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\(\mathbb{C}\)-motivic modular forms. (English) Zbl 1498.14050

The authors give a purely topological construction of a category of \(\Gamma_\star S^0\)-modules (where \(\Gamma_\star S^0\) is an object in the \((\infty,1)\)-category of filtered spectra \(\mathbf{Sp}^{\mathbb{Z}^{\mathrm{op}}}\)) such that \(2\)-complete \(\Gamma_\star S^0\)-modules are equivalent to \(2\)-complete cellular \(\mathbb{C}\)-motivic spectra. Their construction comes equipped with a functor \(\Gamma_\star\) from spectra to \(\Gamma_\star S^0\)-modules, and \(mmf := \Gamma_\star tmf\) has all the expected properties of a \(\mathbb{C}\)-motivic modular forms spectrum. In particular, the mod \(2\) motivic cohomology of \(mmf\) is \(A/\kern-.2em /A(2)\) where \(A\) is Voevodsky’s mod \(2\) motivic Steenrod algebra and \(A(2)\) is the subalgebra generated by \(\mathrm{Sq}^1,\mathrm{Sq}^2,\mathrm{Sq}^4\).
The computation of \(mmf\)’s coefficients predates its (proof of) existence; see [D. C. Isaksen, Homology Homotopy Appl. 11, No. 2, 251–274 (2009; Zbl 1193.55009)]. Additionally, the construction of \(\Gamma_\star S^0\)-modules as a topological replacement for cellular \(\mathbb{C}\)-motivic spectra removes the dependence of recent advances in computational stable homotopy theory on algebraic geometry.
The authors note in Remark 6.13 that \(\Gamma_\star S^0\)-modules are equivalent to Pstragowski’s category of even \(MU\)-synthetic spectra from [P. Pstrągowski, “Synthetic spectra and the cellular motivic category”, Preprint, arXiv:1803.01804]. (In order to demystify this slightly, note that the complex cobordism spectrum \(MU\) is part of the definition of \(\Gamma_\star\).)

MSC:

14F42 Motivic cohomology; motivic homotopy theory
55N34 Elliptic cohomology
55S10 Steenrod algebra
55Q45 Stable homotopy of spheres
55T15 Adams spectral sequences

Citations:

Zbl 1193.55009

References:

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