Cohomology of arithmetic groups

The main objective of this project is the study of the cohomology of arithmetic subgroups of reductive algebraic groups defined over some algebraic number field k, its relationship with the theory of automorphic fonns, and some number theoretical applications. This subject is related in various ways with number theory, geometry and arithmetic algebraic geometry. Its focus is an central questions in pure mathematics. The investigations as proposed pertain to - the use of the principle of Langlands functoriality in constructing cuspidal autounouphic fonns. - the study of Lefschetz numbers to detect cuspidal automorphic forms in their cohomological realization. - the intemal structure of the cohomology constructed by means of the theory of Eisenstein series. - the arithmetic nature of these classes, relations to Euler products attached to automorphic fonns. i.e., automorphic L-functions.

The main objective of this project was the study of the cohomology of arithmetic groups of reductive algebraic groups defined over some number field, its relationship with the theory of automorphic forms, and some number theoretical applications. This subject is related in various ways with number theory, geometry and arithmetic algebraic geometry. Its focus was on central questions in pure mathematics. The investigations pertained to - the internal structure of the cohomology of arithmetic groups constructed by means of derivatives or residues of Eisenstein series - the use of the Langlands functoriality principle to construct cuspidal automorphic representations and their cohomological realizations - vanishing results for the cuspidal cohomology as a consequence of the non-degeneracy of cuspidal automorphic representations - the arithmetic nature of Eisenstein cohomology classes and relations to Euler products attached to automorphic forms