Invariants, cohomology, and automorphic forms of higher order

TL;DR

This paper establishes a general structure theorem for higher order invariants, analyzes the Hecke module structure for arithmetic groups, and introduces higher order cohomology, extending classical results and conjectures.

Contribution

It provides a comprehensive structure theorem for higher order invariants and explores the properties of associated Hecke modules and cohomology theories.

Findings

Hecke modules for higher order invariants lack irreducible submodules

Higher order L-functions do not have Euler products

Classical results are generalized to higher order cohomology

Abstract

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.