Let $K$ be a number field and let $A$ be an abelian variety over $K$ (I'm mostly interested in the case that $A$ is an elliptic curve). We use $v$ to denote places of $K$ and we write $H^i(k, A)$ for the Galois Cohomology $H^i(Gal(k^{al}/k), A(K^{al}))$ where $k=K$ or $K_v$ and $G=Gal(K^{al}/k)$.

The Tate-Shafarevich group $Ш(A/k)$ is defined to be the kernel of the map

\begin{align} H^1(K, A) \to \bigoplus_{v} H^1(K_v, A_v). \end{align}

There is a pairing \begin{align} Ш(A/k) \times Ш(A^{\vee}/k) \to \mathbb{Q}/\mathbb{Z} \end{align}

The construction (which can be found in detail in the reference) goes roughly as follows: For $a \in Ш(A/k) $ we get an $A$-torsor $V_a$ representing $a$. Then there is an isomorphism of $G$-modules $A^{\vee} \to \operatorname{Pic}^0(V_a)$. Let $Q_a=K(V_a)^{\ast}/K^{\ast}$ then there is an exact sequence \begin{align} 0 \to Q \to \operatorname{Div}^0(V_a) \to \operatorname{Pic}^0(A) \to 0 \end{align} which induces a map $Ш(A^{\vee}/k) \to H^2(G,Q_a)$. By some global class field theory there is a map $\phi_a:B \to \mathbb{Q}/\mathbb{Z}$ where $B \subset H^2(G,Q_a)$ and $\operatorname{im}(Ш(A^{\vee}/k)) \subset B$.

I havent been able to show that the pairing is additive in the first coordinate. In other words, if $a, b \in Ш(A/k)$ we get three maps \begin{align} \phi_{a,b,a+b}:Ш(A^{\vee}/k) \to \mathbb{Q}/\mathbb{Z}, \end{align} and I want to show that \begin{align} \phi_a+\phi_b=\phi_{a+b}. \end{align} I have tried looking at the product of torsors in the Weil–Châtelet group but I get stuck comparing $H^2(G, Q_a),H^2(G, Q_b),H^2(G, Q_{a+b})$.

Reference: Milne - 'Arithmetic Duality Theorems' pages 80-81 (http://math.stanford.edu/~conrad/BSDseminar/refs/MilneADT.pdf)


There are alternative definitions of the Cassels-Tate pairing in articles of Stoll, e.g. http://www.mathe2.uni-bayreuth.de/stoll/papers/yoga.pdf.


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