论文标题
设计伪拉普拉斯人
Designed Pseudo-Laplacians
论文作者
论文摘要
我们详细阐述并严格猜测自相邻操作员在$ l $ functions的零位置的自动形式空间的光谱特性上的含义。从1970年代末和1980年代开始，Colin deVerdière，Lax-Phillips和Hejhal的作品中就出现了其中一些想法，更不用说对Pólya和Hilbert的半同伴属性了。例如，给定一个复杂的二次扩展$ k $ $ \ mathbb q $，我们在模块化曲线上对不变的laplacian的限制进行了自然的自我接触扩展，该模块曲线的离散频谱（如果有的话）由值$ s（s-1）$组成的零$ s $ s $ s $ $ s $ $ quam_k（s）。不幸的是，似乎没有理由使这种离散的频谱变得很大。实际上，蒙哥马利的配对相关性以及$ζ（1+it）$的行为，这意味着最多可以在此离散频谱中出现$ 94 \％$ $ζ$的零。不太天真地，从这些考虑因素上遵循了有关零动力学的一些初步阳性结果。
We elaborate and make rigorous various speculations about the implications of spectral properties of self-adjoint operators on spaces of automorphic forms for location of zeros of $L$-functions. Some of these ideas arose in work of Colin de Verdière, Lax-Phillips, and Hejhal, from the late 1970s and early 1980s, not to mention semi-apocryphal attributions to Pólya and Hilbert. For example, given a complex quadratic extension $k$ of $\mathbb Q$, we give a natural self-adjoint extension of a restriction of the invariant Laplacian on the modular curve whose discrete spectrum, if any, consists of values $s(s-1)$ for zeros $s$ of $ζ_k(s)$. Unfortunately, there seems to be no reason for this discrete spectrum to be large. In fact, Montgomery's pair correlation, and the behavior of $ζ(1+it)$, imply that at most $94\%$ of zeros of $ζ(s)$ can appear in this discrete spectrum. Less naively, some preliminary positive results about the dynamics of zeros do follow from these considerations.
