Stable operations and topological modular forms

Abstract

This document expands our structural knowledge of topological modular forms TMF in two directions: the first, by extending the functoriality inherent to the definition of TMF, and the second, being tools to calculate the effect that endomorphisms of TMF have on homotopy groups. These structural statements allow us to lift classical operations on modular forms, such as Adams operations, Hecke operators, and Atkin–Lehner involutions, to stable operations on TMF. Some novel applications of these operations are then found, including a derivation of some congruences of Ramanujan in a purely homotopy theoretic manner, improvements upon known bounds of Maeda’s conjecture, as well as some applications in homotopy theory. These techniques serve as teasers for the potential of these operations.