@reuns Here we work over an algebraically closed field $k$, and you can choose a specific descent to $\mathbb F_{p^2}$ then the result is standard, for example see mathoverflow.net/questions/18982/….

@R.vanDobbendeBruyn Thanks, so the case $X$ is a surface is also true as stably birational smooth projective curves must be isomorphic. For the case $X$ is a threefold, is there any restriction for two smooth projective surfaces to be embedded in a common threefold ?

So for Bianchi modular forms corresponding to QM abelian surfaces, maybe the answer is known.

Thank you. You're right, one shall mean mod $p$, otherwise the question is false for your examples. I notice in a recent paper arxiv.org/abs/1909.07473 where they prove every 2-dimensional abelian scheme over $O_K$ ($K$ can be any number field) admits infinitely many places of geometrically non simple reductions. So it seems interesting that one can't say much for elliptic curves over imaginary quadratic fields.

Thank you. If I have a map between proper varieties with same dimension n, then what is the induced map on the set of irreducible components under identification with $H^2n$? Is it just taking the preimage?

Can you give some details on why it's true for semi-abelian varieties (over positive characteristic fields)?

Thank you! But I think it's true for torus. Is it true for any commutative groups?