Computing the Cassels-Tate Pairing for Genus Two Jacobians with Rational Two Torsion Points

I express my sincere and deepest gratitute to my PhD supervisor, Dr. Tom Fisher, for his patient guidance and insightful comments at every stage during my research.

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 $n^{th}$ 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:

$$\mathcal{C}:y^{2}=f(x)=f_{6}x^{6}+f_{5}x^{5}+f_{4}x^{4}+f_{3}x^{3}+f_{2}x^{2}+f_{1}x+f_{0},$$

where $f_{i}\in K$, $f_{6}\neq 0$ and the discriminant $\triangle(f)\neq 0$, which implies that $f$ has distinct roots in $\bar{K}$.

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 $\mathcal{O}_{J}$. 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 $\omega_{1},...,\omega_{6}$. 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 [thebook, 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

$$e_{2}(P,Q)=(-1)^{|\{P_{1},P_{2}\}\cap\{Q_{1},Q_{2}\}|}.$$

The theta divisor, denoted by $\Theta$, is defined to be the divisor on $J$ that corresponds to the divisor $\{P\}\times\mathcal{C}+\mathcal{C}\times\{P\}$ on $\mathcal{C}\times\mathcal{C}$ under the birational morphism $\text{Sym}^{2}\mathcal{C}\rightarrow J$, for some Weierstrass point $P\in\mathcal{C}$. The Jacobian variety $J$ is principally polarized abelian variety via $\lambda:J\rightarrow J^{\vee}$ sending $P$ to $[\tau_{P}^{*}\Theta-\Theta].$

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 2 on $J$ and these map to the 16 nodal singular points of $\mathcal{K}$ (the nodes). General theory, as in [abelianvarieties, Theorem 11.1], [theta, page 150], shows that the linear system of $n\Theta$ of $J$ has dimension $n^{2}$. Moreover, $|2\Theta|$ is base point free and $|4\Theta|$ is very ample.

Denote a generic point on the Jacobian $J$ of $\mathcal{C}$ by $\{(x,y),(u,v)\}$. Then, following [thebook, Chapter 3, Section 1], the morphism from $J$ to $\mathbb{P}^{3}$ is given by

$$k_{1}=1,k_{2}=(x+u),k_{3}=xu,k_{4}=\beta_{0},$$

where

$$\beta_{0}=\frac{F_{0}(x,u)-2yv}{(x-u)^{2}}$$

with $F_{0}(x,u)=2f_{0}+f_{1}(x+u)+2f_{2}(xu)+f_{3}(x+u)(xu)+2f_{4}(xu)^{2}+f_{5}(x+u)(xu)^{2}+2f_{6}(xu)^{3}.$

We denote the above morphism by $J\xrightarrow{|2\Theta|}\mathcal{K}\subset\mathbb{P}^{3}$ and it maps $\mathcal{O}_{J}$ to $(0:0:0:1)$. It is known that its image in $\mathbb{P}^{3}$ is precisely the Kummer surface $\mathcal{K}$ and is given by the vanishing of the quartic $G(k_{1},k_{2},k_{3},k_{4})$ with explicit formula given in [thebook, Chapter 3, Section 1]. Therefore, the Kummer surface $\mathcal{K}\subset\mathbb{P}^{3}_{k_{i}}$ is defined by $G(k_{1},k_{2},k_{3},k_{4})=0.$

We now look at the embedding of $J$ in $\mathbb{P}^{15}$ induced by $|4\Theta|$. Let $k_{ij}=k_{i}k_{j}$, 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 $\mathcal{L}(2\Theta^{+}+2\Theta^{-})$. The six odd basis elements in $\mathcal{L}(2\Theta^{+}+2\Theta^{-})$ are given explicitly in [explicittwist, 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 $J$.

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

$$\delta:J(K)\rightarrow H^{1}(G_{K},J[2]).$$

(2.1)

The following two propositions are proved in [explicittwist].

We also state and prove the following proposition which is useful for the computation of the Cassels-Tate pairing later in Sections LABEL:sec:explicit-computation-of-d and LABEL:sec:explicit-computation-of-dp-dq-dr-ds. A Brauer-Severi variety is a variety that is isomorphic to a projective space over $\bar{K}$.

We now make some observations and give some notation.

There are four equivalent definitions of the Cassels-Tate pairing stated and proved in [poonenstoll]. In this paper we will only be using the homogeneous space definition of the Cassels-Tate pairing. Suppose $a,a^{\prime}\in\Sh(J)$. Via the polarization $\lambda$, we get $a^{\prime}\mapsto b$ where $b\in\Sh(J^{\vee}).$ 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, $b\in\Sh(J^{\vee})\subset H^{1}(G_{K},J^{\vee})$ represents an element in $H^{1}(G_{K},\text{Pic}^{0}(X_{\bar{K}}))$.

Now consider the exact sequence:

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

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 $H^{3}(G_{K},\bar{K}^{*})=0$, $b^{\prime}$ has a lift $f^{\prime}\in H^{2}(G_{K},\bar{K}(X)^{*})$ via the long exact sequence induced by the short exact sequence $0\rightarrow\bar{K}^{*}\rightarrow\bar{K}(X)^{*}\rightarrow\bar{K}(X)^{*}/\bar{K}^{*}\rightarrow 0:$

$$H^{2}(G_{K},\bar{K}^{*})\rightarrow H^{2}(G_{K},\bar{K}(X)^{*})\rightarrow H^{2}(G_{K},\bar{K}(X)^{*}/\bar{K}^{*})\rightarrow H^{3}(G_{K},\bar{K}^{*})=0.$$

(2.4)

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 $b\in\Sh(J^{\vee})$ is locally trivial which implies its image $b^{\prime}$ is locally trivial. Then the statement is true by the exactness of local version of sequence (2.4).

We then can define

$$\langle a,b\rangle=\sum_{v}\text{inv}_{v}(c_{v})\in\mathbb{Q}/\mathbb{Z}.$$

The Cassels-Tate pairing $\Sh(J)\times\Sh(J)\rightarrow\mathbb{Q}/\mathbb{Z}$ is defined by

$$\langle a,a^{\prime}\rangle_{CT}:=\langle a,\lambda(a^{\prime})\rangle.$$

We sometimes refer to $\text{inv}_{v}(c_{v})$ above as the local Cassels-Tate pairing between $a,a^{\prime}\in\Sh(J)$ 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 for the Cassels-Tate pairing.

Let $\Omega$ represent the set of 6 roots of $f$, denoted by $\omega_{1},...,\omega_{6}$. Recall, as in Proposition 2.3, the isomorphism classes of 2-coverings of $J$ are parameterized 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.