Hecke operators on topological modular forms

TL;DR

This paper extends classical number-theoretic operators like Hecke operators to the realm of topological modular forms (TMF), providing new proofs, properties, and number-theoretic insights using homotopy theory.

Contribution

It develops a stable operator framework for Hecke operators on TMF, enabling simplified proofs and new results in number theory and modular forms.

Findings

Rederived classical Ramanujan congruences using homotopy theory

Constructed new families of Hecke operators satisfying Maeda's conjecture

Provided a unified algebro-geometric approach to classical operators

Abstract

The cohomology theory TMF of topological modular forms is a derived algebro-geometric interpretation of the classical ring of complex modular forms from number theory. In this article, we refine the classical Adams operations, Hecke operators, and Atkin--Lehner involutions from endomorphisms of classical modular forms to stable operators on TMF. Our algebro-geometric formulation of these operators leads to simple proofs of their many remarkable properties and computations. From these properties, we use techniques from homotopy theory to make simple number-theoretic deductions, including a rederivation of some classical congruences of Ramanujan and providing new infinite families of classical Hecke operators which satisfy Maeda's conjecture.