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  "body": "\n## Definition\n\nLet $\\mathcal{M}_{\\mathrm{ell}}$ be the moduli stack of [[elliptic curves|SJ2BVG9F4W]] and $\\overline{\\mathcal{M}}_{\\mathrm{ell}}$ its compactification by generalized elliptic curves.\nAn $E_\\infty$ [[ring spectrum|BAW0T68GWK]] is a ring spectrum with an action of an [[$E_\\infty$ operad|T78YG90VVT]].\nBy the Goerss–Hopkins–Miller theorem there is a sheaf $\\mathcal{O}^{\\mathrm{top}}$ of $E_\\infty$ ring spectra on the étale site of $\\overline{\\mathcal{M}}_{\\mathrm{ell}}$ with the following property.\nFor each étale map $\\operatorname{Spec} R \\to \\mathcal{M}_{\\mathrm{ell}}$, which classifies an elliptic curve $C$ over $R$, the spectrum $\\mathcal{O}^{\\mathrm{top}}(\\operatorname{Spec} R)$ is an [[elliptic cohomology|XH68V2BP4K]] theory for $C$.\nFrom this sheaf one defines three spectra.\n\n1. $\\mathrm{Tmf} = \\mathcal{O}^{\\mathrm{top}}(\\overline{\\mathcal{M}}_{\\mathrm{ell}})$ is the spectrum of global sections.\n2. *Topological modular forms* $\\mathrm{tmf}$ is the connective cover $\\tau_{\\ge 0} \\mathrm{Tmf}$.\n3. Periodic topological modular forms $\\mathrm{TMF} = \\mathcal{O}^{\\mathrm{top}}(\\mathcal{M}_{\\mathrm{ell}})$ is the spectrum of sections over the open substack of smooth curves.\n\n## Remarks\n\nThere is a ring map $\\pi_{2k} \\mathrm{tmf} \\to MF_k$ to the ring of integral [[modular forms|M8NA9X8BTV]] of weight $k$, $MF_* = \\Z[c_4, c_6, \\Delta]/(c_4^3 - c_6^2 - 1728 \\Delta)$.\nIt becomes an isomorphism after tensoring with $\\Q$, so $\\pi_* \\mathrm{tmf} \\otimes \\Q \\cong \\Q[c_4, c_6]$ with $c_4$ in degree $8$ and $c_6$ in degree $12$.\nThe spectrum $\\mathrm{TMF}$ is $576$-periodic: $\\Delta^{24}$ lifts to a unit in $\\pi_{576} \\mathrm{TMF}$.\n\nExample: in degree $8$, $\\pi_8 \\mathrm{tmf} \\otimes \\Q \\cong \\Q$, and the map to modular forms sends a generator to a nonzero multiple of $c_4$, the Eisenstein series of weight $4$; this agrees with $\\dim_\\Q MF_4 \\otimes \\Q = 1$.\n\nNon-example: $\\pi_* \\mathrm{tmf}$ is not the ring $MF_*$ of modular forms.\nThe group $\\pi_3 \\mathrm{tmf} \\cong \\Z/24$ is torsion in odd degree, while $MF_*$ is zero in odd degrees.\n",
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