论文标题
关于Sarnak的密度猜想及其应用
On Sarnak's Density Conjecture and its Applications
论文作者
论文摘要
萨尔纳克（Sarnak）的密度猜想是一种明确的束缚，以一系列共同的一致性一致性算术矩阵在一个半圣母层组中以多种态度表示，这是由萨尔纳克（Sarnak）和Xue的工作激发的。这项工作的目的是讨论类似的假设，它们的相互关系和应用。我们主要关注两种特性 - 光谱球形密度假设和几何弱的注射半径特性。在P-ADIC情况下，我们的结果最强，在那里我们表明这两种特性是等效的，并且都意味着Sarnak的一般密度假设。一种可能的应用是，球形密度假设或弱的注射半径属性意味着Sarnak的最佳起重性能。猜想，所有这些属性都应该具有很大的一般性。我们希望这项工作能够激发他们在新案件中的证据。
Sarnak's Density Conjecture is an explicit bound on the multiplicities of non-tempered representations in a sequence of cocompact congruence arithmetic lattices in a semisimple Lie group, which is motivated by the work of Sarnak and Xue. The goal of this work is to discuss similar hypotheses, their interrelation and applications. We mainly focus on two properties - the spectral Spherical Density Hypothesis and the geometric Weak Injective Radius Property. Our results are strongest in the p-adic case, where we show that the two properties are equivalent, and both imply Sarnak's General Density Hypothesis. One possible application is that either the Spherical Density Hypothesis or the Weak Injective Radius Property imply Sarnak's Optimal Lifting Property. Conjecturally, all those properties should hold in great generality. We hope that this work will motivate their proofs in new cases.
