  We prove effective equidistribution of primitive rational points and of
primitive rational points defined by monomials along long horocycle orbits in
products of the torus and the modular surface. This answers a question posed in
joint work by the first and the last named author with Shahar Mozes and Uri
Shapira. Under certain congruence conditions we prove the joint
equidistribution of conjugate rational points in the two-torus and the modular
surface.
