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>)



Bilinearity of the Cassels-Tate pairing
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>.


