Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Yes, exactly. I wouldn't know how to produce equations with coefficients $\mathbb Q$ for $E \times E'$ (or to prove that it is defined over $\mathbb Q$ in some other way).
Take a CM $\mathbb Q$-curve $E$ such that $\mathbb Q(j(E))$ is a quadratic extension of $\mathbb Q$. Let $E'$ be its Galois conjugate. Then $A = E\times E'$ is a singular abelian surface. You mean that $A$ is a good candidate for what I am looking for? How could one prove that $A$ is defined over $\mathbb Q$?
One can define $f_*$ on $N_1(X)$, the (real) vector space of 1-cycles up to numerical equivalence, which is finite-dimensional. Surjectivity then implies injectivity.
