Topological Hecke eigenforms

TL;DR

This paper investigates topological Hecke eigenforms arising from elliptic homology, establishing a multiplicity one theorem for certain spaces, and introduces a derived eigentheory to understand obstructions in extending eigenforms.

Contribution

It proves a multiplicity one theorem for topological Hecke eigenforms on some spaces and develops a new derived eigentheory to analyze obstructions and eigenform extensions.

Findings

Multiplicity one holds for certain classes of topological spaces.

Explicit calculation of eigenforms for specific 2-cell CW complexes.

Identification of obstructions as derived Hecke eigenforms.

Abstract

We study the eigenforms of the action of A. Baker's Hecke operators on the holomorphic elliptic homology of various topological spaces. We prove a multiplicity one theorem (i.e., one-dimensionality of the space of these "topological Hecke eigenforms" for any given eigencharacter) for some classes of topological spaces, and we give examples of finite CW-complexes for which multiplicity one fails. We also develop some abstract "derived eigentheory" whose motivating examples arise from the failure of classical Hecke operators to commute with multiplication by various Eisenstein series. Part of this "derived eigentheory" is an identification of certain derived Hecke eigenforms as the obstructions to extending topological Hecke eigenforms from the top cell of a CW-complex to the rest of the CW-complex. Using these obstruction classes together with our multiplicity one theorem, we calculate the topological Hecke eigenforms explicitly, in terms of pairs of classical modular forms, on all 2-cell CW complexes obtained by coning off an element in $\pi_n(S^m)$ which stably has Adams-Novikov filtration 1.