On a group closely related with the automorphic Langlands group
J. Korean Math. Soc. 2020 Vol. 57, No. 1, 21-59
https://doi.org/10.4134/JKMS.j180475
Published online January 1, 2020
K\^az\i m \.Ilhan \.Ikeda
Bogazici University
Abstract : Let $L_K$ denote the hypothetical automorphic Langlands gr\-oup of a number field $K$. In our recent study, we briefly introduced a certain unconditional non-commutative topological group $\mathscr {WA}_K^{\underline{\varphi}}$, called the Weil-Arthur id\`ele group of $K$, which, assuming the existence of $L_K$, comes equipped with a natural topological group homomorphism $\mathsf{NR}_K^{\underline\varphi^{\mathrm{Langlands}}}: \mathscr {WA}_K^{\underline{\varphi}}\rightarrow L_K$ that we called the ``Langlands form'' of the global non-abelian norm-residue symbol of $K$. In this work, we present a detailed construction of $\mathscr {WA}_K^{\underline{\varphi}}$ and $\mathsf{NR}_K^{\underline\varphi^{\mathrm{Langlands}}}: \mathscr {WA}_K^{\underline{\varphi}}\rightarrow L_K$, and discuss their basic properties.
Keywords : Automorphic Langlands group, free topological product, local non-abelian class field theory, Weil-Arthur id\`ele group, global non-abelian class field theory
MSC numbers : Primary 11R39; Secondary 11S37
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