Math 2.0

Algebraic topology Definition

Topological modular forms

Builds on Elliptic cohomologyRing spectrumE-infinity operadElliptic curveModular form

Definition

Let ℳell\mathcal{M}_{\mathrm{ell}} be the moduli stack of elliptic curves and ℳ‾ell\overline{\mathcal{M}}_{\mathrm{ell}} its compactification by generalized elliptic curves. An E∞E_\infty ring spectrum is a ring spectrum with an action of an E∞E_\infty operad. By the Goerss–Hopkins–Miller theorem there is a sheaf 𝒪top\mathcal{O}^{\mathrm{top}} of E∞E_\infty ring spectra on the étale site of ℳ‾ell\overline{\mathcal{M}}_{\mathrm{ell}} with the following property. For each étale map Spec⁡R→ℳell\operatorname{Spec} R \to \mathcal{M}_{\mathrm{ell}}, which classifies an elliptic curve CC over RR, the spectrum 𝒪top(Spec⁡R)\mathcal{O}^{\mathrm{top}}(\operatorname{Spec} R) is an elliptic cohomology theory for CC. From this sheaf one defines three spectra.

  1. Tmf=𝒪top(ℳ‾ell)\mathrm{Tmf} = \mathcal{O}^{\mathrm{top}}(\overline{\mathcal{M}}_{\mathrm{ell}}) is the spectrum of global sections.
  2. Topological modular forms tmf\mathrm{tmf} is the connective cover τ≥0Tmf\tau_{\ge 0} \mathrm{Tmf}.
  3. Periodic topological modular forms TMF=𝒪top(ℳell)\mathrm{TMF} = \mathcal{O}^{\mathrm{top}}(\mathcal{M}_{\mathrm{ell}}) is the spectrum of sections over the open substack of smooth curves.

Remarks

There is a ring map π2ktmf→MFk\pi_{2k} \mathrm{tmf} \to MF_k to the ring of integral modular forms of weight kk, MF∗=ℤ[c4,c6,Δ]/(c43−c62−1728Δ)MF_* = \Z[c_4, c_6, \Delta]/(c_4^3 - c_6^2 - 1728 \Delta). It becomes an isomorphism after tensoring with ℚ\Q, so π∗tmf⊗ℚ≅ℚ[c4,c6]\pi_* \mathrm{tmf} \otimes \Q \cong \Q[c_4, c_6] with c4c_4 in degree 88 and c6c_6 in degree 1212. The spectrum TMF\mathrm{TMF} is 576576-periodic: Δ24\Delta^{24} lifts to a unit in π576TMF\pi_{576} \mathrm{TMF}.

Example: in degree 88, π8tmf⊗ℚ≅ℚ\pi_8 \mathrm{tmf} \otimes \Q \cong \Q, and the map to modular forms sends a generator to a nonzero multiple of c4c_4, the Eisenstein series of weight 44; this agrees with dim⁡ℚMF4⊗ℚ=1\dim_\Q MF_4 \otimes \Q = 1.

Non-example: π∗tmf\pi_* \mathrm{tmf} is not the ring MF∗MF_* of modular forms. The group π3tmf≅ℤ/24\pi_3 \mathrm{tmf} \cong \Z/24 is torsion in odd degree, while MF∗MF_* is zero in odd degrees.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectE-infinity operad, concept, no evidence levelE-infinity operadconceptElliptic cohomology, concept, no evidence levelElliptic cohomologyconceptField, concept, no evidence levelFieldconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectHolomorphic function, concept, no evidence levelHolomorphic functionconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptQuotient group, concept, no evidence levelQuotient groupconceptRing spectrum, concept, no evidence levelRing spectrumconceptVector space, definition, no evidence levelVector spacedefinitionElliptic curve, definition, no evidence levelElliptic curvedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectUpper half-plane, object, no evidence levelUpper half-planeobjectModular group, object, no evidence levelModular groupobjectModular form, definition, no evidence levelModular formdefinitionTopological modular forms, definition, no evidence levelTopological modular f…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. E-infinity operad concept
  3. Elliptic cohomology concept
  4. Field concept
  5. General linear group object
  6. Holomorphic function concept
  7. Identity matrix object
  8. Matrix definition
  9. Möbius transformation concept
  10. Quotient group concept
  11. Ring spectrum concept
  12. Vector space definition
  13. Elliptic curve definition
  14. Square matrix definition
  15. Linear map definition
  16. Multilinear map definition
  17. Alternating multilinear map definition
  18. Determinant definition
  19. Special linear group object
  20. Upper half-plane object
  21. Modular group object
  22. Modular form definition
  23. Topological modular forms definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Topological_modular_forms
About this page
TagT6Q7NST7JA
URIm2://T6Q7NST7JA
Kinddefinition
WritingW0 stub
MSC55N34, 55P43, 11F11
DefinesTopological modular forms
UsesElliptic cohomology, Ring spectrum, E-infinity operad, Elliptic curve, Modular form
Changed2 November 2026

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