On a question of Jones on tracelike vectors, cusp forms, and von Neumann algebras

Abstract

Previous work of Vaughan Jones built a bridge between cusp forms and random matrices using the language of von Neumann algebras. Specifically, for a discrete subgroup of $\operatorname{PSL}_2(\mathbb Z)$, there is a natural von Neumann algebra associated to it. Jones showed that for any "tracelike'' vector, there is a corresponding (anti)-isomorphism between this algebra and its commutant. Jones also showed that such vectors exist. As he noted, this connection is not useful without an explicit formula, which he posed as an open question. Here, we resolve Jones' question and discuss its context for a broad number theory audience. As we were finalizing this preprint for submission, we learned of an independent paper of Abreu. Both papers employ a closely-related polar normalisation of a kernel orbit to obtain the Jones tracelike vector at the critical parameter. The novelty of the present work lies in its extension to all $1<s\leq13$ and in the detailed analysis of the resulting von Neumann algebra and cusp form correspondences. This paper also arose out of a near solution of the authors from several years ago, which discussions with ChatGPT helped us finalize into a full solution. The relation to Abreu's work, and the exact nature of our use of AI, are detailed in the introduction.