Serre's Conjecture and Base Change for GL(2)

In this paper, for an odd cyclic totally real extension F/Q, assuming Serre's modularity conjecture of 2-dimension odd mod p Galois representations, we give an elementary proof of the Langlands base change from a space of automorphic forms on the multiplicative group of a definite quaternion algebra B(/Q) to the corresponding space on the multiplicative group of B(F) = B circle times(Q) F (under some mild assumptions). More generally, for a general totally real Galois extension, we state a conjecture describing the action of Gal(F/Q) on the 0-dimensional automorphic variety of B(F)(x) which implies the existence of base-change relative to F/Q.