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arXiv · 2402.18488

Von Neumann Dimensions and Trace Formulas II: A Jacquet-Langlands correspondence for Arithmetic Group Algebras in $\rm{GL}(2)$

Jun Yang
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Abstract

We propose a global Jacquet-Langlands correspondence for the modules over the von Neumann algebras of $S$-arithmetic subgroups of $\rm{GL}(2)$ and of a quaternion algebra $D$, which are both defined over a totally real number field $F$. If a representation $π'=\otimesπ'_v$ of $D^{\times}(\mathbb{A}_F)$ corresponds to a representation $π=\otimes π_v$ of $\rm{GL}(2,\mathbb{A}_F)$, we have $\frac{\dim_{\mathcal{L}(\rm{SL}(2,\mathcal{O}_S))}π_S}{\dim_{\mathbb{C}}π'_S}=|\frac{ζ_{D}(0)}{ζ_{F}(0)}|$, where $ζ_F,ζ_{D}$ are the zeta functions of $F,D$ respectively.

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BibTeXRIS

Jun Yang. 2024-02-28. Von Neumann Dimensions and Trace Formulas II: A Jacquet-Langlands correspondence for Arithmetic Group Algebras in $\rm{GL}(2)$. https://arxiv.org/abs/2402.18488

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