Building some kernel of functoriality? The case of unramified automorphic induction from GL\(_1\) to GL\(_2\)
type: Item
instance of: scholarly article
title: Building some kernel of functoriality? The case of unramified automorphic induction from GL\(_1\) to GL\(_2\)
zbMATH Open document ID: 1221.11221
author: Laurent Lafforgue
publication date: 2010-05-05T00:00:00Z
full work available at URL: https://eudml.org/doc/116274
review text: 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.
reviewed by: J. G. M. Mars
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 11F03
zbMATH DE Number: 5703822
zbMATH Keywords: Langlands' functoriality
zbMATH Keywords: automorphic induction
MaRDI profile type: Publication
OpenAlex ID: W2320249379
cites work: Converse theorems for \(\text{GL}_n\)
cites work: Automorphic forms on GL (2)
cites work: Q4662310
cites work: The multiplicity one theorem for \(\mathrm{GL}_n\)
cites work: On an explicit formula for class-$1$ “Whittaker functions” on $GL_n $ over $\beta $-adic fields
cites work: Q5094828
DOI: 10.5802/AIF.2518
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5703822