On a question of Jones on tracelike vectors, cusp forms, and von Neumann algebras
Abstract
Using the theory of von Neumann algebras, Vaughan Jones showed abstractly that there is a bridge between holomorphic cusp forms and random matrices. This bridge is supplied by a distinguished function he called a {\it tracelike vector}. He posed the following problem: construct a tracelike vector to make this bridge explicit. We carry out Jones' vision for the full modular group. We construct holomorphic tracelike vectors for every finite-covolume Fuchsian group throughout the permissible Bergman range. For the modular group at the critical parameter s=13, we use the resulting unit tracelike vector to assign explicit group coefficients to cusp form multiplication operators, and form positive finite random matrices from a shared Haar-distributed unitary matrix. Almost surely, their normalised traces recover all joint operator moments, first as the matrix size tends to infinity, and then as the group-word cutoff tends to infinity. We also identify the limiting mixed second moment with a series of Petersson norms that involve Maass-Shimura operators of cusp forms. The exact nature of our use of AI is detailed in the introduction.
Related
- Non-isomorphism of q-Gaussian von Neumann Algebras in Infinite Variables
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- Group actions on von Neumann algebras with compact open subgroups
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BibTeX
@misc{diamantis2026question,
title = {{On a question of Jones on tracelike vectors, cusp forms, and von Neumann algebras}},
author = {Nikolaos Diamantis and Larry Rolen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39904},
eprint = {2609.39904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39904},
}
RIS
TY - UNPB
TI - On a question of Jones on tracelike vectors, cusp forms, and von Neumann algebras
AU - Nikolaos Diamantis
AU - Larry Rolen
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.39904
UR - https://arxiv.org/abs/2609.39904
AB - Using the theory of von Neumann algebras, Vaughan Jones showed abstractly that there is a bridge between holomorphic cusp forms and random matrices. This bridge is supplied by a distinguished function he called a {\it tracelike vector}. He posed the following problem: construct a tracelike vector to make this bridge explicit. We carry out Jones' vision for the full modular group. We construct holomorphic tracelike vectors for every finite-covolume Fuchsian group throughout the permissible Bergman range. For the modular group at the critical parameter s=13, we use the resulting unit tracelike vector to assign explicit group coefficients to cusp form multiplication operators, and form positive finite random matrices from a shared Haar-distributed unitary matrix. Almost surely, their normalised traces recover all joint operator moments, first as the matrix size tends to infinity, and then as the group-word cutoff tends to infinity. We also identify the limiting mixed second moment with a series of Petersson norms that involve Maass-Shimura operators of cusp forms. The exact nature of our use of AI is detailed in the introduction.
ER -