Solvable base change and Rankin-Selberg convolutions

Let π and π′ be unitary automorphic cuspidal representations of GLn(AE) and GLm(AF), and let E and F be solvable Galois extensions of Q of degrees l and l′, respectively. Using the fact that the automorphic induction and base change maps exist for E and F, and assuming an invariance condition under the actions of the Galois groups, we attach to the pair (π, π′) a Rankin-Selberg L-function L(s, π ×E,F xxxxxx) for which we prove a prime number theorem. This gives a method for comparing two representations that could be defined over completely different extensions, and the main results give a measure of how many cuspidal components the two representations π and π′ have in common when automorphically induced down to the rational numbers. The proof uses the structure of the Galois group of the composite extension EF and the character groups attached to the fields via class field theory. The second main theorem also gives an indication of when the base change of π up to the composite extension EF remains cuspidal.