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

Jun Yang

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.

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

Abstract

We propose a global Jacquet-Langlands correspondence for the modules over the von Neumann algebras of -arithmetic subgroups of and of a quaternion algebra , which are both defined over a totally real number field . If a representation of corresponds to a representation of , we have , where are the zeta functions of respectively.
Paper Structure (4 sections, 15 theorems, 12 equations)

This paper contains 4 sections, 15 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Trace formulas of $\mathop{\mathrm{GL}}\nolimits(2)$ and the correspondence for $S$-adelic groups
  3. Modules over twisted group von Neumann algebras
  4. Dimensions over the principal $S$-arithmetic groups

Key Result

Theorem 1

Given a global Jacquet-Langlands correspondence $\pi'=\otimes\pi'_v$$\mapsto$$\pi=\otimes \pi_v$, we have

Theorems & Definitions (26)

  • Theorem
  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Proposition 3.1
  • ...and 16 more