On a group closely related with the automorphic Langlands group
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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.
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.
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