Invariants, cohomology, and automorphic forms of higher order

A general structure theorem on higher order invariants is proven. For an arithmetic group, the structure of the corresponding Hecke module is determined. It is shown that the module does not contain any irreducible submodule. This explains the fact that L-functions of higher order forms have no Euler product. Higher order cohomology is introduced, classical results of Borel are generalized, and a higher order version of Borel’s conjecture is stated.