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Now let $k$ be a finite field. If $E$ is an elliptic curve over $k$ then $End(E)\otimes\mathbf{Q}$ is either an imaginary quadratic field or a definite quaternion algebra over $\mathbf{Q}$. Warning: $E$ may be supersingular even if $End(E)\otimes\mathbf{Q}$ is an imaginary quadratic field; this means that not all endomorphisms of $E$ are defined over $k$...
If $A$ is not simple then it is isogenous to a product $E_1 \times E_2$ of elliptic curves $E_1$ and $E_2$, and the Picard numbers of $A$ and $E_1\times E_2$ do coincide. If $E_1$ is not isogenous to $E_2$ then $\rho(E_1 \times E_2)=2,$ which is not the case. Therefore, $A$ is isogenous to a square $E^2$ of an elliptic curve $E=E_1$. It is known that $End(E)... 3 There are alternative definitions of the Cassels-Tate pairing in articles of Stoll, e.g. http://www.mathe2.uni-bayreuth.de/stoll/papers/yoga.pdf. 6 I'm a bit late to the party, but since these question are clearly still getting views, I'll answer the second question a bit. Given a nice category, one can form the pointed category$C$, and consider functors$F:C\to Ab$. There are cannonical maps$\beta:F(X_0\times\dots \times X_n)\to \prod_i F(X_0\times \dots \times X_{i-1}\times X_{i+1}\times \dots \...