2·1. The set-up

In this paper, we are working over a number field K. For any field k, we let $\bar{k}$ denote its algebraic closure and let $\mu_n \subset \bar{k}$ denote the $n^{\mathrm{th}}$ roots of unity in $\bar{k}$ . We let $G_k$ denote the absolute Galois group $\text{Gal}(\bar{k}/k)$ .

Let $\mathcal{C}$ be a general genus two curve defined over K, which is a smooth projective curve. It can be given in the following hyperelliptic form:

\begin{align*}\mathcal{C}\,:\, y^2 = f(x)= f_6x^6 +f_5x^5 +f_4x^4 +f_3x^3 +f_2x^2 +f_1x +f_0,\end{align*}

where $f_i \in K$ , $f_6 \neq 0$ and the discriminant $\triangle(f) \neq 0$ , which implies that f has distinct roots in $\bar{K}$ . Note that we choose to not use the quintic representation of the genus 2 curve because using the degree 6 form simplifies some computations and makes the final formula slightly more symmetric and elegant.

We let J denote the Jacobian variety of $\mathcal{C}$ , which is an abelian variety of dimension two defined over K that can be identified with $\text{Pic}^0(\mathcal{C})$ . We denote the identity element of J by $\mathcal{O}_J$ . We also denote the two points at infinity on $\mathcal{C}$ by $\infty^+, \infty^-$ . Via the natural isomorphism $\text{Pic}^2(\mathcal{C}) \rightarrow \text{Pic}^0(\mathcal{C})$ sending $[P_1+P_2] \mapsto [P_1 + P_2 - \infty^+ - \infty^-]$ , a point $P \in J$ can be identified with an unordered pair of points of $\mathcal{C}$ , $\{P_1, P_2\}$ . This identification is unique unless $P= \mathcal{O}_J$ , in which case it can be represented by any pair of points on $\mathcal{C}$ in the form $\{(x, y), (x, -y)\}$ or $\{\infty^+, \infty^-\}$ . Suppose the roots of f are denoted by $\omega_1, ..., \omega_6$ and let $(\omega_1, 0), ..., (\omega_6, 0)$ be all the Weierstrass points on $\mathcal{C}$ . Then $J[2] = \{\mathcal{O}_J, \{(\omega_i, 0), (\omega_j, 0)\}\text{ for } i\neq j\}$ . Also, for a point $P \in J$ , we let $\tau_P\,:\,J \rightarrow J$ denote the translation by P on J.

As described in [ Reference Cassels and FlynnCF96 , chapter 3, section 3], suppose $\{P_1, P_2\} $ and $\{Q_1, Q_2\}$ represent $P, Q \in J[2]$ where $P_1, P_2, Q_1, Q_2$ are Weierstrass points. Then the Weil paring is given by the formula

\begin{align*}e_2(P, Q)= ({-}1)^{|\{P_1, P_2\} \cap \{Q_1, Q_2\} |}.\end{align*}

2·2. Theta divisor and Kummer surface

A theta divisor, denoted by $\Theta$ , is defined to be any divisor on J that is the image of the divisor $\{P\}\times \mathcal{C}+ \mathcal{C} \times \{P\}$ on $\text{Sym}^2\mathcal{C}$ under the birational morphism $\text{Sym}^2\mathcal{C} \rightarrow J$ , for some Weierstrass point $P \in \mathcal{C}$ . The Jacobian variety J is a principally polarised abelian variety via $\lambda\,:\, J \rightarrow J^{\vee}$ sending $Q\in J$ to $[\tau_Q^*\Theta-\Theta].$ It can be checked that $2\Theta \sim \Theta^+ +\Theta^-$ , where $\Theta^+$ denotes the divisor on J that corresponds to the divisor $\{\infty^+\}\times \mathcal{C}+ \mathcal{C} \times \{\infty^+\}$ on $\mathcal{C} \times \mathcal{C}$ and similarly for $\Theta^-$ . In particular, this implies that the divisor class of $2n\Theta$ is defined over the base field K, for any positive integer n.

The Kummer surface, denoted by $\mathcal{K}$ , is the quotient of J via the involution $[{-}1]\,:\, P \mapsto -P$ . The fixed points under the involution are the 16 points of order dividing 2 on J and these map to the 16 nodal singular points of $\mathcal{K}$ (the ${nodes}$ ). General theory, as in [ Reference MilneMil08 , theorem 11·1], [ Reference MumfordMum70 , page 150], shows that the linear system of $n\Theta$ on J has dimension $n^2$ . Moreover, $|2\Theta|$ is base point free and induces a morphism from J to $\mathbb{P}^3$ defined over K while $|4\Theta|$ is very ample and induces an embedding from J to $\mathbb{P}^{15}$ defined over K.

2·3. Explicit embeddings of J and $\mathcal{K}$

From Section 2·2, we know that $\dim\mathcal{L}(2\Theta)=4$ and $2\Theta \sim \Theta^+ +\Theta^-$ . Let $\{k_1, k_2, k_3, k_4\}$ denote the basis of $\mathcal{L}(\Theta^+ +\Theta^-)$ as defined in [ Reference Cassels and FlynnCF96 , chapter 3, section 1]. We will restate the definition here for completeness. Denote a generic point on the Jacobian J of $\mathcal{C}$ by $\{(x, y), (u, v)\}$ . There is then a morphism from J to $\mathbb{P}^3$ given by

\begin{align*}k_1 = 1, k_2 = (x+u), k_3 = xu, k_4 = \beta_0,\end{align*}

where

\begin{align*}\beta_0 = \frac{F_0(x, u)-2yv}{(x-u)^2}\end{align*}

with $F_0(x, u)= 2f_0 + f_1(x+u) +2f_2(xu) + f_3(x+u)(xu) + 2f_4(xu)^2 + f_5(x+u)(xu)^2 + 2f_6(xu)^3$ . The image of this morphism is the Kummer surface $\mathcal{K} \subset \mathbb{P}^3$ defined by the vanishing of the quartic polynomial $G(k_1,k_2,k_3,k_4)$ as specified in [ Reference Cassels and FlynnCF96 , chapter 3, section 1]. From now on, we denote this morphism by $J \stackrel{|2\Theta|}{\to} \mathcal{K} \subset \mathbb{P}^3$ . We note that it maps ${\mathcal O}_J$ to $(0\,:\,0\,:\,0\,:\,1)$ .

We now look at the embedding of J in $\mathbb{P}^{15}$ induced by $|4\Theta|$ . Let $k_{ij} = k_ik_j$ , for $1 \leq i\leq j \leq 4$ . Since $\mathcal{K}$ is irreducible and defined by a polynomial of degree 4, $k_{11}, k_{12}, ..., k_{44}$ are 10 linearly independent even elements in $\mathcal{L}(2\Theta^+ +2 \Theta^-)$ . Six further odd basis elements $b_1,b_2,...b_6$ in $\mathcal{L}(2\Theta^+ + 2\Theta^-)$ are given explicitly in [ Reference Flynn, Testa and van LuijkFTvL12 , section 3]. A function g on J is even when it is invariant under the involution $[{-}1]\,:\, P \mapsto -P$ and is odd when $g\circ [{-}1]=-g$ .

Unless stated otherwise, we will use the basis $k_{11}, k_{12}, ..., k_{44}, b_1, ..., b_6$ for $\mathcal{L}(2\Theta^+ + 2\Theta^-)$ to embed J in $\mathbb{P}^{15}$ . The following theorem gives the defining equations of J.

2·4. Principal homogeneous spaces and 2-coverings

A principal homogeneous space or torsor for J defined over a field K is a variety V together with a morphism $\mu \,:\, J \times V \rightarrow V$ , both defined over K, that induces a simply transitive action on the $\bar{K}$ -points.

We say $(V_1, \mu_1)$ and $(V_2, \mu_2)$ are isomorphic over a field extension $K_1$ of K if there is an isomorphism $\phi\,:\, V_1 \rightarrow V_2$ defined over $K_1$ that respects the action of J.

A 2-covering of J is a variety X defined over K together with a morphism $\pi \,:\, X \rightarrow J$ defined over K, such that there exists an isomorphism $\phi \,:\, X \rightarrow J$ defined over $\bar{K}$ with $\pi = [2] \circ \phi$ . An isomorphism $(X_1, \pi_1) \rightarrow (X_2, \pi_2)$ between two 2-coverings is an isomorphism $h\,:\, X_1 \rightarrow X_2$ defined over K with $\pi_1 = \pi_2 \circ h$ . We sometimes denote $(X, \pi)$ by X when the context is clear.

It can be checked that a 2-covering is a principal homogeneous space. The short exact sequence $0 \rightarrow J[2] \rightarrow J \xrightarrow{2} J \rightarrow 0$ induces the connecting map in the long exact sequence

(2·1) \begin{equation}\delta\,:\, J(K) \rightarrow H^1(G_K,J[2]).\end{equation}

The following two propositions are proved in [ Reference Flynn, Testa and van LuijkFTvL12 ].

We also state and prove the following proposition which is useful for the computation of the Cassels–Tate pairing later in Sections 4·2 and 4·3. A Brauer–Severi variety is a variety defined over K that is isomorphic to a projective space over $\bar{K}$ .

Proposition 2·5. Let $(X, \pi)$ be a 2-covering of J, with $\phi \circ [2]= \pi$ . Then the linear system $|\phi^*(2\Theta)|$ determines a map $X \rightarrow S$ defined over K, where S is a Brauer–Severi variety. Also, there exists an isomorphism $\psi$ defined over $\bar{K}$ making the following diagram commute:

(2·2)

In particular, if $(X, \pi)$ corresponds to a Selmer element via the correspondence in Proposition 2·3, then the Brauer–Severi variety S is isomorphic to $\mathbb{P}^{3}$ .

Proof. Since $(X, \pi)$ is a 2-covering of J, by Proposition 2·3, we have that for each $\sigma \in G_K$ , $\phi \circ (\phi^{-1})^{\sigma} = \tau_{P}$ for some $P \in J[2]$ . By Remark 2·1, we have $\tau_P^*(2\Theta) \sim 2\Theta$ which implies that $\phi^*(2\Theta) \sim (\phi^{\sigma})^*(2\Theta)$ , hence the morphism induced by $|\phi^*(2\Theta)|$ is defined over K.

Now if $(X, \pi)$ corresponds to a Selmer element, then X everywhere locally has a point by Proposition 2·4, and hence S everywhere locally has a point. Since the Hasse principle holds for Brauer–Severi varieties by [ Reference Coray and ManoilCM96 , corollary 2·6], we know that S has a point over K and hence it is isomorphic to $\mathbb{P}^{3}$ by [ Reference Gille and SzamuelyGS06 , theorem 5·1·3].

We now make some observations and give some notation.

Remark 2·6. Let $\epsilon \in \text{Sel}^2(J)$ , and let $(J_{\epsilon}, \pi_{\epsilon})$ denote the 2-covering corresponding to $\epsilon$ . There exists an isomorphism $\phi_{\epsilon}$ defined over $\bar{K}$ such that $[2] \circ \phi_{\epsilon}=\pi_{\epsilon}$ . Then, by Proposition 2·5, we have the following commutative diagram:

(2·3)

The image of $J_{\epsilon}$ under the morphism induced by $|\phi_{\epsilon}^*(2\Theta)|$ is a surface, denoted by $\mathcal{K}_{\epsilon}$ , which we call the twisted Kummer surface corresponding to $\epsilon$ . Also $\psi_{\epsilon}$ is a linear isomorphism $\mathbb{P}^3 \rightarrow \mathbb{P}^3$ defined over $\bar{K}$ , and $\psi_{\epsilon}|_{\mathcal{K}_{\epsilon}}\,:\, \mathcal{K}_{\epsilon}\rightarrow \mathcal{K}$ is also an isomorphism over $\bar{K}$ . For simplicity of notation later, we may also refer to this map from $\mathcal{K}_{\epsilon}\rightarrow \mathcal{K}$ as $\psi_{\epsilon}$

Notation 2·7 Suppose $(J_{\epsilon}, \pi_{\epsilon})$ is the 2-covering of J corresponding to $\epsilon \in H^1(G_K, J[2])$ . The involution $[{-}1]\,:\, P \mapsto -P$ on J induces an involution $\iota_{\epsilon}$ on $J_{\epsilon}$ such that $\phi_{\epsilon} \circ \iota_{\epsilon}=[{-}1] \circ \phi_{\epsilon}$ , where $[2] \circ \phi_{\epsilon}=\pi_{\epsilon}$ . Moreover, the degree two morphism $J_{\epsilon} \xrightarrow{|\phi_{\epsilon}^*(2\Theta)|} \mathcal{K}_{\epsilon} \subset \mathbb{P}^3$ in (2·3) is precisely the quotient by $\iota_{\epsilon}$ and so an alternative definition of $\mathcal{K}_{\epsilon}$ is as the quotient of $J_{\epsilon}$ by $\iota_{\epsilon}$ . We call a function g on $J_{\epsilon}$ even if it is invariant under $\iota_{\epsilon}$ and odd if $g \circ \iota_{\epsilon}=-g$ .

2·5. Definition of the Cassels–Tate Pairing

There are four definitions of the Cassels–Tate pairing stated and proved equivalent in [ Reference Poonen and StollPS99 ]. In this paper we will only be using the homogeneous space definition of the Cassels–Tate pairing. We recall this definition now. Suppose $a, a^{\prime} $ . Via the polarization $\lambda$ , we get $a^{\prime} \mapsto b$ where Let X be the (locally trivial) principal homogeneous space defined over K representing a. Then $\text{Pic}^0(X_{\bar{K}})$ is canonically isomorphic as a $G_K$ -module to $\text{Pic}^0(J_{\bar{K}}) = J^{\vee}(\bar{K}).$ Therefore, represents an element in $H^1(G_K, \text{Pic}^0(X_{\bar{K}}))$ .

Now consider the exact sequence:

\begin{align*}0 \rightarrow \bar{K}(X)^*/\bar{K}^* \rightarrow \text{Div}^0(X_{\bar{K}}) \rightarrow \text{Pic}^0(X_{\bar{K}}) \rightarrow 0.\end{align*}

We can then map b to an element $b^{\prime} \in H^2(G_K, \bar{K}(X)^*/\bar{K}^*)$ using the long exact sequence associated to the short exact sequence above. Since K is a number field, we know that $H^3(G_K, \bar{K}^*)=0$ as proved in [ Reference Cassels and FrohlichCF67 , chapter VII, section 11·4]. So b ^′ has a lift $f^{\prime} \in H^2(G_K, \bar{K}(X)^*) $ via the long exact sequence induced by the short exact sequence $0 \rightarrow \bar{K}^* \rightarrow \bar{K}(X)^* \rightarrow \bar{K}(X)^*/\bar{K}^* \rightarrow 0\,:\,$

(2·4) \begin{equation}H^2(G_K, \bar{K}^*) \rightarrow H^2(G_K, \bar{K}(X)^*)\rightarrow H^2(G_K, \bar{K}(X)^*/\bar{K}^*) \rightarrow H^3(G_K, \bar{K}^*)=0.\end{equation}

Next we show that $f^{\prime}_v \in H^2(G_{K_v}, \bar{K_v}(X)^*)$ is the image of an element $c_v \in H^2(G_{K_v}, \bar{K_v}^*).$ This is because is locally trivial which implies its image b ^′ is locally trivial. Then the statement is true by the exactness of local version of sequence (2·4).

We then can define

\begin{align*}\langle a, b\rangle = \sum_v \text{inv}_v(c_v) \in \mathbb{Q}/\mathbb{Z}.\end{align*}

The Cassels–Tate pairing is defined by

\begin{align*}\langle a, a^{\prime} \rangle_{CT} := \langle a, \lambda(a^{\prime})\rangle.\end{align*}

We sometimes refer to $\text{inv}_v(c_v)$ above as the local Cassels–Tate pairing between $a, a^{\prime}$ for a place v of K. Note that the local Cassels–Tate pairing depends on the choice of $f^{\prime}\in H^2(G_K, \bar{K}(X)^*)$ . We make the following remarks that are useful for the computation of the Cassels–Tate pairing.

2·6. Explicit 2-coverings of J

Let $\omega_1, ..., \omega_6$ denote the 6 roots of f. Recall, as in Proposition 2·3, that the isomorphism classes of 2-coverings of J are parameterised by $H^1(G_K, J[2])$ . For the explicit computation of the Cassels–Tate pairing, we need the following result on the explicit 2-coverings of J corresponding to elements in $\text{Sel}^2(J)$ . We note that this theorem in fact works over any field of characteristic different from 2.

4·1. Computing the twist of the Kummer surface

We describe a practical method for computing a linear isomorphism $\psi_{\epsilon}\,:\, \mathbb{P}^3 \rightarrow \mathbb{P}^3$ corresponding to $\epsilon$ , which maps $\mathcal{K}_{\epsilon}\rightarrow \mathcal{K}$ . More explicitly, we need to compute $\psi_{\epsilon}$ such that $\psi_{\epsilon}(\psi_{\epsilon}^{-1})^{\sigma}$ is the action of translation by $\epsilon_{\sigma} \in J[2]$ on $\mathcal{K}$ and $(\sigma \mapsto \epsilon_{\sigma})$ is a cocycle representing $\epsilon$ . Since all points in J[2] are defined over K, the coboundaries in $B^1(G_K, J[2])$ are trivial. Therefore, these conditions determine $\psi_{\epsilon}$ uniquely up to precomposing by a linear change of coordinates defined over K.

For each $T \in J[2]$ , we have an explicit formula for $M_T \in \text{GL}_4(K)$ , given in [ Reference Cassels and FlynnCF96 , chapter 3, section 2], representing the action of translation by $T\in J[2]$ on the Kummer surface $\mathcal{K} \subset \mathbb{P}^3$ . It can be checked that $\{M_T, T\in J[2]\}$ form a basis of $\text{Mat}_4(K)$ . Let $c_P,c_Q,c_R,c_S\in K$ be such that $M_P^2 = c_PI, M_Q^2 = c_QI,M_R^2 = c_RI, \; \text{and} \; M_S^2 = c_SI.$ The explicit formulae for $c_P,c_Q,c_R,c_S$ can also be found in [ Reference Cassels and FlynnCF96 , chapter 3, section 2]. Moreover, by [ Reference Cassels and FlynnCF96 , chapter 3, section 3] and the Weil pairing relationship among the generators P, Q, R, S of J[2] specified by (3·1), we know that $[M_P, M_Q] = [M_R, M_S] = -I$ and the commutators of the other pairs are trivial.

Suppose $(a, b, c, d) \in (K^*/(K^*)^2)^4$ represents $\epsilon$ . Let $A \in \text{GL}_4(\bar{K})$ represent the linear isomorphism $\psi_{\epsilon}$ and let $M^{\prime}_T=A^{-1}M_TA \in \text{GL}_4(\bar{K})$ represent the action of T on the twisted Kummer $\mathcal{K}_{\epsilon}$ . It can be checked, see [ Reference YanYan , lemma 3·2·1] for details, that the set of matrices in $\text{PGL}_4(\bar{K})$ that commute with $M_T$ in $\text{PGL}_4(\bar{K})$ for all $T \in J[2]$ is $\{[M_T], T \in J[2]\}$ . This implies that any $B \in \text{GL}_4(\bar{K})$ such that $[M^{\prime}_T]=[B^{-1}M_TB] \in \text{PGL}_4(\bar{K})$ for all $T \in J[2]$ is equal to a multiple of $M_{T_0}$ composed with A for some $T_0 \in J[2]$ and so also represents $\psi_{\epsilon}$ . Hence, it will suffice to compute the matrices $M^{\prime}_T$ .

Consider $[M^{\prime}_T] \in \text{PGL}_4(\bar{K})$ and $\sigma \in G_K$ . We have

\begin{align*}[M^{\prime}_T]([M^{\prime}_T]^{-1})^{\sigma}=[A^{-1}M_TA(A^{-1})^{\sigma}M_T^{-1}A^{\sigma}]\in \text{PGL}_4(\bar{K}).\end{align*}

Recall that for each element $x \in K^*/(K^*)^2$ , we define $\tilde{x}_{\sigma} \in \{0, 1\}$ such that $({-}1)^{\tilde{x}_{\sigma}}=\sigma(\sqrt{x})/\sqrt{x}$ . Since $[A(A^{-1})^{\sigma}]=[M_P^{\tilde{b}_{\sigma}}M_Q^{\tilde{a}_{\sigma}}M_R^{\tilde{d}_{\sigma}}M_S^{\tilde{c}_{\sigma}}]$ and $M_T$ commutes with $M_P,M_Q,M_R,M_S$ in $\text{PGL}_4(K)$ , we get that $[M^{\prime}_T]([M^{\prime}_T]^{-1})^{\sigma} = I \in \text{PGL}_4(\bar{K})$ for all $\sigma \in G_K$ . Therefore we have $[M^{\prime}_T]$ is in $\text{PGL}_4(K)$ . This means that we can redefine $M^{\prime}_T=\lambda_TA^{-1}M_TA$ for some $\lambda_T\in \bar{K}$ such that $M^{\prime}_T\in \text{GL}_4(K)$ , by choosing a K defined representative.

Let $N_P=1/\sqrt{c_P}M_P, N_Q=1/\sqrt{c_Q}M_Q, N_R=1/\sqrt{c_R}M_R, N_S=1/\sqrt{c_S}M_S$ . Then $N_T^2=I$ for $T=P, Q, R, S$ . Define $N^{\prime}_T=A^{-1}N_TA\in \text{GL}_4(\bar{K})$ for $T=P, Q, R, S$ . We note that $N^{\prime}_T, M^{\prime}_T$ represent the same element in $\text{PGL}_4(\bar{K})$ and ${N^{\prime}}_T^2=I$ for each $T=P, Q, R, S$ . Suppose ${M^{\prime}}_P^2=\alpha_PI, {M^{\prime}}_Q^2=\alpha_QI, {M^{\prime}}_R^2=\alpha_RI, {M^{\prime}}_S^2=\alpha_SI.$ Then $N^{\prime}_T=1/\sqrt{\alpha_T}M^{\prime}_T$ for each $T=P, Q, R, S$ . Note that there might be some sign issues here but they will not affect the later computation. Since

\begin{align*}N^{\prime}_P({N^{\prime}}_P^{-1})^{\sigma}=A^{-1}N_PA(A^{-1})^{\sigma}(N_P^{-1})^{\sigma}A^{\sigma},\end{align*}

via $N_P=1/\sqrt{c_P}M_P$ , $N^{\prime}_P=1/\sqrt{\alpha_P}M^{\prime}_P$ with $M_P, M^{\prime}_P \in \text{GL}_4(K)$ , the fact that $[A(A^{-1})^{\sigma}]=[M_P^{\tilde{b}_{\sigma}}M_Q^{\tilde{a}_{\sigma}}M_R^{\tilde{d}_{\sigma}}M_S^{\tilde{c}_{\sigma}}]$ , and the fact that $M_P$ commutes with $M_P,M_R, M_S\in GL_4(K)$ and $M_P M_Q = -M_Q M_P$ , we can derive that

\begin{align*}\frac{\sigma(\sqrt{\alpha_P})}{\sqrt{\alpha_P}}=({-}1)^{{\tilde{a}_{\sigma}}}\frac{\sigma(\sqrt{c_P})}{\sqrt{c_P}}=\frac{\sigma(\sqrt{a})}{\sqrt{a}}\frac{\sigma(\sqrt{c_P})}{\sqrt{c_P}},\end{align*}

and similar equations hold for Q,R,S too.

This implies that $\alpha_P=c_P a$ up to squares in K and so via rescaling $M^{\prime}_P$ by elements in K, we have ${M^{\prime}}_P^2 = c_P aI$ . Similarly, ${M^{\prime}}_Q^2 = c_Q bI, {M^{\prime}}_R^2 = c_R cI, {M^{\prime}}_S^2 = c_S dI$ . We note that we also have $[M^{\prime}_P, M^{\prime}_Q] = [M^{\prime}_R, M^{\prime}_S] = -I$ and the commutators of the other pairs are trivial. This implies that

\begin{align*}\text{Mat}_4(K) \cong (c_P a, c_Q b) \otimes (c_R c, c_S d)\end{align*}

\begin{align*}M^{\prime}_P \mapsto i_1 \otimes 1, M^{\prime}_Q \mapsto j_1 \otimes 1, M^{\prime}_R \mapsto 1 \otimes i_2, M^{\prime}_S \mapsto 1\otimes j_2,\end{align*}

where $(c_P a, c_Q b)$ and $(c_R c, c_S d)$ are quaternion algebras with generators $i_1, j_1$ and $i_2, j_2$ respectively. In Section 5, we will interpret this isomorphism as saying that the image of $\epsilon$ via the obstruction map is trivial.

Let $A = (c_P a, c_Q b), B= (c_R c, c_S d)$ . By the argument above, we know $A \otimes B$ represents the trivial element in $\text{Br}(K)$ and an explicit isomorphism $A \otimes B \cong \text{Mat}_4(K)$ will give us the explicit matrices $M^{\prime}_P, M^{\prime}_Q, M^{\prime}_R, M^{\prime}_S$ we seek. Since the classes of A, B are in $\text{Br}[2]$ , we have A, B representing the same element in $\text{Br}(K)$ . This implies that $A \cong B$ over K, by Wedderburn’s Theorem. We have the following lemma.

Lemma 4·1. Consider a tensor product of two quaternion algebras $A \otimes B$ , where $A =(\alpha, \beta), \; B=(\gamma, \delta),$ with generators $i_1, j_1$ and $i_2, j_2$ respectively. Suppose there is an isomorphism $\psi\,:\, B \xrightarrow{\sim} A$ given by

\begin{align*} {}i_2 \mapsto a_1 \cdot 1 + b_1 \cdot i_1 + c_1 \cdot j_1 + d_1 \cdot i_1j_1,\\ {}j_2 \mapsto a_2 \cdot 1 + b_2 \cdot i_1 + c_2 \cdot j_1 + d_2 \cdot i_1j_1. {}\end{align*}

Then there is an explicit isomorphism

\begin{align*}A \otimes B \cong \text{Mat}_4(K)\end{align*}

given by

\begin{align*}i_1\otimes 1\mapsto M_{i_1} := \begin{bmatrix} {}0 & \quad \alpha & \quad 0 & \quad 0\\ {}1 & \quad 0 & \quad 0 & \quad 0\\ {}0 & \quad 0 & \quad 0 & \quad \alpha\\ {}0 & \quad 0 & \quad 1 & \quad 0\\ {}\end{bmatrix}.\end{align*}

\begin{align*}j_1 \otimes 1\mapsto M_{j_1} := \begin{bmatrix}0 & \quad 0 & \quad \beta & \quad 0\\0 & \quad 0 & \quad 0 & \quad -\beta\\1 & \quad 0 & \quad 0 & \quad 0\\0 & \quad -1 & \quad 0 & \quad 0\\\end{bmatrix}\end{align*}

\begin{align*}1 \otimes i_2 \mapsto M_{i_2} := \begin{bmatrix}a_1 & \quad b_1 \cdot \alpha & \quad c_1 \cdot \beta & \quad - d_1 \cdot \alpha\beta\\b_1 & \quad a_1 & \quad -d_1 \cdot \beta & \quad c_1 \cdot \beta\\c_1 & \quad d_1 \cdot \alpha & \quad a_1 & \quad - b_1 \cdot \alpha\\d_1 & \quad c_1 & \quad - b_1 & \quad a_1\\\end{bmatrix}\end{align*}

\begin{align*}1 \otimes j_2 \mapsto M_{j_2} := \begin{bmatrix}a_2 & \quad b_2 \cdot \alpha & \quad c_2 \cdot \beta & \quad - d_2 \cdot \alpha\beta\\b_2 & \quad a_2 & \quad -d_2 \cdot \beta & \quad c_2 \cdot \beta\\c_2 & \quad d_2 \cdot \alpha & \quad a_2 & \quad - b_2 \cdot \alpha\\d_2 & \quad c_2 & \quad - b_2 & \quad a_2\\\end{bmatrix}\end{align*}

Proof. We have that $A \otimes A^{op}$ is isomorphic to a matrix algebra. More specifically, $A \otimes A^{op} \cong \text{End}_{K}(A)$ via $u \otimes v \mapsto (x \mapsto uxv)$ , which makes $A \otimes A^{op} \cong \text{Mat}_4(K)$ after picking a basis for A. Hence,

\begin{align*}A \otimes B^{op} &\cong \text{Mat}_4(K)\\u \otimes v & \mapsto (x \mapsto ux\psi(v)).\end{align*}

More explicitly, fixing the basis of A to be $\{1, i_1, j_1, i_1j_1\}$ , the isomorphism is as given in the statement of the lemma.

By taking $A = (c_P a, c_Q b), B= (c_R c, c_S d)$ in Lemma 4·1, we know that the matrices $M^{\prime}_P, M^{\prime}_Q, M^{\prime}_R, M^{\prime}_S$ and $M_{i_1}, M_{j_1}, M_{i_2}, M_{j_2}$ are equal up to conjugation by a matrix $C \in \text{GL}_4(K)$ via the Noether Skolem Theorem. After a change of coordinates for $\mathcal{K}_{\epsilon}\subset \mathbb{P}^3$ according to C, we have that $M^{\prime}_P, M^{\prime}_Q, M^{\prime}_R, M^{\prime}_S$ are equal to $M_{i_1}, M_{j_1}, M_{i_2}, M_{j_2}$ . Lemma 4·1 therefore reduces the problem of computing the $M^{\prime}_T$ to that of computing an isomorphism between the two quaternion algebras. See [ Reference YanYan , corollary 4·2·3] for the description of an explicit algorithm. Finally we solve for a matrix A such that $M^{\prime}_T=\lambda_TA^{-1}M_TA$ for some $\lambda_T \in \bar{K}$ and $T=P, Q, R, S$ by linear algebra.

4·2. Explicit computation of D

In this section, we explain a method for computing the K-rational divisor D on $J_{\epsilon}$ representing the divisor class $\phi_{\epsilon}^*(2\Theta)$ . The idea is to compute it via the commutative diagram (2·3) in Remark 2·6.

By Theorem 2·9, there is an explicit isomorphism $\phi_\epsilon \,:\, J_\epsilon \to J$ defined over $\overline{K}$ given by a linear change of coordinates on $\mathbb{P}^{15}$ . We write $u_0, ..., u_9, v_1, ..., v_6$ for the coordinates on the ambient space of $J_{\epsilon} \subset \mathbb{P}^{15}$ and write $k_{11}, k_{12}, ..., k_{44}, b_1, ..., b_6$ for the coordinates on the ambient space of $J \subset \mathbb{P}^{15}$ . By the same theorem, $\phi_{\epsilon}$ is represented by a block diagonal matrix consisting of a block of size 10 corresponding to the even basis elements and a block of size 6 corresponding to the odd basis elements. Following Section 4·1, we can compute an explicit isomorphism $\psi_{\epsilon} \,:\, \mathbb{P}^3 \rightarrow \mathbb{P}^3$ , that maps $\mathcal{K}_{\epsilon} \rightarrow \mathcal{K}$ , where $\mathcal{K}_{\epsilon}$ is the twisted Kummer surface corresponding to $\epsilon$ . We write $k^{\prime}_1, ..., k^{\prime}_4$ for the coordinates on the ambient space of $\mathcal{K}_{\epsilon} \subset \mathbb{P}^3$ . Recall that since all points in J[2] are defined over K, all coboundaries in $B^1(G_K, J[2])$ are trivial. So, we have that $\phi_{\epsilon}(\phi_{\epsilon}^{-1})^{\sigma}$ and $\psi_{\epsilon} (\psi_{\epsilon}^{-1})^{\sigma}$ both give the action of translation by some $\epsilon_{\sigma} \in J[2]$ such that $(\sigma \mapsto \epsilon_{\sigma})$ represents $\epsilon \in \text{Sel}^2(J)$ .

Define $k^{\prime}_{ij} =k^{\prime}_i k^{\prime}_j$ . The isomorphism $\psi_{\epsilon}\,:\, \mathbb{P}_{k^{\prime}_i}^3 \rightarrow \mathbb{P}_{k_i}^3$ , induces a natural isomorphism $\tilde{\psi_{\epsilon}}\,:\, \mathbb{P}^9_{k^{\prime}_{ij}} \rightarrow \mathbb{P}^9_{k_{ij}}$ . More explicitly, suppose $\psi_{\epsilon}$ is represented by the $4 \times 4$ matrix A where $\left(k^{\prime}_1\,:\,..., k^{\prime}_4\right) \mapsto \left(\sum_{i=1}^4A_{1i}k^{\prime}_i\,:\,...\,:\,\sum_{i=1}^4A_{4i}k^{\prime}_i\right)$ . Then $\tilde{\psi_{\epsilon}}\,:\, \mathbb{P}^9_{k^{\prime}_{ij}} \rightarrow \mathbb{P}^9_{k_{ij}}$ is given by $\left(k^{\prime}_{11}\,:\, k^{\prime}_{12}\,:\,... \,:\,k^{\prime}_{44}\right)\mapsto \left(\sum_{i, j=1}^4A_{1i}A_{1j}k^{\prime}_{ij}\,:\,\sum_{i, j=1}^4A_{1i}A_{2j}k^{\prime}_{ij}\,:\,...\,:\,\sum_{i, j=1}^4A_{4i}A_{4j}k^{\prime}_{ij}\right)$ .

On the other hand, the isomorphism $\phi_{\epsilon}\,:\, \mathbb{P}^{15}_{\{u_i, v_i\}} \rightarrow \mathbb{P}^{15}_{\{k_{ij}, b_i\}}$ induces a natural isomorphism $\tilde{\phi_{\epsilon}}\,:\, \mathbb{P}^9_{u_i} \rightarrow \mathbb{P}^9_{k_{ij}}$ represented by the $10\times 10$ block of the matrix representing $\phi_{\epsilon}$ . Since $\phi_{\epsilon}(\phi_{\epsilon}^{-1})^{\sigma}$ and $\psi_{\epsilon} (\psi_{\epsilon}^{-1})^{\sigma}$ both give the action of translation by some $\epsilon_{\sigma} \in J[2]$ , we get $\tilde{\phi_{\epsilon}}(\tilde{\phi_{\epsilon}}^{-1})^{\sigma}=\tilde{\psi_{\epsilon}}(\tilde{\phi_{\epsilon}}^{-1})^{\sigma}$ . Therefore, $\tilde{\psi_{\epsilon}}^{-1}\tilde{\phi_{\epsilon}}$ is defined over K.

Now let $v\,:\, \mathbb{P}^3 \rightarrow \mathbb{P}^9$ be the Veronese embedding that sends $(x_0\,:\,x_1\,:\,x_2\,:\,x_3)$ to $(x_0^2\,:\, x_0x_1\,:\,...\,:\,x_3^2)$ and let $\mathcal{K}_{\mathbb{P}^9_{k_{ij}}}$ and $\mathcal{K}_{\epsilon, \mathbb{P}_{k^{\prime}_{ij}}^9}$ be the corresponding image of $\mathcal{K}$ and $\mathcal{K}_{\epsilon}$ in $\mathbb{P}^9_{k_{ij}}$ and $\mathbb{P}^9_{k^{\prime}_{ij}}$ via the Veronese embedding. Furthermore, let $\mathcal{K}_{\epsilon, \mathbb{P}_{u_i}^9}$ be the image of $J_{\epsilon}$ via the projection map $\mathbb{P}^{15}_{u_i,v_i}\rightarrow \mathbb{P}^9_{u_i}$ . The various maps between projective spaces are summarised in the following commutative diagram.

(4·1)

Restricting these maps to J, $\mathcal{K}$ and their twists (and using the same names for these restricted maps) we obtain the following commutative diagram that decomposes the standard commutative diagram (2·3).

(4·2)

Here $g_1\,:\,(k_{11}\,:\,...\,:\,k_{44}) \mapsto (k_{11}\,:\, ...\,:\, k_{14})$ and $g_2\,:\,(k^{\prime}_{11}\,:\,...\,:\,k^{\prime}_{44}) \mapsto (k^{\prime}_{11}\,:\, ...\,:\, k^{\prime}_{14})$ are the projection maps, which are the one sided inverse of Veronese embedding. The composition of the morphisms on the bottom row gives the standard morphism $J \xrightarrow{|2\Theta|} \mathcal{K}\subset \mathbb{P}^3$ and the composition of the morphisms on the top row gives $J_{\epsilon} \xrightarrow{|\phi_{\epsilon}^*(2\Theta)|} \mathcal{K}_{\epsilon}\subset \mathbb{P}^3$ .

Let D be the pull back on $J_{\epsilon}$ via the horizontal map, as in diagram (4·2), $J_{\epsilon} \rightarrow \mathcal{K}_{\epsilon}$ of the hyperplane section given by $k^{\prime}_1=0$ . This implies that D is a K-rational divisor on $J_{\epsilon}$ representing the class of $\phi_{\epsilon}^*(2\Theta)$ . Moreover, the pull back on $J_{\epsilon}$ via the map $J_{\epsilon} \rightarrow \mathcal{K}_{\epsilon, \mathbb{P}_{k^{\prime}_{ij}}^9}$ as in (4·2) of the hyperplane section given by $k^{\prime}_{11}=0$ is 2D.

4·3. Explicit computation of $D_P, D_Q, D_R, D_S$

In this section, we explain how to compute the K-rational divisors $D_P, D_Q, D_R, D_S$ defined in Remark 3·2. More explicitly, for $T \in J[2]$ , we give a method for computing a K-rational divisor $D_T$ on $J_{\epsilon}$ representing the divisor class of $\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))$ for some $T_1$ on J such that $2T_1=T$ . Recall that we assume all points in J[2] are defined over K and we have an explicit isomorphism $\phi_{\epsilon}\,:\, J_{\epsilon} \rightarrow J$ such that $(J_{\epsilon}, [2]\circ \phi_{\epsilon})$ is the 2-covering of J corresponding to $\epsilon\in \text{Sel}^2(J)$ . Recall $\delta\,:\, J(K) \rightarrow H^1(G_K, J[2])$ in (2·1). We first prove the following lemma.

Proof. Using the same argument as in the proof of Lemma 3·1(i), we know that $(J_{\epsilon}, [2] \circ \tau_{T_1} \circ \phi_{\epsilon})$ is the 2-covering of J corresponding to $\epsilon+\delta(T) \in H^1(G_K, J[2])$ . Since all points in J[2] are defined over K, we have $\tau_{T_1} \circ \phi_{\epsilon} \circ ((\tau_{T_1} \circ \phi_{\epsilon} )^{-1})^{\sigma} = \phi_{\epsilon+ \delta(T)} \circ (\phi_{\epsilon+ \delta(T)}^{-1})^{\sigma}$ , as required.

Let $T \in J(K)$ with $2T_1 = T$ . Consider the commutative diagram below which is formed of two copies of the standard diagram (2·3). In this diagram all the horizontal maps are defined over K. By Lemma 4·2, the composition of the three vertical maps on the left is defined over K, even though the individual maps are only defined over $\overline{K}$ . The composition of the four wavy arrows is therefore defined over K. Pulling back a hyperplane section on $\mathcal{K}_{\epsilon + \delta(T)} \subset \mathbb{P}^3$ via this composition gives a K-rational divisor $D_T$ on $J_\epsilon$ representing the divisor class $\phi_{\epsilon}^{*}(\tau_{T_1}^{*}(2 \Theta))$ . If we further assume that $T \in J[2]$ then the composition of the three vertical maps on the left is given by a change of coordinates on $\mathbb{P}^{15}$ and so by a $16 \times 16$ matrix defined over K.

(4·3)

The bottom horizontal morphism $J_{\epsilon+\delta(T)} \xrightarrow{|\phi^*_{\epsilon+ \delta(T)}(2\Theta)|} \mathcal{K}_{\epsilon+\delta(T)} \subset \mathbb{P}^3$ can be explicitly computed using the algorithm in Section 4·2 with the Selmer element $\epsilon$ replaced by $\epsilon+\delta(T)$ . Also, by Theorem 2·9, we have explicit formulae for $\phi_{\epsilon}$ and $\phi_{\epsilon+ \delta(T)}.$ Hence, to explicitly compute $D_T$ , we need to compute a pullback by $\tau_{T_1}$ , for some $T_1$ such that $2T_1=T$ .

Since we need to apply the above argument for T running over our basis P, Q, R, S for J[2], it would suffice to compute the translation maps $\tau_{T_1}\,:\, J \rightarrow J $ when $T_1 \in J[4]$ . Each of these translation maps is given by a change of coordinates on $\mathbb{P}^{15}$ and so by a $16 \times 16$ matrix. We show how to compute a $10 \times 16$ submatrix in the following proposition. We then explain below why this is sufficient for our purposes.

Proposition 4·3. Suppose $T_1 \in J[4]$ . Given the coordinates of $T_1 \in J \subset \mathbb{P}^{15}_{\{k_{ij},b_i\}}$ , we can compute a $10 \times 16$ matrix representing the composition of maps (say $\Psi$ ) in the second row of the following commutative diagram.

(4·4)

Proof. Let $T =2T_1 \in J[2]$ . Recall that we let $M_T$ denote the action of translation by T on $\mathcal{K} \subset \mathbb{P}^3$ . Then for any $P \in J$ , we have $k_i(P+T)=\sum_{j=1}^4(M_T)_{ij}k_j(P)$ projectively as a vector of length 4, and projectively as a vector of length 10, $k_{ij}(P+T_1)$ is equal to

\begin{align*}k_i(P+T_1)k_j(P+T_1)=k_i(P+T_1) \sum_{l=1}^4(M_T)_{jl}k_l(P-T_1)=\sum_{l=1}^4(M_T)_{jl}k_l(P-T_1)k_i(P+T_1).\end{align*}

By [ Reference FlynnFly93 , theorem 3.2], there exists a $4 \times 4$ matrix of bilinear forms $\phi_{ij}(P, T_1)$ that is projectively equal to the matrix $k_i(P-T_1)k_j(P+T_1)$ . Moreover explicit formulae are given for these bilinear forms. Since we have an explicit formula for $M_T$ in in [ Reference Cassels and FlynnCF96 , chapter 3, section 2], we can compute a $10\times16$ matrix representing $\Psi$ as claimed in the statement of the proposition.

Consider $T \in J[2]$ with $T_1 \in J[4]$ such that $2T_1=T$ . We follow the discussion in Section 4·2 with $\epsilon$ replaced by $\epsilon+\delta(T)$ . This gives a diagram analogous to (4·2). Let $k^{\prime}_{1, T}, ..., k^{\prime}_{4, T}$ be the coordinates on the ambient space of $\mathcal{K}_{\epsilon+\delta(T)}\subset \mathbb{P}^3$ and let $u_{0, T}, ..., u_{9, T}, v_{1, T}, ..., v_{6, T}$ be the coordinates on the ambient space of $J_{\epsilon+\delta(T)}\subset \mathbb{P}^{15}$ . Let $k^{\prime}_{ij, T}=k^{\prime}_{i, T} k^{\prime}_{j, T} $ . Decomposing the lower half of the diagram (4·3) gives the commutative diagram below.

(4·5)

Recall Proposition 4·3 explains how $\Psi$ can be explicitly computed and the composition of the wavy arrows in (4·5) is defined over K by Lemma 4·2. Let $D_T$ be the pull back on $J_{\epsilon}$ via the wavy arrows in (4·5) of the hyperplane section given by $k^{\prime}_{1, T} =0$ . This implies that $D_T$ is a K-rational divisor on $J_{\epsilon}$ representing the class of $\phi_{\epsilon}^*(\tau_{T_1}^*(2\Theta))$ . Moreover the pull back on $J_\epsilon$ via the composition of the first three wavy arrows in (4·5) of the hyperplane section given by $k^{\prime}_{11,T} = 0$ is $2 D_T$ .

We now apply the above discussion with $T=P, Q, R, S$ and obtain the divisors $D_P, D_Q, D_R, D_S$ on $J_{\epsilon}$ described in Remark 3·2 as required.