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Constructing and calculating Adams operations on dualisable topological modular forms

Jack Morgan Davies

TL;DR

The article extends Adams operations to the cohomology theory of topological modular forms (Tmf) by exploiting spectral-algebraic geometry and Goerss–Hopkins obstruction theory, yielding the first stable, multiplicative endomorphisms on Tmf. It then computes the action of these operations on the p-completed theory Tmf_p via descent spectral sequences and Anderson duality, and relates the results to classical K-theory through cusp evaluations. Building on these foundations, the paper constructs height-2 connective Adams summands u and U, and height-2 image-of-J spectra j^2, showing when tmf_p splits into sums of u and auxiliary summands depending on p, and producing connectivetopology analogues of Adams summands and image-of-J spectra. The results provide a concrete height-2 analogue of the Adams framework, with explicit calculations and applications to cofibre sequences and dualities, offering new computational tools for stable homotopy theory and the study of TMF-based phenomena. Collectively, these contributions open avenues for height-2 Behrens-Q-type constructions and deepen connections between modular forms, spectral algebraic geometry, and stable homotopy theory.

Abstract

We construct Adams operations on the cohomology theory Tmf of topological modular forms; the first such stable operations on this cohomology theory. These Adams operations are then calculated on the Tmf-cohomology of spheres using a combination of descent spectral sequences and Anderson duality. Applications of these operations are then given, including constructions of connective height 2 analogues of Adams summands and image of J spectra.

Constructing and calculating Adams operations on dualisable topological modular forms

TL;DR

The article extends Adams operations to the cohomology theory of topological modular forms (Tmf) by exploiting spectral-algebraic geometry and Goerss–Hopkins obstruction theory, yielding the first stable, multiplicative endomorphisms on Tmf. It then computes the action of these operations on the p-completed theory Tmf_p via descent spectral sequences and Anderson duality, and relates the results to classical K-theory through cusp evaluations. Building on these foundations, the paper constructs height-2 connective Adams summands u and U, and height-2 image-of-J spectra j^2, showing when tmf_p splits into sums of u and auxiliary summands depending on p, and producing connectivetopology analogues of Adams summands and image-of-J spectra. The results provide a concrete height-2 analogue of the Adams framework, with explicit calculations and applications to cofibre sequences and dualities, offering new computational tools for stable homotopy theory and the study of TMF-based phenomena. Collectively, these contributions open avenues for height-2 Behrens-Q-type constructions and deepen connections between modular forms, spectral algebraic geometry, and stable homotopy theory.

Abstract

We construct Adams operations on the cohomology theory Tmf of topological modular forms; the first such stable operations on this cohomology theory. These Adams operations are then calculated on the Tmf-cohomology of spheres using a combination of descent spectral sequences and Anderson duality. Applications of these operations are then given, including constructions of connective height 2 analogues of Adams summands and image of J spectra.

Paper Structure

This paper contains 20 sections, 19 theorems, 96 equations.

Table of Contents

  1. Constructions
  2. Algebro-geometric background
  3. Constructions using $p$-divisible groups
  4. Proof of \ref{['thaversiontwo']}
  5. (For $p\neq 2$)
  6. (For $p=2$)
  7. (Back to general $p$)
  8. Proof of \ref{['tha']}
  9. Calculations
  10. Defining the subgroup ${\mathfrak{F}\mathrm{ree}}$ of $\pi_\ast\mathop{\mathrm{Tmf}}\nolimits$
  11. Anderson duality
  12. Proof of \ref{['thb']}
  13. Applications
  14. Connective height $2$ Adams summands and \ref{['thd']}
  15. (The $p\equiv_{12}1$ case)
  16. ...and 5 more sections

Key Result

Theorem A

For every integer $k$, there is a morphism of $\mathbf{E}_\infty$-rings $\psi^k\colon \mathop{\mathrm{Tmf}}\nolimits[\frac{1}{k}]\to \mathop{\mathrm{Tmf}}\nolimits[\frac{1}{k}]$ and a commutative diagram of $\mathbf{E}_\infty$-rings \begin{tikzcd} {\Tmf[\frac{1}{k}]}\ar[r, "{\psi^k}"]\ar[d] & {\Tmf[

Theorems & Definitions (50)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Definition 1.1
  • Definition 1.2
  • Theorem 1.4
  • Corollary 1.5
  • Definition 1.6
  • ...and 40 more