$GL(2,\mathbb{Q}_p)$上的微局域提升和量子唯一遍历性
Microlocal lifts and quantum unique ergodicity on $GL(2,\mathbb{Q}_p)$
摘要：
证明了对于属于主列的自守形式，在$GL(2,\mathbb{Q}_p)$的紧算术子上算术量子唯一遍历性成立.我们解释了这一结论的特征函数的均匀分布的覆盖一个固定的正则图或沿嵌套序列的正则图。我们的结果是第一次他们的任何p-adic算术商。它们可以被理解为类似的林登施特劳斯定理的等距分布的马斯形式在一个紧凑的算术表面。这里的新成分包括介绍了一个代表性理论的概念“p-adic microlocal升降机”与有利的性质，如对角不变性的限制措施，证明正熵的限制措施在一个p-adic方面，以下的方法Bourgain-Lindenstrauss，和一些分析本地Rankin-Selberg积分涉及这里介绍的microlocal升降机以及经典newvectors。一个重要的输入是Einsiedler-Lindenstrauss的测量分类结果。
英文摘要：
We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any p-adic arithmetic quotient. They may be understood as analogues of Lindenstrauss's theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a representation-theoretic notion of "p-adic microlocal lifts" with favorable properties, such as diagonal invariance of limit measures, the proof of positive entropy of limit measures in a p-adic aspect, following the method of Bourgain--Lindenstrauss, and some analysis of local Rankin--Selberg integrals involving the microlocal lifts introduced here as well as classical newvectors. An important input is a measure-classification result of Einsiedler--Lindenstrauss.
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