E \ E[n] \ar[d]^{r} \\ R^{×} \ar[r]^{∂} & S^{×} \ar[r]^{·ρ} & S^{×}}

Proposition 3.7. Given a Weierstrass equation for E and an element ρ ∈ H, with corresponding n-covering π : C → E, we can explicitly compute a set of n^{2}(n^{2} − 3)/2 linearly independent quadrics over K which define the image of

g_{C} : C ⟶ ℙ(R) ≅ ℙ^{n^{2}−1}.

If E[n](K̄) = E[n](K), then the z_{T} are coordinate functions on R, and the defining quadrics can be split into two groups as follows. For all T_{1}, T_{2} ∈ E[n](K̄) \ {O}, we have

(x(T_{1}) − x(T_{2}))z_{O}^{2} + ρ(T_{1}, −T_{1})z_{T_{1}}z_{−T_{1}} − ρ(T_{2}, −T_{2})z_{T_{2}}z_{−T_{2}},

and for all T_{11}, T_{12}, T_{21}, T_{22} ∈ E[n](K̄) \ {O} such that

T_{11} + T_{12} = T_{21} + T_{22} = T ≠ O,

we have

(λ(T_{21}, T_{22})−λ(T_{11}, T_{12}))z_{O}z_{T} −ρ(T_{11}, T_{12})z_{T_{11}}z_{T_{12}} +ρ(T_{21}, T_{22})z_{T_{21}}z_{T_{22}}.

4. A Multiplication Table for the Obstruction Algebra

As noted in [11, Section III.8], for each T ∈ E[n](K̄) there is a rational function F_{T} ∈ K̄(E) with

div(F_{T}) = n(T) − n(O).

In the last section we defined rational functions G_{T}. We now scale the F_{T} so that F_{T} ∘ [n] = G_{T}^{n}. It is equivalent to demand that the leading coefficient of each F_{T}, when expanded as a Laurent series in the local parameter x/y at O, should be 1. (See ‘Step 1’ in Section 5.3 of [3].) It turns out that the F_{T} are rather easy to compute; this will be explained in [4].

Following ‘Step 2’ (loc. cit.), we now define ε ∈ (R̄ ⊗K̄ R̄)^{×} by

ε(T_{1}, T_{2}) = \frac{F_{T_{1}+T_{2}}(P)}{F_{T_{1}}(P)F_{T_{2}}(P − T_{1})}.

for any P ∈ E(K̄) \ {O, T_{1}, T_{1} + T_{2}}. By the discussion in Section 3 of [3], this does not depend on P and satisfies ε(T_{1}, T_{2})ε(T_{2}, T_{1})^{−1} = e_{n}(T_{1}, T_{2}). Since the map defining ε is Galois-equivariant, we obtain an element ε ∈ (R ⊗_{K} R)^{×}. The subgroup H ⊂ (R ⊗_{K} R)^{×} was defined in Definition 3.4.

Proposition 4.1. There is a map

H ⟶ {central simple K-algebras of dimension n^{2}}

sending ρ to an algebra A_{ρ} such that

Ob_{n}(ξ) = [A_{ρ}] ∈ Br(K)[n],

where ξ ∈ H^{1}(K, E[n]) is the element represented by ρ ∈ H. In particular, if ρ′ = ρ ∂z for some z ∈ R^{×}, then [A_{ρ}] = [A_{ρ′}]; in fact, the algebras A_{ρ} and A_{ρ′} are isomorphic.

Proof. In ‘Step 3’ (loc. cit.) we defined A_{ρ} = (R, +, ∗_{ερ}), where ∗_{ερ} is a new multiplication on R. To define it we view R ⊗_{K} R as an R-algebra via the comultiplication Δ : R → R ⊗_{K} R. Recall that this is defined by

Δ(α)(T_{1}, T_{2}) = α(T_{1} + T_{2}).

The corresponding trace map

Tr : R ⊗_{K} R ⟶ R

is given by (Tr z)(T) = ∑_{T_{1}+T_{2}=T} z(T_{1}, T_{2}). Then we define

x ∗_{ερ} y = Tr(ερ · x ⊗ y)

for all x, y ∈ R. The stated properties of A_{ρ} were established in Section 4 of [3]. If ρ′ = ρ ∂z, then the isomorphism A_{ρ′} → A_{ρ} is given by α ↦ zα where the multiplication takes place in R. □

Definition 4.2. Let φ_{ρ} : R → A_{ρ} be the isomorphism of underlying K-vector spaces, inherent in the proof of Proposition 4.1.

The construction in the proof provides us with explicit structure constants of A_{ρ} in terms of a K-basis of R. In [4], we will discuss how to compute the structure constants in practice.

5. Trivialisation of the obstruction algebra

Recall that when ρ ∈ H has trivial obstruction there is a trivialisation isomorphism τ_{ρ} : A_{ρ} → Mat_{n}(K), where A_{ρ} is the central simple algebra in Proposition 4.1. Our algorithm will need to make this trivialisation explicit. In general this will be carried out by the ‘Black Box’ to be discussed further in [4]; however, when ρ = 1 we can write down a standard trivialisation τ_{1} : A_{1} → Mat_{n}(K). In fact τ_{1} will depend on a choice of morphism f_{E} : E → ℙ^{n−1} determined by the complete linear system |n(O)|. We make this choice now.

Let f_{E}^{∨} : E → (ℙ^{n−1})^{∨} be the dual map of f_{E}, i.e., the map that takes P ∈ E(K̄) to the osculating hyperplane at f_{E}(P). The elements of ℙ^{n−1} will be written as column vectors, and the elements of (ℙ^{n−1})^{∨} as row vectors. For each T ∈ E[n](K̄) there is a matrix M_{T} ∈ GL_{n}(K̄) such that translation by T on E extends to the automorphism of ℙ^{n−1} defined by M_{T}.

Proposition 5.1. We may fix the scalings of the M_{T} ’s so that

(5)  F_{T} (P) = \frac{f_{E}^{∨}(O) · M_{T}^{−1} · f_{E}(P)}{f_{E}^{∨}(O) · f_{E}(P)}

where the F_{T} ’s are the rational functions on E defined in Section 4. In particular, M_{O} = I.

Proof. The right hand side is well-defined (the undetermined scalings of f_{E}^{∨}(O) and of f_{E}(P) cancel out) and has divisor n(T) − n(O). □

Let δ_{T} ∈ R̄ be the characteristic function of T, i.e., the map that takes T to 1 but sends all other elements of E[n](K̄) to 0. It is clear that the set of δ_{T} for T ∈ E[n](K̄) are a basis for R̄ as a K̄-vector space.

Definition 5.2. Recall the isomorphism φ_{1} : R → A_{1} of underlying K-vector spaces. Let τ_{1} : A_{1} → Mat_{n}(K) be the linear map of K-vector spaces given by

τ_{1}(φ_{1}(α)) = ∑_{T ∈E[n](K̄)} α(T)M_{T} .

(Since T ↦ M_{T} is G_{K}-equivariant, the map τ_{1}, though a priori defined as a map of K̄-vector spaces, is G_{K}-equivariant.) Note that τ_{1} sends φ_{1}(δ_{T}) ∈ A_{1} ⊗_{K} K̄ to M_{T} ∈ Mat_{n}(K̄).

Proposition 5.3. The map τ_{1} : A_{1} → Mat_{n}(K) is an isomorphism of K-algebras.

Proof. (See also [3, Prop. 5.8.(ii)].) By [3, Lemma 4.8] the set of M_{T} for T ∈ E[n](K̄) form a basis for Mat_{n}(K̄). So it is clear that τ_{1} is an isomorphism of K-vector spaces. We must show that it is also a ring homomorphism.

We recall that A_{1} = (R, +, ∗_{ε}). The new multiplication ∗_{ε} extends to a multiplication on R̄ given by

δ_{T_{1}} ∗_{ε} δ_{T_{2}} = Tr(ε · δ_{T_{1}} ⊗ δ_{T_{2}}) = ε(T_{1}, T_{2})δ_{T_{1}+T_{2}}.

Applying τ_{1} to both sides, it is apparent that what we have to show is that

M_{T_{1}}M_{T_{2}} = ε(T_{1}, T_{2})M_{T_{1}+T_{2}}

for all T_{1}, T_{2} ∈ E[n](K̄). In any case it is clear that M_{T_{1}}M_{T_{2}} = c M_{T_{1}+T_{2}} for some constant c ∈ K̄^{×}.

Substituting T = T_{1} + T_{2} in (5), we get

(6)  F_{T_{1}+T_{2}}(P) = c \frac{f_{E}^{∨}(O) · M_{T_{2}}^{−1}M_{T_{1}}^{−1} · f_{E}(P)}{f_{E}^{∨}(O) · f_{E}(P)}.

We arbitrarily lift f_{E} and f_{E}^{∨} to rational maps E −−→ 𝔸^{n}. The definition of M_{T} gives

(7)  M_{T_{1}}^{−1} · f_{E}(P) = h(P)f_{E}(P − T_{1})

for some rational function h ∈ K̄(E). Premultiplying by f_{E}^{∨}(O) we get

(8)  h(P) = \frac{f_{E}^{∨}(O) · M_{T_{1}}^{−1} · f_{E}(P)}{f_{E}^{∨}(O) · f_{E}(P − T_{1})}.

Substituting (7) into (6) gives

F_{T_{1}+T_{2}}(P) = c \frac{f_{E}^{∨}(O) · M_{T_{2}}^{−1} · f_{E}(P − T)}{f_{E}^{∨}(O) · f_{E}(P)} h(P).

Then by (5) and (8) we obtain

F_{T_{1}+T_{2}}(P) = c F_{T_{2}}(P − T_{1})F_{T_{1}}(P).

Comparing with the definition of ε in Section 4, it follows that c = ε(T_{1}, T_{2}) as required. □

6. The Segre Factorisation for E

Recall our convention that for V a vector space over K, we write ℙ(V) = Proj(K[V]) where K[V] = ⊕_{d≥0} Sym^{d}(V^{∗}) is the ring of polynomial functions on V. We abbreviate ℙ(Mat_{n}(K)) as ℙ(Mat_{n}). The K-points of ℙ(Mat_{n}) may be identified with the set Mat_{n}(K)/K^{×}.

In the last section we fixed a morphism f_{E} : E → ℙ^{n−1} determined by the complete linear system |n(O)|, and wrote f_{E}^{∨} : E → (ℙ^{n−1})^{∨} for the dual map. We now consider the composite map

λ_{E} : E ⟶^{f_{E}×f_{E}^{∨}} ℙ^{n−1} × (ℙ^{n−1})^{∨} ⟶^{Segre} ℙ(Mat_{n}).

We recall that we represent elements of ℙ^{n−1} as column vectors, and elements of (ℙ^{n−1})^{∨} as row vectors. The Segre map is then given by matrix multiplication. In particular, its image is the locus of rank 1 matrices. Since each point of E lies on its own osculating hyperplane, for P ∈ E(K̄) we have f_{E}^{∨}(P) · f_{E}(P) = 0. Then

Tr(λ_{E}(P)) = Tr(f_{E}(P) · f_{E}^{∨}(P)) = Tr(f_{E}^{∨}(P) · f_{E}(P)) = 0.

That is, the image of λ_{E} is contained in the locus of trace zero matrices, which is a hyperplane in ℙ(Mat_{n}). There is a direct sum decomposition Mat_{n}(K) = ⟨I_{n}⟩⊕{Tr = 0}. Note that the trace zero subspace contains all the matrices M_{T} for T ≠ O. We write proj for the rational map ℙ(Mat_{n}) −−→ ℙ(Mat_{n}) induced by the second projection.

We defined maps g_{E} : E → ℙ(R), φ_{1} : R → A_{1} and τ_{1} : A_{1} → Mat_{n}(K) in Sections 3, 4 and 5, respectively.

Theorem 6.1. The following diagram commutes.

\xymatrix{E \ar[rr]^{λ_{E}} \ar[d]^{g_{E}} &  & ℙ(Mat_{n}) \\ ℙ(R) \ar[r]^{∼}_{φ_{1}} & ℙ(A_{1}) \ar[r]^{∼}_{τ_{1}} & ℙ(Mat_{n}) \ar@{-->}[u]^{proj}}

The maps g_{E} and τ_{1} were defined using the G_{T} ’s and the M_{T} ’s, respectively. Note that the matrices M_{T} depend on our choice of the embedding f_{E} : E → ℙ^{n−1}.

The proof of Theorem 6.1 is based on the following result.

Theorem 6.2. For P ∈ E(K̄) \ E[n](K̄) we have

λ_{E}(P) = ∑_{T ≠O} G_{T} (P)M_{T} .

Let us show how Theorem 6.1 follows from this. Since the commutativity of the diagram is a geometric question, we are free to work over K̄. From the definitions we have τ_{1} ∘ φ_{1} ∘ g_{E}(P) = ∑_{T} G_{T} (P)M_{T}. Then composing with the projection map to the trace zero subspace we get P ↦ ∑_{T ≠O} G_{T} (P)M_{T}, which equals λ_{E}(P) by Theorem 6.2. The proof of Theorem 6.2 is split into a series of lemmas. The following notation will be used throughout. Let Q_{1}, …, Q_{n} be n points in E(K̄) and let P = ∑_{i=1}^{n} Q_{i}. We define morphisms h_{E} : E^{n} → ℙ^{n−1} and h_{E}^{∨} : E^{n} → (ℙ^{n−1})^{∨} as follows. For the first map we put

h_{E}(Q_{1}, …, Q_{n}) = f_{E}(P).

The second map takes (Q_{1}, …, Q_{n}) to the hyperplane meeting E (or rather the image of f_{E}) in the divisor (P − nQ_{1}) + … + (P − nQ_{n}). Notice that since this divisor has sum O, it is indeed linearly equivalent to the hyperplane section. The first lemma can be seen as a multi-variable generalisation of formula (5) in Proposition 5.1.

Lemma 6.3. If Q_{1}, …, Q_{n} ∈ E(K̄) \ E[n](K̄) then

G_{T} (Q_{1}) … G_{T} (Q_{n}) = \frac{h_{E}^{∨}(Q_{1}, …, Q_{n}) · M_{T}^{−1} · h_{E}(Q_{1}, …, Q_{n})}{h_{E}^{∨}(Q_{1}, …, Q_{n}) · h_{E}(Q_{1}, …, Q_{n})}

for all T ∈ E[n](K̄).

Proof. We view each side as a rational function on E^{n}. The strategy of the proof is first to compare divisors, and then to check scalings by specialising to the case Q_{1} = Q_{2} = … = Q_{n}. Let pr_{i} : E^{n} → E be projection to the ith factor. The left hand side has divisor

∑_{i=1}^{n} ∑_{nx=T} pr_{i}^{∗}(x) − ∑_{i=1}^{n} ∑_{nx=O} pr_{i}^{∗}(x).

From the definitions of h_{E} and h_{E}^{∨} the right hand side has a zero whenever nQ_{i} = T for some i, and a pole whenever nQ_{i} = O for some i. Therefore the right hand side has divisor

∑_{i=1}^{n} ∑_{nx=T} a_{x} pr_{i}^{∗}(x) − ∑_{i=1}^{n} ∑_{nx=O} b_{x} pr_{i}^{∗}(x)

where the a_{x} and b_{x} are positive integers. If we replace Q_{1} by Q_{1} + S for S ∈ E[n](K̄), then the right hand side is multiplied by a non-zero scalar (the commutator of M_{S} and M_{T}). It follows that the integers a_{x} and b_{x} do not depend on x. So the right hand side has divisor

a ∑_{i=1}^{n} ∑_{nx=T} pr_{i}^{∗}(x) − b ∑_{i=1}^{n} ∑_{nx=O} pr_{i}^{∗}(x)

where a and b are positive integers. Since this divisor is principal, its pull-back by any morphism E → E^{n} has degree 0. This enables us to show that a = b. Since E^{n} is a projective variety, it follows that the right hand side is

c G_{T} (Q_{1})^{a} ⋯ G_{T} (Q_{n})^{a}

for some constant c ∈ K̄^{×}. We now specialise by taking Q_{1} = Q_{2} = … = Q_{n} (= Q say). Then P = nQ and

c G_{T} (Q)^{na} = \frac{f_{E}^{∨}(O) · M_{T}^{−1} · f_{E}(P)}{f_{E}^{∨}(O) · f_{E}(P)}.

The definition of F_{T} in Section 4 gives F_{T} (P) = F_{T} (nQ) = G_{T} (Q)^{n}. Finally we compare with (5) to get c = 1 and a = 1. □

Lemma 6.4. If Q_{1}, …, Q_{n} ∈ E(K̄) \ E[n](K̄) then

\frac{1}{n} ∑_{T ∈E[n](K̄)} G_{T} (Q_{1}) … G_{T} (Q_{n})M_{T} = \frac{h_{E}(Q_{1}, …, Q_{n}) · h_{E}^{∨}(Q_{1}, …, Q_{n})}{h_{E}^{∨}(Q_{1}, …, Q_{n}) · h_{E}(Q_{1}, …, Q_{n})}

Proof. Since the M_{T} form a basis for Mat_{n}(K̄), we can write the right hand side as n^{−1} ∑_{T} a_{T} (Q_{1}, …, Q_{n})M_{T} for some rational functions a_{T} on E^{n}. To compute the a_{T} ’s we premultiply by M_{T}^{−1} and take the trace. Since

Tr(M_{T}) = \begin{cases}n & if T = O, \\ 0 & otherwise, \end{cases}

on the left hand side we get G_{T} (Q_{1}) … G_{T} (Q_{n}). On the right hand side, using Tr(AB) = Tr(BA) gives

a_{T} (Q_{1}, …, Q_{n}) = Tr(M_{T}^{−1} · \frac{h_{E}(Q_{1}, …, Q_{n}) · h_{E}^{∨}(Q_{1}, …, Q_{n})}{h_{E}^{∨}(Q_{1}, …, Q_{n}) · h_{E}(Q_{1}, …, Q_{n})})

= Tr(\frac{h_{E}^{∨}(Q_{1}, …, Q_{n}) · M_{T}^{−1} · h_{E}(Q_{1}, …, Q_{n})}{h_{E}^{∨}(Q_{1}, …, Q_{n}) · h_{E}(Q_{1}, …, Q_{n})})

= G_{T} (Q_{1}) … G_{T} (Q_{n}), where we have used Lemma 6.3 in the last equality. □

Lemma 6.5. There is a commutative diagram of morphisms

[figure: E^{n}; h_{E}×h_{E}^{∨}; ∏ g_{E}; ℙ^{n−1} × (ℙ^{n−1})^{∨}; Segre; ℙ(Mat_{n}); ℙ(R); ∼; φ_{1}; ℙ(A_{1}); ∼; τ_{1}; ℙ(Mat_{n})]

where (∏ g_{E})(Q_{1}, …, Q_{n}) = ∏_{i=1}^{n} g_{E}(Q_{i}).

Proof. From the definitions we have

τ_{1} ∘ φ_{1} ∘ (∏ g_{E})(Q_{1}, …, Q_{n}) = ∑_{T} G_{T} (Q_{1}) … G_{T} (Q_{n})M_{T} .

So the commutativity is already clear from Lemma 6.4.

It only remains to check that ∏ g_{E} is a morphism. To do this we write it as a composite

E^{n} ⟶^{g_{E}^{n}} ℙ(R)^{n} \underrightarrow{∏} ℙ(R)

where the second map is induced by multiplication in R. We check that the image of the first map is contained in the domain of definition of the second.

If Q ∈ E(K̄)\E[n](K̄), then g_{E}(Q) is the class of T ↦ G_{T} (Q), whereas if Q ∈ E[n](K̄), then g_{E}(Q) is the class of T ↦ res_{Q}(G_{T}), where the residue is taken with respect to a local parameter at Q. (The choice of local parameter does not matter.)

Now suppose Q_{1}, …, Q_{n} ∈ E(K̄) and ∏_{i=1}^{n} g_{E}(Q_{i}) is undefined as an element of ℙ(R). Then for each T ≠ O there exists 1 ≤ i ≤ n such that G_{T} (Q_{i}) = 0, and hence nQ_{i} = T. But there are n^{2} − 1 such choices of T and only n choices of i. So this is impossible. It follows that ∏ g_{E} is a morphism as claimed. □

To complete the proof of Theorem 6.2, and hence of Theorem 6.1, we put Q_{1} = Q and Q_{2} = … = Q_{n} = O in Lemma 6.5. Notice that h_{E}(Q, O, …, O) = f_{E}(Q) and h_{E}^{∨}(Q, O, …, O) = f_{E}^{∨}(Q). Also, with the G_{T} ’s scaled as in Proposition 3.1(iii), g_{E}(O) is the class of ∑_{T ≠O} δ_{T}.

So for Q ∈ E(K̄) \ E[n](K̄) we obtain

λ_{E}(Q) = Segre ∘ (f_{E} × f_{E}^{∨})(Q)

= Segre ∘ (h_{E} × h_{E}^{∨})(Q, O, …, O)

= τ_{1} ∘ φ_{1}(g_{E}(O)^{n−1}g_{E}(Q))

= ∑_{T ≠O} G_{T} (Q)M_{T}

7. The Segre Factorisation for C

Let A be a central simple algebra over K of dimension n^{2}. Let S and S^{∨} be the Brauer-Severi varieties given by the minimal right and left ideals of A, respectively (see [10, p. 160]). There is a natural map Segre : S × S^{∨} → ℙ(A) given by intersecting ideals. We say an element a ∈ A has rank r if the map x ↦ ax is an endomorphism of A (as a K-vector space) of rank rn.

Lemma 7.1. The Segre map S × S^{∨} → ℙ(A) is an embedding, with image the locus of rank 1 elements in ℙ(A).

Proof. If A ≅ Mat_{n}(K) then S ≅ ℙ^{n−1} and the Segre map reduces to that studied in Section 6. The description of the image is no more than the observation that a (non-zero) matrix has rank 1 if and only if it can be written as a column vector times a row vector. In general we use that there is an isomorphism of K̄-algebras A ⊗_{K} K̄ ≅ Mat_{n}(K̄).

The Segre map is an embedding since, on the rank 1 locus in ℙ(A), an inverse is given by sending the class of a ∈ A to the pair of ideals (aA, Aa). □

From now on, we call the Segre map the (generalised) Segre embedding. We write 1_{A} for the multiplicative identity of A, and Trd : A → K for the reduced trace. There is a decomposition of K-vector spaces A = ⟨1_{A}⟩ ⊕ {Trd = 0}. As in Section 6 we write proj for projection onto the second factor.

The subgroup H ⊂ (R ⊗_{K} R)^{×} was defined in Definition 3.4.

Theorem 7.2. Let ρ ∈ H, and let π : C → E be the corresponding n-covering. Let S and S^{∨} be the Brauer-Severi varieties given by the minimal right and left ideals in ℙ(A_{ρ}). Then there is a morphism f_{C} : C → S with dual f_{C}^{∨} : C → S^{∨} such that (i) the following diagram commutes:

\xymatrix{C \ar[r]^{f_{C}×f_{C}^{∨}} \ar[d]^{g_{C}} & S × S^{∨} \ar[r]^{Segre} & ℙ(A_{ρ}) \\ ℙ(R) \ar[rr]^{∼}_{φ_{ρ}} &  & ℙ(A_{ρ}) \ar@{-->}[u]^{proj}}

(ii) the Brauer-Severi diagram [C → S] and the class of ρ in H correspond to the same element of H^{1}(K, E[n]).

The theorem is proved by combining Theorem 6.1 with the next lemma. First we recall how the element ρ ∈ H and n-covering π : C → E are related. In Section 3 we fixed an element γ ∈ R̄^{×} with ρ = ∂γ and defined a cocycle ξ ∈ H^{1}(K, E[n]) via w(ξ_{σ}) = σ(γ)/γ for all σ ∈ G_{K}. Then we let π : C → E be the ξ-twist of the trivial n-covering. Thus there is an isomorphism ϕ : C → E defined over K̄ with π = [n] ∘ ϕ and σ(ϕ) ∘ ϕ^{−1} = τ_{ξ_{σ}} for all σ ∈ G_{K}.

Lemma 7.3. There is an isomorphism of K̄-algebras β : A_{ρ} ⊗_{K} K̄ → A_{1} ⊗_{K} K̄ making the following diagram commute.

(9)  \xymatrix{C \ar[r]^{g_{C}} \ar[d]^{ϕ} & ℙ(R) \ar[r]^{∼}_{φ_{ρ}} \ar[d]^{·γ} & ℙ(A_{ρ}) \ar[d]^{β} \ar@{-->}[dr]^{τ_{1}∘β} \ar@{-->}[r]^{proj} & ℙ(A_{ρ}) \ar[dr]^{τ_{1}∘β} \\ E \ar[r]^{g_{E}} & ℙ(R) \ar[r]^{∼}_{φ_{1}} & ℙ(A_{1}) \ar[r]^{τ_{1}} & ℙ(Mat_{n}) \ar@{-->}[r]^{proj} & ℙ(Mat_{n})}

Proof. The first square commutes by Proposition 3.5(v). We define β : A_{ρ} ⊗_{K} K̄ → A_{1} ⊗_{K} K̄ to make the second square commute. It is an isomorphism of K̄-algebras by [3, Lemma 4.6]. We recall from Proposition 5.3 that τ_{1} is an isomorphism of K-algebras. Finally, since proj is defined purely in terms of the algebra structure, it commutes with the algebra isomorphism τ_{1} ∘ β. □

Theorem 6.1 identifies the composite of the second row of (9) as λ_{E}, where we recall

λ_{E} : E ⟶^{f_{E}×f_{E}^{∨}} ℙ^{n−1} × (ℙ^{n−1})^{∨} ⟶^{Segre} ℙ(Mat_{n}).

Since λ_{E} factors via the Segre embedding, its image belongs to the rank 1 locus of ℙ(Mat_{n}). Let λ_{C} = proj ∘φ_{ρ} ∘ g_{C} be the composite of the first row of (9). Recalling that τ_{1} ∘ β is an isomorphism of K̄- algebras, it follows that λ_{C} has image belonging to the rank 1 locus of ℙ(A_{ρ}). By Lemma 7.1 it therefore factors via the Segre embedding.

In other words, there are morphisms f_{C} : C → S and f_{C}^{∨} : C → S^{∨} making the diagram in the first part of Theorem 7.2 commute. It remains to show that [C → S] is a Brauer-Severi diagram, that f_{C}^{∨} is dual to f_{C} and that [C → S] and ρ correspond to the same element of H^{1}(K, E[n]).

The isomorphism of K̄-algebras τ_{1} ∘ β : A_{ρ} ⊗_{K} K̄ → Mat_{n}(K̄) induces isomorphisms ψ : S → ℙ^{n−1} and ψ^{∨} : S^{∨} → (ℙ^{n−1})^{∨} defined over K̄ making the following diagram commute.

\xymatrix{C \ar[r]^{f_{C}×f_{C}^{∨}} \ar[d]_{ϕ} & S × S^{∨} \ar[r]^{Segre} \ar[d]_{ψ×ψ^{∨}} & ℙ(A_{ρ}) \ar[d]^{τ_{1}∘β} \\ E \ar[r]^{f_{E}×f_{E}^{∨}} & ℙ^{n−1} × (ℙ^{n−1})^{∨} \ar[r]^{Segre} & ℙ(Mat_{n})}

This shows that via (ϕ, ψ), the morphism f_{C} : C → S is isomorphic over K̄ to f_{E} : E → ℙ^{n−1}. Hence [C → S] is a Brauer-Severi diagram. Since σ(ϕ) ∘ ϕ^{−1} is translation by ξ_{σ} this diagram is the ξ-twist of [E → ℙ^{n−1}], and therefore the Brauer-Severi diagram corresponding to ρ. At the same time, we see that f_{C}^{∨} is dual to f_{C} (since f_{E}^{∨} is dual to f_{E}). This completes the proof of Theorem 7.2.

8. Finding Equations for C in ℙ^{n−1}

We continue to represent elements of H^{1}(K, E[n]) by elements ρ ∈ H, where the subgroup H ⊂ (R ⊗_{K} R)^{×} was defined in Section 3. The obstruction algebra A_{ρ} was introduced in Section 4. If ρ represents an element of H^{1}(K, E[n]) with trivial obstruction, then there is an explicit trivialisation isomorphism of K-algebras τ_{ρ} : A_{ρ} → Mat_{n}(K).

Theorem 8.1. Let ρ ∈ H, and let π : C → E be the corresponding n-covering. Suppose given an isomorphism of K-algebras τ_{ρ} : A_{ρ} → Mat_{n}(K). Then there is a morphism f′_{C} : C → ℙ^{n−1} with dual f′_{C}^{∨} such that

(i) the following diagram commutes:

\xymatrix{C \ar[r]^{f′_{C}×f′_{C}^{∨}} \ar[d]^{g_{C}} & ℙ^{n−1} × (ℙ^{n−1})^{∨} \ar[r]^{Segre} & ℙ(Mat_{n}) \\ ℙ(R) \ar[r]_{φ_{ρ}}^{∼} & ℙ(A_{ρ}) \ar[r]_{τ_{ρ}}^{∼} & ℙ(Mat_{n}) \ar@{-->}[u]^{proj}}

(ii) the Brauer-Severi diagram [C → ℙ^{n−1}] and the class of ρ in H correspond to the same element of H^{1}(K, E[n]).

Proof. Let S and S^{∨} be the Brauer-Severi varieties given by the minimal right and left ideals in ℙ(A_{ρ}). The isomorphism τ_{ρ} induces Kisomorphisms ψ′ : S → ℙ^{n−1} and ψ′^{∨} : S^{∨} → (ℙ^{n−1})^{∨}. We modify the maps f_{C} and f_{C}^{∨} of Theorem 7.2 to give maps f′_{C} and f′_{C}^{∨} making the following diagram commute.

[figure: C; f′_{C}×f′_{C}^{∨}; ℙ^{n−1} × (ℙ^{n−1})^{∨}; Segre; ℙ(Mat_{n}); C; f_{C}×f_{C}^{∨}; S × S^{∨}; Segre; ψ′×ψ′^{∨}; ℙ(A_{ρ}); τ_{ρ}]

Since proj is defined purely in terms of the algebra structure, it commutes with the algebra isomorphism τ_{ρ}. The theorem now follows by combining the above diagram with that in Theorem 7.2. □

In Section 3 we explained how to write down equations for C as a curve of degree n^{2} in ℙ(R) ≅ ℙ^{n^{2}−1}. Theorem 8.1 now tells us how to convert these to equations for C as a curve of degree n in ℙ^{n−1}. First we use τ_{ρ} to get equations for C in ℙ(Mat_{n}). Then we project onto the trace zero matrices. Next we write x_{11}, x_{12}, …, x_{nn} for our coordinate functions on ℙ(Mat_{n}) and substitute x_{ij} = x_{i}y_{j}, where the x_{i} and y_{j} are new indeterminates (2n in total). This corresponds to pulling the image of C in ℙ(Mat_{n}) back under the Segre map. To project to the first factor ℙ^{n−1}, we eliminate the y_{j}. We are left with equations in the x_{i}, and these define the image of f′_{C} : C → ℙ^{n−1}. We will describe in [4] how this computation can be reduced to linear algebra for any specific value of n.

For aesthetic as well as practical reasons, it is desirable to find a change of coordinates on ℙ^{n−1} so that the equations for C have small coefficients. The necessary minimisation and reduction procedures will be described in a forthcoming paper [5] for the cases n = 3 and n = 4.

It is also desirable to have explicit equations for the covering map π : C → E. In principle these could be obtained using the methods of Section 3. However in the cases n = 2, 3, 4 equations for π are already given by classical formulae, reproduced in [1].

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, Cambridge University Press, Cambridge, 1991.

[3] J.E. Cremona, T.A. Fisher, C. O’Neil, D. Simon and M. Stoll, Explicit ndescent on elliptic curves, I Algebra, submitted for publication. See ArXiv math.NT/0606580.

[4] J.E. Cremona, T.A. Fisher, C. O’Neil, D. Simon and M. Stoll, Explicit ndescent on elliptic curves, III Algorithms, in preparation.

[5] J.E. Cremona, T.A. Fisher and M. Stoll, Minimisation and reduction for 3and 4-coverings of elliptic curves, in preparation.

[6] M. Hindry and J.H. Silverman, Diophantine geometry, Graduate Texts in Mathematics 201, Springer-Verlag, New York, 2000.

[7] MAGMA is described in W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24, 235–265 (1997).

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

[8] D. Mumford, Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), pp. 29–100, Edizioni Cremonese, Rome, 1970.

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

[10] J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer-Verlag, New York, 1979.

[11] J.H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1992.

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

E-mail address: John.Cremona@nottingham.ac.uk

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

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

Barnard College, Columbia University, Department of Mathematics, 2990 Broadway MC 4418, New York, NY 10027-6902,USA

E-mail address: oneil@math.columbia.edu

Université de Caen, Campus II - Boulevard Maréchal Juin, BP 5186– 14032, Caen, France

E-mail address: Denis.Simon@math.unicaen.fr

School of Engineering and Science, International University Bremen (Jacobs University Bremen as of spring 2007), P.O. Box 750561, 28725 Bremen, Germany

E-mail address: M.Stoll@iu-bremen.de