From elliptic curves to homotopy theory

Hopkins, M. J.; Mahowald, Mark

doi:10.1090/surv/201/15

Cited by 32 publications

(18 citation statements)

References 14 publications

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“…We have the following; see . Proposition The homotopy ring

π_{*} E^{h G_{48}}

is periodic of period 192 with periodicity generator detected by

Δ^{8}

.…”

Section: The Homotopy Groups Of Homotopy Fixed Point Spectramentioning

confidence: 98%

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Topological resolutions in K(2)‐local homotopy theory at the prime 2

Bobkova

Goerss

2018

Journal of Topology

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We provide a topological duality resolution for the spectrum E hS 1 2 2 , which itself can be used to build the K(2)-local sphere. The resolution is built from spectra of the form E hF 2 where E2 is the Morava spectrum for the formal group of a supersingular curve at the prime 2 and F is a finite subgroup of the automorphisms of that formal group. The results are in complete analogy with the resolutions of Goerss, Henn, Mahowald and Rezk (Ann. of Math.(2) 162 (2005) 777-822) at the prime 3, but the methods are of necessity very different. As in the prime 3 case, the main difficulty is in identifying the top fiber; to do this, we make calculations using Henn's centralizer resolution.

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“…We have the following; see . Proposition The homotopy ring

π_{*} E^{h G_{48}}

is periodic of period 192 with periodicity generator detected by

Δ^{8}

.…”

Section: The Homotopy Groups Of Homotopy Fixed Point Spectramentioning

confidence: 98%

“…Remarks and yield an isomorphism of spectral sequences

Therefore, we focus on the case of

G_{48}

. Here the standard sources are although it requires some translation in each case to get the results we want.…”

Section: The Homotopy Groups Of Homotopy Fixed Point Spectramentioning

confidence: 99%

Topological resolutions in K(2)‐local homotopy theory at the prime 2

Bobkova

Goerss

2018

Journal of Topology

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“…Theorem 5.8 (Hopkins-Mahowald [HM98]). tmf (2) ∧ DA(1) ≃ tmf 1 (3) (2) ≃ τ ≥0 (Tmf (2) ∧ DA(1)) is a form of BP 2 .…”

Section: 1mentioning

confidence: 99%

“…The computation is carried out by exhibiting a 2-local eight cell complex DA(1) and demonstrating an equivalence (also due to [HM98]),…”

mentioning

confidence: 99%

The homology of $\mathrm{tmf}$

Mathew

2016

Homology, Homotopy and Applications

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We compute the mod 2 homology of the spectrum tmf of topological modular forms by proving a 2-local equivalence tmf ∧ DA(1) ≃ tmf 1 (3) ≃ BP 2 , where DA(1) is an eight cell complex whose cohomology doubles the subalgebra A(1) of the Steenrod algebra generated by Sq 1 and Sq 2 . To do so, we give, using the language of stacks, a modular description of the elliptic homology of DA(1) via level three structures. We briefly discuss analogs at odd primes and recover the stack-theoretic description of the Adams-Novikov spectral sequence for tmf.Date: December 21, 2015.

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“…Precisely similar statements hold for tmf . This is based on results of Hopkins-Mahowald [14], with an improved formal context of Hill-Lawson [13] giving maps in the category of commutative tmf algebras. The results about finite cell complexes are proved by Mathew [21].…”

Section: Dmentioning

confidence: 99%

Ausoni–Bökstedt duality for topological Hochschild homology

Greenlees

2016

Journal of Pure and Applied Algebra

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Abstract. We consider the Gorenstein condition for topological Hochschild homology, and show that it holds remarkably often. More precisely, if R is a commutative ring spectrum and R −→ k is a map to a field of characteristic p then, provided k is small as an R-module, T HH(R; k) is Gorenstein in the sense of [11]. In particular, this holds if R is a (conventional) regular local ring with residue field k of characteristic p.Using only Bökstedt's calculation of T HH(k), this gives a non-calculational proof of dualities observed by Bökstedt [9]

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From elliptic curves to homotopy theory

Abstract: Abstract. A surprising connection between elliptic curves over finite fields and homotopy theory has been discovered by Hopkins. In this note we will follow this development for the prime 2 and discuss the homotopy which developed from this.

Cited by 32 publications

References 14 publications

Topological resolutions in K(2)‐local homotopy theory at the prime 2

Topological resolutions in K(2)‐local homotopy theory at the prime 2

The homology of $\mathrm{tmf}$

Ausoni–Bökstedt duality for topological Hochschild homology

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