On the image and fibres of solvable base change

Abstract

Our aim is to describe the image and fibres of the base change lift constructed by Langlands, for solvable extensions of number fields. A particular instance of the principle of functoriality enunciated by Lang- lands, is the base change map relating automorphic forms on GL2(Ak) to forms on GL2(AK), for an extension K/k of number fields. Using the trace formula and following earlier workby Saito and Shintani, Langlands in (L), established the existence of the base change lift BCK/k sending cuspidal automorphic rep- resentations of GL2(Ak) to automorphic representations of GL2(AK), provided K/k is a cyclic extension of prime degree. Langlands further characterized the image and the fibres of the base change map BCK/k. Given an automorphism σ of K, and an automorphic representation Π of GL2(AK), let Π σ denote the automorphic representation of GL2(AK) defined by Π σ (g )=Π (σ −1 (g)). The descent property for invariant, automorphic representations established by Langlands is that if Π is a cuspidal automorphic representation of GL2(AK), which is invariant with respect to the action of the Galois group G(K/k )o f a cyclic extension K/k of prime degree, then Π lies in the image of the base change map BCK/k from automorphic representations on GL2(Ak) to automor- phic representations on GL2(AK). Further if π1 and π2 are distinct, cuspidal automorphic representations of GLn(Ak) which base change to Π, then