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The descent spectral sequence for topological modular forms

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arxiv 2412.01640 v2 pith:DE6DWXLH submitted 2024-12-02 math.AT

Christian Carrick , Jack Morgan Davies , Sven van Nigtevecht This is my paper · ORCID
classification math.AT
keywords formsmathrmmodulartopologicalapproachcircularitycomputationconfirming
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abstract

We prove the Gap Theorem for the spectrum of topological modular forms $\mathrm{Tmf}$. This removes a longstanding circularity in the literature, thereby confirming the computation of $\pi_\ast \mathrm{tmf}$ from over two decades ago by Hopkins and Mahowald. Our approach is crucially a modern one, developing and refining many techniques in synthetic spectra.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Affineness and reconstruction in complex-periodic geometry

    math.AT 2025-10 accept novelty 8.0 of 10

    A new spectral-stack framework shows that many moduli stacks in complex-periodic homotopy theory, including bounded-height oriented formal groups and oriented elliptic curves, are determined by their global sections.

  2. On periodic families in the stable stems of height two

    math.AT 2025-06 unverdicted novelty 7.0 of 10

    Theorem A establishes 125 nonzero v_2^32-periodic families in the 2-primary stable stems, 50 new, all vanishing in TMF yet detected by the Atkin-Lehner fixed point spectrum J_0(3).

  3. Cellularity of Chromatic Synthetic Spectra

    math.AT 2025-05 conditional novelty 7.0 of 10

    Synthetic spectra based on Morava E-theory are generated by bigraded spheres and are equivalent to modules over a filtered ring spectrum.

  4. Revisiting the $\beta_1$-action on the $3$-primary stable homotopy groups of spheres

    math.AT 2025-11 conditional novelty 6.0 of 10

    Products of the 3-primary β₁-periodic family in the stable homotopy groups of spheres are nonzero for up to five factors and zero for six or more, with analogous sharp cutoffs for β₂-, α₁β₂-, and [α₁β₃/3]-twisted products.

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