关于Jones的迹类向量、尖点形式与von Neumann代数问题
On a question of Jones on tracelike vectors, cusp forms, and von Neumann algebras
- University of Nottingham（诺丁汉大学）
- Vanderbilt University（范德堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文解决了Jones关于迹类向量存在的开放问题，通过极坐标归一化方法在临界参数处构造迹类向量，并将结果推广到所有$1<s\leq13$，详细分析了von Neumann代数与尖点形式的对应关系。
AI中文摘要：
Vaughan Jones的先前工作利用von Neumann代数的语言在尖点形式与随机矩阵之间建立了桥梁。具体而言，对于$\operatorname{PSL}_2(\mathbb Z)$的一个离散子群，存在一个与之关联的自然von Neumann代数。Jones证明了对于任何“迹类”向量，该代数与其交换子之间存在一个相应的（反）同构。Jones还证明了此类向量的存在性。正如他所指出的，如果没有显式公式，这种联系并无用处，因此他将其作为一个开放问题提出。在此，我们解决了Jones的问题，并为广泛的数论读者讨论其背景。在我们准备提交此预印本时，我们得知Abreu有一篇独立论文。两篇论文都采用了密切相关的核轨道的极坐标归一化方法，在临界参数处获得Jones迹类向量。本工作的新颖之处在于将其扩展到所有$1<s\leq13$，并对由此产生的von Neumann代数与尖点形式对应关系进行了详细分析。本文还源于作者数年前的一个近乎解决的方案，与ChatGPT的讨论帮助我们将其完善为完整解决方案。与Abreu工作的关系以及我们使用AI的确切性质，在引言中有详细说明。
英文摘要：
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.
