Journal of the
Korean Mathematical Society JKMS

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

Supported by: An outline of this work was presented as a part of our talk in the Short Communications Session of the ICM-Seoul 2014. The author was supported by a NANUM 2014 Travel Grant and by Yeditepe University.