Hecke operators on topological modular forms

The cohomology theory TMF of topological modular forms is a derived algebro-geometric interpretation of the 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 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 Hecke operators which satisfy Maeda's conjecture.