Invariants, cohomology, and automorphic forms of higher order

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.

Modular iterated integrals associated with cusp forms

“…To define our extended higher-order modular forms, it will be necessary to describe a general framework involving a family of representations. See [6,7] for two alternative general definitions of higher-order objects, which are built on only one representation and which use the formalism of the augmentation ideal. Exceptionally, we will give the next definition for general Fuchsian groups Γ of the first kind acting on H with non-compact quotient Γ\H.…”

Higher order invariants in the case of compact quotients

“…For Ω = C ∞ (G) this is Theorem 2.3.5 of [6]. For Ω = L 2 loc (G) the result follows from the latter by density arguments.…”

“…The paper [6] contains a structure theorem showing that spaces of higher order automorphic forms are in a natural way subspaces of tensor products of automorphic representation spaces. In this paper we restrict to the simple case of a compact quotient in which case we are able to • remove the "subspace" part from the assertion, i.e., get a more precise statement on the structure of higher order forms, and…”

Untitled

“…Remark. As A. Deitmar has pointed out, the last two propositions should also follow from [Deitmar 2008]. We have opted for explicit methods of proof because they are necessary for later parts of the paper.…”

“…Although higher-order invariants have been classified in several cases and from various perspectives [Chinta et al 2002;Diamantis and O'Sullivan 2008;Diamantis and Sim 2008;Deitmar 2008;, the real-analytic case to which the important function E * (−, s) belongs has not been fully addressed up to now. This is perhaps not surprising, given that such functions can contain very rich and complex information, as the example of E * (−, s) shows.…”