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Building some kernel of functoriality? The case of unramified automorphic induction from GL$$_1$$ to GL$$_2$$. (Construire un noyau de la fonctorialité? Le cas de l’induction automorphe sans ramification de GL$$_1$$ à GL$$_2$$.) (French. English summary) Zbl 1221.11221
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.

##### MSC:
 11R39 Langlands-Weil conjectures, nonabelian class field theory 11F03 Modular and automorphic functions
##### Keywords:
Langlands’ functoriality; automorphic induction
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##### References:
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