Character varieties, A–polynomials and the AJ conjecture

We establish some facts about the behavior of the rational-geometric subvariety of the SL₂() or PSL₂() character variety of a hyperbolic knot manifold under the restriction map to the SL₂() or PSL₂() character variety of the boundary torus, and use the results to get some properties about the A-polynomials and to prove the AJ conjecture for certain class of knots in S³ including in particular any $2$-bridge knot over which the double branched cover of S³ is a lens space of prime order.