Building some kernel of functoriality? The case of unramified automorphic induction from GL1 to GL2

The purpose of this work is to present a new adelic method for realising Langlands' functoriality principle in the case of unramified automorphic induction from GL1 to GL2 on function fields. A kernel of functoriality is built on the product of the adelic groups GL1 and GL2. It is some kind of "family" local version of the construction for global Whittaker models, which is classically used in the "converse theorems" of Weil and Piatetski-Shapiro. Essential use is made of the Fourier transform on adele groups and of the Poisson formula, just as in Tate's thesis.