Automorphic spectrum and arithmetic groups

The cohomology of an arithmetic subgroup of a reductive algebraic group G defined over some number field k can be interpreted in terms of its automorphic spectrum. Various techniques have been used in some cases, on the one hand, to detect the internal automorphic structure of certain analytically defined subspaces of the cohomology (cuspidal, Eisenstein) and to establish the actual existence of specific types of automorphic representations in the automorphic spectrum, on the other. Various applications of these investigations to questions in arithmetic and geometry have been given. It is the main objective of this project to study this circle of ideas in the case of the general linear group over some finite-dimensional division algebra D defined over k. Firstly, using a refined stabilized form of Arthur`s trace formula, this pertains to an analysis of the cuspidal cohomology. Secondly, it is intended to study cohomology classes originating with residues of Eisenstein series in this case, that is, to understand the contribution of the residual spectrum to the cohomology.

The cohomology of an arithmetic subgroup of a reductive algebraic group G de?ned over some number ?eld k can be interpreted in terms of its automorphic spectrum. Various techniques have been used in this project, on the one hand, to detect the internal automorphic structure of certain analytically de?ned subspaces of the cohomology (cuspidal, Eisenstein) and to establish the actual existence of speci?c types of automorphic representations in the automorphic spectrum, on the other. The methods used or developed were an in-depth analysis of arithmetic data involved or the geometric construction of non-vanishing cohomology classes for arithmetic groups. The latter classes are then related to automorphic representations contributing to the spectrum.Various applications of these investigations to questions in arithmetic and geometry have been given.