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Hecke operators on topological modular forms

Jack Morgan Davies

TL;DR

This work constructs three families of stable TMF operations—stable Adams operations $\psi^k$, stable Atkin–Lehner involutions $w_Q$, and stable Hecke operators $\mathrm{T}_n$—as genuine endomorphisms of the cohomology theory TMF by elevating classical modular-form operations to stable, functorial maps via Lurie’s and spectral Mackey formalisms. It shows these operations agree with their classical modular-form counterparts on TMF-cohomology of spheres and commute coherently up to higher homotopies, with explicit composition rules that recover the familiar Hecke algebra structure after appropriate localization. The paper proves the necessity of inverting primes for certain stable operators, and uses these operators to rederive Ramanujan-type congruences and to provide new infinite families of Hecke operators that satisfy Maeda’s conjecture, thus linking stable homotopy theory to deep number-theoretic phenomena. Overall, the results illuminate how a refined, functorial TMF framework yields both computational tools in homotopy theory and new number-theoretic insights, including congruences and irreducibility/ Galois properties of Hecke actions. The methods open avenues for further applications to TMF with level structures and Behrens–Laures-type spectra, potentially advancing our understanding of Maeda’s conjecture across primes.

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.

Hecke operators on topological modular forms

TL;DR

This work constructs three families of stable TMF operations—stable Adams operations , stable Atkin–Lehner involutions , and stable Hecke operators —as genuine endomorphisms of the cohomology theory TMF by elevating classical modular-form operations to stable, functorial maps via Lurie’s and spectral Mackey formalisms. It shows these operations agree with their classical modular-form counterparts on TMF-cohomology of spheres and commute coherently up to higher homotopies, with explicit composition rules that recover the familiar Hecke algebra structure after appropriate localization. The paper proves the necessity of inverting primes for certain stable operators, and uses these operators to rederive Ramanujan-type congruences and to provide new infinite families of Hecke operators that satisfy Maeda’s conjecture, thus linking stable homotopy theory to deep number-theoretic phenomena. Overall, the results illuminate how a refined, functorial TMF framework yields both computational tools in homotopy theory and new number-theoretic insights, including congruences and irreducibility/ Galois properties of Hecke actions. The methods open avenues for further applications to TMF with level structures and Behrens–Laures-type spectra, potentially advancing our understanding of Maeda’s conjecture across primes.

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.
Paper Structure (21 sections, 44 theorems, 140 equations, 3 tables)

This paper contains 21 sections, 44 theorems, 140 equations, 3 tables.

Table of Contents

  1. Extending the functorality of $\mathcal{O}^\mathrm{top}$
  2. Lurie's theorem and topological modular forms
  3. Extension to isogenies of invertible degree
  4. Extension to a spectral Mackey functor
  5. Into the zoo of stable operations
  6. Definitions and comparisons
  7. Interrelations between stable operators
  8. Decomposing the stacks $\mathcal{M}_n$
  9. Hecke composition formula
  10. Factorisations into $p$-primary components
  11. Nonexistence of integral stable Hecke operators
  12. ($p=2$ case)
  13. ($p=3$ case)
  14. Applications to number theory
  15. Congruences between coefficients in $q$-expansions
  16. ...and 6 more sections

Key Result

Theorem 1

There is a sheaf $\mathcal{O}^\mathrm{top}_\mathrm{GHM}$ of $\mathbf{E}_\infty$-rings on the moduli stack $\mathcal{M}$ of elliptic curves and an isomorphism of sheaves of abelian groups $\pi_{2d}\mathcal{O}^\mathrm{top}_\mathrm{GHM}\simeq \omega^{\otimes d}$.

Theorems & Definitions (106)

  • Theorem : [Goerss--Hopkins--Miller, Lurie (see bourbakigoerss or ec2name)]
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G: [Congruences of Ramanujan]
  • Theorem H
  • Theorem 1.1
  • ...and 96 more