On the $K(1)$-local homotopy of $\mathrm{tmf} \wedge \mathrm{tmf}$

Dominic Leon Culver; Paul VanKoughnett

REVIEW 2 major objections 4 minor 25 references

On the $K(1)$-local homotopy of $\mathrm{tmf} \wedge \mathrm{tmf}$

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper computes the K(1)-local homotopy of tmf∧tmf at primes p≤5, describing it as two inverted j-invariants and one θ-algebra generator with one relation.

desk verdict A genuinely useful computation of K(1)-local tmf cooperations, but the advertised j-invariant form of the main theorem is not proved as written. read the letter →

arxiv 1908.01904 v1 pith:OBABOVTH submitted 2019-08-05 math.AT

Dominic Leon Culver , Paul VanKoughnett This is my paper · ORCID

classification math.AT MSC 55P4255N3455T25

keywords K(1)-localhomotopytheorytopologicalmodularformstmfcooperationstheta-algebrasL-completeHopfalgebroidsAdamsspectralsequencep-adicHopkinspresentation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper computes the height-one (K(1)-local) layer of the cooperations algebra of topological modular forms: the homotopy groups of $L_{K(1)}(\mathrm{tmf} \wedge \mathrm{tmf})$ at primes $p \le 5$. The answer is a completed ring built from two inverted copies of the modular $j$-invariant, $j^{-1}$ and $\bar j^{-1}$, together with one extra $\theta$-algebra generator $\lambda$ subject to the single relation $\psi^p(\lambda) - \lambda = j^{-1} - \bar j^{-1}$, a 'second-order' analogue of the classical presentation of $KO_*KO$. This closes the last uncomputed chromatic layer of $\mathrm{tmf}_*\mathrm{tmf}$ at small primes, and it shows that the $K(1)$-local $\mathrm{tmf}$-based Adams spectral sequence for the sphere collapses immediately, with $v_1$-periodic classes only on the 0 and 1 lines. A sympathetic reader should care because explicit cooperations are the algebraic input for Adams spectral sequences, and this result replaces an intractable object with a finite presentation.

What carries the argument

The central object is Hopkins's $E_\infty$-cone $T_\zeta$ on $\zeta$, the $K(1)$-local $E_\infty$-algebra obtained by killing $\zeta$ as an $E_\infty$-cell. Its $KO$-homology is $KO_* \otimes T(b)$ with $\psi^g(b) = b+1$, and the element $f = \psi^p(b)-b$ plays the role of $j^{-1}$. The mechanism that carries the argument is a mix of $\theta$-algebra and $\lambda$-ring structure: a Hopf-algebra splitting argument, using Mahler's binomial basis for $\mathrm{Maps}_{\mathrm{cts}}(\mathbb{Z}_p,\mathbb{Z}_p)$, shows that $KO_*T_\zeta$ is an extended $KO_*KO$-comodule, which gives $\pi_*T_\zeta = KO_* \otimes T(f)$ and then the $T_\zeta$ cooperations formula. The pro-freeness machinery for $L$-complete Hopf algebroids turns these algebraic extendedness statements into Künneth isomorphisms and a change-of-rings theorem. The final identification $L_{K(1)}\mathrm{tmf} = T_\zeta/(\theta(f)-h(f))$ rests on the q-expansion congruence $f \equiv j^{-1} \bmod p$.

What would settle it

Compute the q-expansion of $f=\psi^p(b)-b$ in $KO_0T_\zeta$ at $p=2$ or $p=3$ to enough order and check the congruence $f \equiv j^{-1} \bmod p$ together with the identity $\theta(f)=h(f)$ for a power series $h$; Proposition 5.2 cites this calculation to [13, 7.1] rather than performing it, so a mismatch would invalidate the presentation of $L_{K(1)}\mathrm{tmf}$ and hence Theorem A.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is Theorem A: at primes $p \le 5$, there is an isomorphism of completed rings $$\pi_*L_{K(1)}(\mathrm{tmf} \wedge \mathrm{tmf}) \cong \left(KO_*[$j^{{-1}}$,\bar $j^{{-1}}$] \otimes T(\$\lambda$)/(\psi^p(\$\lambda$) - \$\lambda$ - $j^{{-1}}$ + \bar $j^{{-1}}$)\right)_p^\wedge.$$ The proof first computes the cooperations of Hopkins's $E_\infty$-cone $T_\zeta$ on the class $\zeta \in \pi_{-1}L_{K(1)}S$, giving $KO_* \otimes T(f,\bar f,\ell)/(\psi^p(\ell)-\ell - f + \bar f)$, and then imposes the single relation $\theta(f)=h(f)$ that converts $T_\zeta$ into $L_{K(1)}\mathrm{tmf}$, with $f$ identified with $j^{-1}$ via q-expansions. The result identifies $\pi_*L_{K(1)}(\mathrm{tmf} \wedge \mathrm{tmf})$ as a ring of ordinary two-variable $p$-adic modular functions, generated over the one-variable functions $j^{-1}$ and $\bar j^{-1}$ by a single class $\lambda$. Theorem B then gives, for any spectrum $X$, a conditionally convergent $\mathrm{tmf}$-based Adams spectral sequence with $E_2 = \mathrm{Ext}_{\mathrm{tmf}_*\mathrm{tmf}}(\mathrm{tmf}_*, \mathrm{tmf}_*X)$; for $X = S^0$ the $E_2$-page is exactly $\mathrm{Ext}_{KO_*KO}(KO_*,KO_*) \cong H^*_{\mathrm{cts}}(\mathbb{Z}_p^\times/\mu, KO_*)$, which is concentrated on the 0 and 1 lines, so the spectral sequence collapses.

Load-bearing premise

The computation rests on the cited q-expansion identification (Proposition 5.2) that the algebra generator $f$ in Hopkins's presentation is congruent to $j^{-1}$ modulo $p$ and generates the same completed subring; if that identification fails at the relevant completed rings, the defining relation $\theta(f)=h(f)$ for $L_{K(1)}\mathrm{tmf}$ and every cooperations formula derived from it would collapse.

Editorial extensions

If this is right

At primes $p \le 5$, the $K(1)$-local layer of $\mathrm{tmf}$ cooperations is completely described by the completed ring of Theorem A, removing it as an obstacle in the global $\mathrm{tmf}$-based Adams spectral sequence.
The $K(1)$-local $\mathrm{tmf}$-based Adams spectral sequence for the sphere collapses at $E_2$, with its $v_1$-periodic classes occurring only on the 0 and 1 lines.
For any $K(1)$-local spectrum $X$, there is a conditionally convergent $\mathrm{tmf}$-based Adams spectral sequence with $E_2 = \mathrm{Ext}_{\mathrm{tmf}_*\mathrm{tmf}}(\mathrm{tmf}_*, \mathrm{tmf}_*X)$.
The ring $\pi_0L_{K(1)}(\mathrm{tmf} \wedge \mathrm{tmf})$ is the ring of ordinary two-variable $p$-adic modular functions on the moduli problem of pairs of elliptic curves with an isomorphism of formal groups, generated over the one-variable functions by a single new generator.
As a $\theta$-algebra, $\mathrm{tmf}_*\mathrm{tmf}$ is generated over $\mathrm{tmf}_*$ by a single element $\ell$ with a relation linking $\ell$, $\theta(\ell)$, and $p\theta^2(\ell)$, giving a second-order analogue of the presentation of $KO_*KO$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the collapse of the $K(1)$-local $\mathrm{tmf}$-Adams spectral sequence makes a testable prediction about the global $\mathrm{tmf}$-Adams spectral sequence: every $v_1$-periodic class should live on the 0 or 1 line, which can be checked against low-stem computations of $\mathrm{tmf}_*\mathrm{tmf}$.
Beyond the paper, a direct q-expansion verification of Proposition 5.2 at $p=2$ and $p=3$ would independently certify the single external input on which the presentation rests, since the paper cites the calculation rather than carrying it out.
Beyond the paper, one could seek an explicit formula for the generator $\lambda$ as a $p$-adic analytic function on the moduli of pairs of ordinary elliptic curves, interpolating formal-group isomorphisms; a concrete expression for $\lambda$ would turn Theorem A into a computational tool for evaluating $\mathrm{tmf}$-Adams differentials.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this paper does a genuinely useful thing: it computes the K(1)-local homotopy of tmf ∧ tmf at p = 2, 3, 5 through a detailed, mostly self-contained presentation, and it corrects a real error in Hopkins's calculation of π_*T_ζ (Remark 3.28). That correction, and the careful cooperative computation for T_ζ in Section 4, are the honest core of the paper. Second, the headline Theorem A is not proved as written. The problem sits in Remark 6.2, not in the earlier machinery. The stress-test note is correct: setting λ = α(ℓ) does not convert the relation of Theorem 6.1 into the relation of Theorem A. Since ψ^p is a continuous ring map, ψ^p(λ) − λ = α(ℓ + f − fbar) − α(ℓ), a series that depends on ℓ; it is not the constant j^{-1} − jbar^{-1}. A corrected change of variables would need λ = ℓ + t with t ∈ Z_p[[f, fbar]] solving ψ^p(t) − t = (j^{-1} − jbar^{-1}) − (f − fbar). The linear part of ψ^p − id on the maximal ideal is −1, so such a t should exist, but the paper supplies no argument. As it stands, the equivalence between the f-based presentation (Theorem 6.1) and the modular j-invariant presentation (Theorem A) is unproved. This is not a fatal blow to the computations: Theorem 6.1 and the cooperations algebra for T_ζ are the substance, and they look solid. The reliance on the cited q-expansion identification in Proposition 5.2 is standard practice, though it is another external input that the paper does not verify. If I were editing, I would send this to a knowledgeable referee with a request to fix or downgrade Remark 6.2, and to consider making Theorem 6.1 the main result. The paper deserves serious refereeing; the flaw is localized and likely repairable, and the corrected T_ζ computation alone is worth publishing.

Referee Report

2 major / 4 minor

Summary. The paper computes, at primes p ≤ 5, the K(1)-local homotopy of tmf ∧ tmf. The main tool is Hopkins's presentation of L_K(1)tmf as a pushout of the E∞-cone Tζ, together with a detailed development of L-complete Hopf algebroids, pro-freeness criteria, a relative Künneth theorem, and a change-of-rings theorem. Sections 3 and 4 compute π_*Tζ and the cooperations of Tζ; Section 5 reviews the presentation of K(1)-local tmf; Section 6 derives the cooperations of tmf and the tmf-based Adams spectral sequence. Theorem A gives a modular presentation of π_*L_K(1)(tmf ∧ tmf), and Theorem B identifies the E2-term of the K(1)-local tmf-based Adams spectral sequence for the sphere with the continuous group cohomology of Z_p^×/μ with coefficients in KO_*.

Significance. If Theorem A is established, it gives a complete and surprisingly simple algebraic description of the K(1)-local cooperations of tmf, controlled by two inverted modular j-invariants and a single extra θ-algebra generator subject to one second-order relation. The paper's treatment of L-completeness, pro-freeness, and relative Ext is careful and mostly self-contained; the pro-freeness lemmas, the computation of Tζ cooperations, and the pushout presentation of tmf are supported by detailed proofs. The deduction of π_* from the stated presentations does not appear to use circular or fitted input. The paper also correctly identifies that the resulting tmf-based Adams spectral sequence collapses at E2 and is concentrated on the 0 and 1 lines. The main caveat is that the advertised modular presentation, Theorem A, is not deduced from Theorem 6.1 by the argument given in Remark 6.2.

major comments (2)

[§6, Remark 6.2] The coordinate change claimed here is not valid as written. Since ψ^p is a continuous ring endomorphism and α has coefficients in Z_p, we have ψ^p(α(ℓ)) = α(ψ^p(ℓ)) = α(ℓ + f − \bar f). Hence the element λ = α(ℓ) satisfies ψ^p(λ) − λ = α(ℓ + f − \bar f) − α(ℓ), which is not equal to j^{-1} − \bar j^{-1} in general. The relation in Theorem A therefore does not follow from Theorem 6.1 by the substitution given. A corrected proof would need to introduce a new generator λ = ℓ + t, with t ∈ Z_p[[f,\bar f]] chosen so that ψ^p(t) − t = α^{-1}(f) − α^{-1}(\bar f) − (f − \bar f), or give an equivalent argument; no such t is constructed. Since Theorem A is one of the paper's main claims, this is a load-bearing gap.
[§5, Proposition 5.2] The identification of the formal generator f with the inverse modular j-invariant is quoted from [13, 7.1] as a calculation using q-expansions and is not carried out in the paper. This identification is the only place where the algebraic generator f is tied to the modular j-invariant, and it is used to define h(f), the pushout presentation of tmf, and hence all subsequent cooperations formulas. I am not asking that all of [13] be reproved, but since this particular assertion is load-bearing for Theorem A, the authors should either give a proof, or state the precise theorem from [13] with enough detail for the reader to verify the integrality and congruence claims.

minor comments (4)

[§2.3, Proposition 2.20] The statement of the change-of-rings theorem refers to a map B ⊗_A Γ → C without first introducing C; the hypotheses should be stated in full.
[§2.3, Lemma 2.12] The forgetful functor in the statement should presumably land in Mod^∧_{R_*}, not in Mod^∧_{Γ_*}; the current wording is a typo.
[§5, proof of Theorem 5.6] The displayed chain ending in '= Z_p[f] = π_*tmf' should be KO_*[f] in nonzero degrees; as written it appears to refer only to the degree-zero part.
[Appendix A.2, Definition A.8] There is a typo: 'operaitons' should be 'operations'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the cooperations algebra is derived from the stated K(1)-local tmf presentation and a direct computation of T_zeta cooperations; the only external debt is a cited q-expansion identification, not a fitted input or self-citation.

full rationale

The derivation chain is self-contained against independent inputs. Theorem 4.4 computes pi_*(T_zeta wedge T_zeta) from the presentation KO_*T_zeta = KO_* tensor T(b) and the definition f = psi^p(b) - b; the generator l is introduced as b - b' and its relation psi^p(l) - l = f - fbar follows by direct substitution, so no parameter is fitted to the target ring. Theorem 6.1 then forms tmf wedge tmf as a pushout over P(S^0), killing theta(f)-h(f) and theta(fbar)-h(fbar), which is exactly the Hopkins presentation of tmf adopted in Theorem 5.6. The relation for l is inherited, not adjusted to match a precomputed answer. The identification f with an invertible power series in j^{-1} (Proposition 5.2) is quoted from Hopkins [13, 7.1] as a q-expansion computation; this is an external, published, falsifiable calculation and is not a self-citation by the present authors. Theorems B and 6.6 follow by a change-of-rings theorem once the Hopf algebroid is known, and no 'prediction' is produced from fitted data. The possible mathematical gap in Remark 6.2's change of variables lambda = alpha(l) is a correctness concern, not a circularity: even if the displayed modular form of the relation were not proved as written, Theorem 6.1 itself is an independent computation. No circular step is exhibited.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard K(1)-local algebra from Hovey, Strickland, Baker, and Bousfield, plus two external inputs: Hopkins's small cell presentation of K(1)-local tmf with its q-expansion identification, and Laures's additive equivalence. None of these are free parameters fitted to the target ring. No new spectra, particles, forces, or abstract entities are postulated; the generator lambda is a computed algebraic variable, not an invented object.

assumptions (8)

domain assumption K-theory cooperations are given by K_*K isomorphic to Maps_cts(Z_p^times, K_*) with the standard Hopf algebroid structure (Theorem 2.22, citing Hovey).
Used throughout Sections 3 and 4 to compute KO-cooperations and Adams operations; accepted from the literature and not reproved.
domain assumption For flat K_*X, the K-homology of the free K(1)-local E-infinity algebra P(X) is the free theta-algebra T(K_*X) (Theorem A.6, citing Barthel-Frankland and Rezk).
Used in Corollary 3.12 to identify KO_*T_zeta and in Theorem 4.4; this is the central bridge from topology to theta-algebra algebra.
standard math Bousfield's description of the free theta-algebra T(x) as a completed polynomial ring on theta^n(x) for n >= 0 (Theorem A.5).
Used in Lemmas 3.16 and 5.3 to prove pro-freeness and to compute the quotients defining tmf.
domain assumption The q-expansion identification f congruent to j^{-1} mod p and Zp[f] isomorphic to Zp[j^{-1}] (Proposition 5.2, citing Hopkins [13, 7.1]).
Not proved in this paper; load-bearing for the pushout presentation of tmf and for renaming f as j^{-1} in the main theorem.
domain assumption K(1)-local tmf is the E-infinity pushout T_zeta wedge_{P(S^0)} S^0 with generator theta(f) - h(f) (Theorem 5.6, due to Hopkins).
This presentation is the starting point for the computation of tmf and tmf; it rests on Hopkins's construction of K(1)-local tmf as a small cell complex.
domain assumption Laures's additive equivalence tmf_*X isomorphic to KO_*X[j^{-1}], hence tmf is a wedge of copies of KO as K(1)-local spectra (Proposition 6.4 proof, citing [18, Cor 3]).
Used in Proposition 6.4 to prove the left unit tmf_* to tmf_*tmf is pro-free, hence that (tmf_*, tmf_*tmf) is an L-complete Hopf algebroid for Theorem B.
standard math Mahler's basis theorem: continuous functions Z_p to Z_p are the completed span of binomial functions beta_k(x) = binomial(x,k).
Used in Proposition 3.21 and Lemma 3.24 to identify Maps_cts(Z_p, Z_p) and construct the coalgebra splitting.
standard math The profinite group Z_p^times / mu has cohomological dimension 1 for continuous group cohomology.
Used in Corollary 6.7 to show that the K(1)-local tmf-based Adams spectral sequence for the sphere collapses at E2.

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Pith. "Pith review of On the $K(1)$-local homotopy of $\mathrm{tmf} \wedge \mathrm{tmf}$." pith.science (2026). https://pith.science/paper/OBABOVTH

@misc{pith2026190801904,
  author       = {Pith},
  title        = {Pith review of: On the $K(1)$-local homotopy of $\mathrmtmf \wedge \mathrmtmf$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBABOVTH}},
  note         = {Machine review of arXiv:1908.01904}
}

abstract

As a step towards understanding the $\mathrm{tmf}$-based Adams spectral sequence, we compute the $K(1)$-local homotopy of $\mathrm{tmf} \wedge \mathrm{tmf}$, using a small presentation of $L_{K(1)}\mathrm{tmf}$ due to Hopkins. We also describe the $K(1)$-local $\mathrm{tmf}$-based Adams spectral sequence.

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[13]

Hopkins, K(1)-local E∞ -ring spectra, Topological modular forms, 2014, pp

Michael J. Hopkins, K(1)-local E∞ -ring spectra, Topological modular forms, 2014, pp. 287–302

work page 2014
[1]

Andrew Baker, L-complete Hopf algebroids and their comodules , Alpine perspectives on alge- braic topology, 2009, pp. 1–22

work page 2009
[2]

4, 2065–2131

Tobias Barthel and Martin Frankland, Completed power operations for Morava E-theory, Alge- braic & Geometric Topology 15 (2015), no. 4, 2065–2131

work page 2015
[3]

Tobias Barthel and Drew Heard, The E2-term of the K(n)-local En-Adams spectral sequence, Topology and its Applications 206 (2016), 190–214

work page 2016
[4]

Behrens, K

M. Behrens, K. Ormsby, N. Stapleton, and V. Stojanoska, On the ring of cooperations for 2- primary connective topological modular forms , Journal of Topology 12 (2019Mar), no. 2, 577– 657

work page
[5]

Mark Behrens, The construction of tmf , Topological modular forms, 2014, pp. 131–188

work page 2014
[6]

A. K. Bousﬁeld, The localization of spectra with respect to homology , Topology 18 (1979), no. 4, 257–281

work page 1979
[7]

, On λ-rings and the K-theory of inﬁnite loop spaces , K-Theory 10 (1996), 1–30

work page 1996

Show all 25 references

[8]

R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, H∞ Ring Spectra and their Applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, 1986

1986
[9]

E. S. Devinatz and M. J. Hopkins, Homotopy ﬁxed point spectra for closed subgroups of the morava stabilizer groups , Topology 43 (2004), no. 1, 1–47

2004
[10]

Paul Goerss, Hans-Werner Henn, Mark Mahowald, and Charles Rezk, A resolution of the K(2)- local sphere at the prime 3 , Annals of Mathematics 162 (2005), 777–8222

2005
[11]

preprint

Paul Goerss and Mike Hopkins, Moduli problems for structured ring spectra , 2005. preprint. 40 DOMINIC LEON CUL VER AND PAUL V ANKOUGHNETT

2005
[12]

Paul G Goerss, Topological modular forms [after Hopkins, Miller and Lurie ], Ast´ erisque332 (2010), 221–255

2010
[14]

6 (2004), no

Mark Hovey, Operations and co-operations in Morava E-theory, Homology Homotopy Appl. 6 (2004), no. 1, 201–236. [15] , Some spectral sequences in Morava E-theory, Vol. 29, 2004. Preprint, available online at http://hopf.math.purdue.edu/Hovey/morava-E-SS.pdf

2004
[16]

1, 284–302

Mark Hovey and Hal Sadofsky, Invertible spectra in the E(n)-local stable homotopy category , Journal of the London Mathematical Society 60 (1999), no. 1, 284–302

1999
[17]

666, American Mathematical Society, 1 999

Mark Hovey and Neil Strickland, Morava K-theories and Localization, Memoirs of the American Mathematical Society, vol. 666, American Mathematical Society, 1 999
[18]

2, 371–403

Gerd Laures, K(1)-local topological modular forms, Inventiones mathematicae 157 (2004), no. 2, 371–403

2004
[19]

2, 593–623

Wolfgang Lellmann and Mark Mahowald, The bo-Adams spectral sequence, Transactions of the American Mathematical Society 300 (1987), no. 2, 593–623

1987
[20]

Mark Mahowald, bo-resolutions, Paciﬁc J. Math. 92 (1981), no. 2, 365–383

1981
[21]

Mark Mahowald and Charles Rezk, Topological modular forms of level 3 , Pure and Applied Math Quarterly 5 (2009)

2009
[22]

Milnor and John C

John W. Milnor and John C. Moore, On the structure of Hopf algebras , Annals of Mathematics, Second Series 81 (1965), 211–264

1965
[23]

Ravenel, Complex cobordism and stable homotopy groups of spheres , 2nd ed., Amer- ican Mathematical Society, 2004

Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres , 2nd ed., Amer- ican Mathematical Society, 2004

2004
[24]

2, 327–379

Charles Rezk, The congruence criterion for power operations in Morava E-theory, Homology, Homotopy and Applications 11 (2009), no. 2, 327–379

2009
[25]

Alain Robert, A Course in p-Adic Analysis 198 (2000)

2000
[26]

3, 311–328

Clarence Wilkerson, Lambda-rings, binomial domains, and vector bundles over CP (∞), Com- munications in Algebra 10 (1982Jan), no. 3, 311–328. University of Illinois, Urbana-Champaign E-mail address : dculver@illinois.edu Purdue University E-mail address : pvankoug@purdue.edu

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [13]

Hopkins, K(1)-local E∞ -ring spectra, Topological modular forms, 2014, pp

Michael J. Hopkins, K(1)-local E∞ -ring spectra, Topological modular forms, 2014, pp. 287–302

work page 2014

[2] [1]

Andrew Baker, L-complete Hopf algebroids and their comodules , Alpine perspectives on alge- braic topology, 2009, pp. 1–22

work page 2009

[3] [2]

4, 2065–2131

Tobias Barthel and Martin Frankland, Completed power operations for Morava E-theory, Alge- braic & Geometric Topology 15 (2015), no. 4, 2065–2131

work page 2015

[4] [3]

Tobias Barthel and Drew Heard, The E2-term of the K(n)-local En-Adams spectral sequence, Topology and its Applications 206 (2016), 190–214

work page 2016

[5] [4]

Behrens, K

M. Behrens, K. Ormsby, N. Stapleton, and V. Stojanoska, On the ring of cooperations for 2- primary connective topological modular forms , Journal of Topology 12 (2019Mar), no. 2, 577– 657

work page

[6] [5]

Mark Behrens, The construction of tmf , Topological modular forms, 2014, pp. 131–188

work page 2014

[7] [6]

A. K. Bousﬁeld, The localization of spectra with respect to homology , Topology 18 (1979), no. 4, 257–281

work page 1979

[8] [7]

, On λ-rings and the K-theory of inﬁnite loop spaces , K-Theory 10 (1996), 1–30

work page 1996