Analytic and Reidemeister torsion for non-compact locally symmetric spaces

数论研究领域的一个新发展是基于算术群的上同调扭转应对应于有限域上的伽罗瓦表示的认识。由于这种表示是数论的中心兴趣，因此关于上同扭存在的问题是极其重要的。这个问题通常是在渐近意义上的。对于紧算术群，在几种情况下得到了上同调扭转的渐近性的结果。然而，许多算术群是不紧的。甚至对于那些最自然产生的群也是如此从伽罗瓦表示的角度来看，它们是最有趣的，比如整数上的主同余子群。对于这类群，上同扭的存在性和大小问题一般来说是开放的。我的研究项目的主要目标是证明，对于不紧的算术群，上同调扭转也呈指数增长。作为一个精确的定量陈述，我想确定这个渐近增长中的领先项与相应的L2扭转，其渐近行为是已知的。为了达到这些目的，我想研究有限体积非紧局部对称空间的解析扭转，特别是它与Reidemeister扭转的关系以及它在高阶情况下的一些基本性质。我的主要研究方法将基于几何分析技术在非紧和奇异流形上的应用，这些技术是由斯坦福大学的Rafe Mazzeo教授特别开发的。由于这些技术是为非常普遍的情况而开发的，我认为它们也适用于几何和解析结构的各种变形，我想在我的证明中使用这些变形。此外，我打算将几何分析的方法应用到局部对称空间的情况下，作为进一步研究项目的基础，这些项目不是本应用程序的内容。我想特别提到我的目标是用几何散射理论的方法研究拉普拉斯算子的连续谱。

A new development in the research area of Number Theory is based on the insight that cohomological torsion of arithmetic groups should correspond to Galois representations over finite fields. Since such representations are of central interest in Number Theory, the question about the existence of cohomological torsion is therefore extremely important. This question is usually meant in an asymptotic sense.For cocompact arithmetic groups, results about the asymptotic behaviour of cohomological torsion were obtained in several situations. However, a lot of arithmetic groups are not cocompact. This is the case even for those groups which arise most naturally and which are most interesting from the point of view of Galois representations, for example principal congruence subgroups over the integers. For such groups the question about the existence and size of cohomological torsion is in general open. The main goal of my research project is to show that also for arithmetic groups which are not cocompact the cohomological torsion grows exponentially. As a precise quantitative statement I want to identify the leading term in this asymptotic growth with the corresponding L2 torsion, whose asymptotic behaviour is already known. In order to achieve these goals I want to investigate the analytic torsion of non-compact locally symmetric spaces of finite volume, in particular its relation to Reidemeister torsion as well as some of its basic properties in the higher rank situation.The main method of my research will be based on the application of techniques from Geometric Analysis on non compact and singular manifolds, which were in particular developed by Professor Rafe Mazzeo from Stanford. Since these techniques were developed for very general situations, they are on my opinion also applicable to the various deformations of the geometric and analytic structure that I want to use in my proof at several places.Furthermore, I intend to use the approach of applying methods from Geometric Analysis to the case of locally symmetric spaces also as a basis for further research projects which are not the content of this application. I would like to mention in particular my goal to study the continuous spectrum of the Laplace operator using methods of geometric scattering theory.