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Exotic periodic phenomena in the cohomology of the moduli stack of $1$-dimensional formal group laws

Daniel C. Isaksen, Hana Jia Kong, Guchuan Li, Yangyang Ruan, Heyi Zhu

TL;DR

This work uncovers exotic $w_1$-periodic phenomena in the $ ext{C}$-motivic stable homotopy groups and in the cohomology of the moduli stack of 1-dimensional formal group laws, i.e., the Adams–Novikov $E_2$-page. By integrating the $ ext{C}$-motivic Adams spectral sequence with the Burklund–Xu spectral sequence, the authors construct infinite families of $ ext{eta}$-torsion and $w_1$-periodic elements whose degrees form arithmetic progressions along lines of slope $1/5$, and they translate these algebraic structures to the classical Adams–Novikov $E_2$-page. A central achievement is the precise description of $ ext{eta}$-exponents in coweights $4n-1$, including the Mahowald family analogies and 2-adic valuation patterns, aided by Massey-product computations and infinite $d_2$-differentials like $d_2(h_3 g^k) = h_0 h_2 h_2 g^k$. The work further demonstrates the power of the Burklund–Xu spectral sequence to compute Chow-degree-one phenomena and provides a concrete computation of the cohomology of $ ext{C}$-motivic $ ext{A}(2)$ in Chow degree one, illustrating the practical reach of these methods for unraveling the $w_1$-periodic landscape in motivic and classical contexts.

Abstract

We describe some periodic structure in the cohomology of the moduli stack of 1-dimensional formal group laws, also known as the $E_2$-page of the classical Adams--Novikov spectral sequence. This structure is distinct from the familiar $v_n$-periodicities, and it displays interesting number-theoretic properties. Our techniques involve the $\mathbb{C}$-motivic Adams spectral sequence, and we obtain analogous periodic structure in $\mathbb{C}$-motivic stable homotopy.

Exotic periodic phenomena in the cohomology of the moduli stack of $1$-dimensional formal group laws

TL;DR

This work uncovers exotic -periodic phenomena in the -motivic stable homotopy groups and in the cohomology of the moduli stack of 1-dimensional formal group laws, i.e., the Adams–Novikov -page. By integrating the -motivic Adams spectral sequence with the Burklund–Xu spectral sequence, the authors construct infinite families of -torsion and -periodic elements whose degrees form arithmetic progressions along lines of slope , and they translate these algebraic structures to the classical Adams–Novikov -page. A central achievement is the precise description of -exponents in coweights , including the Mahowald family analogies and 2-adic valuation patterns, aided by Massey-product computations and infinite -differentials like . The work further demonstrates the power of the Burklund–Xu spectral sequence to compute Chow-degree-one phenomena and provides a concrete computation of the cohomology of -motivic in Chow degree one, illustrating the practical reach of these methods for unraveling the -periodic landscape in motivic and classical contexts.

Abstract

We describe some periodic structure in the cohomology of the moduli stack of 1-dimensional formal group laws, also known as the -page of the classical Adams--Novikov spectral sequence. This structure is distinct from the familiar -periodicities, and it displays interesting number-theoretic properties. Our techniques involve the -motivic Adams spectral sequence, and we obtain analogous periodic structure in -motivic stable homotopy.

Paper Structure

This paper contains 39 sections, 47 theorems, 51 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The cohomology of the moduli stack of 1-dimensional formal group laws
  3. Motivation from topological modular forms
  4. w1-periodicity
  5. Some w1-periodic homotopy
  6. The Adams--Novikov E2-page
  7. Relationship to Mahowald's eta j elements
  8. Relationship to Andrews's work
  9. The Burklund--Xu spectral sequence
  10. Future directions
  11. Outline
  12. Background
  13. Submodules, cogenerators, and coexponents
  14. Classical v1-periodicity and C-motivic w1-periodicity in algebra
  15. v1-periodicity in the cohomology of the classical Steenrod algebra
  16. ...and 24 more sections

Key Result

Theorem 1.2

(see thm:C-homotopy and thm:C-homotopy-beta) For all $n \geq 1$, there is a non-zero element in $\mathbb{C}$-motivic stable homotopy such that:

Figures (11)

  • Figure 1: Some $w_1$-periodic elements in the $\mathbb{C}$-motivic Adams $E_\infty$-page. Gray elements were previously studied by Andrews. Red elements are studied in this manuscript.
  • Figure 2: The classical $v_1$-periodic motif of period $(8,4)$. Gray dots detect $2$-torsion; red dots detect elements that are not $2$-torsion.
  • Figure 3: A $\mathbb{C}$-motivic $w_1$-periodic motif of period $(20,4,12)$. Gray dots detect $h_1$-torsion; red dots detect elements that are not $h_1$-torsion.
  • Figure 4: Some $v_1$-periodic $h_0$-submodules in the classical Adams $E_2$-page, or some $w_1$-periodic $h_1$-submodules in the $\mathbb{C}$-motivic Adams $E_2$-page. Numbers along the top are $(\mathrm{stem}+ 1)$, resp., $(\mathrm{coweight} + 1)$. Numbers along the bottom are coexponents.
  • Figure 5: Some $v_1$-periodic $h_0$-submodules in the classical Adams $E_\infty$-page. Numbers along the top are $(\mathrm{stem}+ 1)$. Numbers along the bottom are coexponents.
  • ...and 6 more figures

Theorems & Definitions (123)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 2.1
  • Proposition 2.3
  • proof
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • ...and 113 more