Building some kernel of functoriality? The case of unramified automorphic induction from GL\(_1\) to GL\(_2\)
Let \(H\) denote the group \(\text{GL}_2\) over a global field \(F\), let \(E\) be a quadratic extension of \(F\) and let \(G = \text{Res}_{E/F}\text{GL}_1\). In the present article the author realizes the well-known transfer of automorphic forms on \(G(F_{\mathbb A}) = E_{\mathbb A}^{\times}\) to automorphic forms on \(H(F_{\mathbb A})\) by a new method. More precisely, it is the unramified case to which the method is applied: the automorphic forms are eigenfunctions of the spherical Hecke algebra at all places, \(E/F\) is an everywhere unramified extension of function fields. The transfer is achieved by the construction of kernels, which are spherical functions on the product of \(E^{\times}\backslash E_{\mathbb A}^{\times}\) and \(\text{GL}_2(F)\backslash \text{GL}_2(F_{\mathbb A})\). Local kernels are defined for all places using Whittaker functions. For the construction of a global kernel a sum over \(E^{\times}\times F^{\times}\) of the product of local kernels over all places is considered. In order to make it invariant under \(\text{GL}_2(F)\) complementary terms must be added. Here the Poisson formula on \(E_{\mathbb A}\) is used. The proofs are elementary computations at all places.
scientific article; zbMATH DE number 5703822