Hecke operators on topological modular forms
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913720591974400 |
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| author | Davies, Jack Morgan |
| author_facet | Davies, Jack Morgan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_06208 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hecke operators on topological modular forms Davies, Jack Morgan Algebraic Topology Algebraic Geometry Number Theory 11F23, 11F25, 14D23, 55N34, 55N22, 55P43, 55S25 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. |
| title | Hecke operators on topological modular forms |
| topic | Algebraic Topology Algebraic Geometry Number Theory 11F23, 11F25, 14D23, 55N34, 55N22, 55P43, 55S25 |
| url | https://arxiv.org/abs/2212.06208 |