{
  "id": "2212.06208",
  "version": "v1",
  "published": "2022-12-12T19:29:23.000Z",
  "updated": "2022-12-12T19:29:23.000Z",
  "title": "Hecke operators on topological modular forms",
  "authors": [
    "Jack Morgan Davies"
  ],
  "comment": "53 pages, comments welcome",
  "categories": [
    "math.AT",
    "math.AG",
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-12T19:29:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F23",
      "11F25",
      "14D23",
      "55N34",
      "55N22",
      "55P43",
      "55S25"
    ],
    "keywords": [
      "topological modular forms",
      "hecke operators",
      "complex modular forms",
      "satisfy maedas conjecture",
      "simple number-theoretic deductions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}