Analytic and Reidemeister torsion for non-compact locally symmetric spaces

• Applicant: Dr. Jonathan Pfaff
• Subject Area: Mathematics
• Term: from 2013 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 250392313

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.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Dr. Rafe Mazzeo