### BibTeX

@MISC{Behrens07noteson,

author = {Mark Behrens},

title = { Notes on the construction of tmf},

year = {2007}

}

### OpenURL

### Abstract

### Citations

166 | Arithmetic moduli of elliptic curves - Katz, Mazur - 1985 |

159 | The Arithmetic of Elliptic Curves, Graduate Texts - Silverman - 1986 |

145 | Model categories. Mathematical Surveys and Monographs, 63 - Hovey - 1999 |

126 | Les schémas de modules de courbes elliptiques - Deligne, Rapoport - 1973 |

82 | p-divisible groups, in - Tate - 1966 |

77 | Motivic symmetric spectra
- Jardine
(Show Context)
Citation Context ... For a small Grothendieck site C with enough points, let PreSp C denote the category of presheaves of symmetric spectra of simplicial sets. The category PreSp C has a Jardine model category structure =-=[Jar00]-=-, where (1) The cofibrations are the sectionwise cofibrations of symmetric spectra, (2) The weak equivalences are the stalkwise stable equivalences of symmetric spectra, (3) The fibrant objects are th... |

41 |
Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups
- Devinatz, Hopkins
(Show Context)
Citation Context ...N) ) K(1) Consider the finite (Z/p) × Galois extension E h(1+pZp) 1 � (tmf K(2)) K(1). of S K(1) given by the homotopy fixed points of E1-theory with respect to the open subgroup 1 + pZp ⊂ Z × p (see =-=[DH04]-=-, [Rog08]). Note that we have (8.3) (E h(1+pZp) 1 ) h(Z/p)× � SK(1). Define spectra (tmf (N, p) K(2)) K(1) := (tmf (N) K(2)) K(1) ∧S K(1) E h(1+pZp) 1 (tmf (p) K(2)) K(1) := (tmf K(2)) K(1) ∧S K(1) E ... |

35 |
Homotopy commutative diagrams and their realizations
- Dwyer, Kan
- 1989
(Show Context)
Citation Context ...mponents. We have constructed a functor Ō top K(1) : ((Mord ell )et,aff ) op → Ho(Commutative tmf K(1)-algebras). Since the mapping spaces are contractible, this functor lifts to give a presheaf (see =-=[DKS89]-=-) O top K(1) : ((Mord ell )et,aff ) op → Commutative tmf K(1)-algebras. The same argument used to prove part (2) of Theorem 7.7 proves the following. Proposition 7.15. Suppose that Spf(R) → Mord ell i... |

34 | Homotopy spectral sequences and obstructions - Bousfield - 1989 |

33 | Hypercovers and simplicial presheaves
- Dugger, Hollander, et al.
(Show Context)
Citation Context ... (3) The fibrant objects are those objects which are fibrant in the injective model structure of the underlying diagram model category structure, and which satisfy descent with respect to hypercovers =-=[DHI04]-=-. The following lemma will be useful. Lemma 2.1. (1) If F ∈ PreSp C satisfies homotopy descent with respect to hypercovers, then the fibrant replacement in the Jardine model structure F → F ′ is a sec... |

25 |
Moduli spaces of commutative ring spectra. Structured ring spectra, 151–200
- Goerss, Hopkins
- 2004
(Show Context)
Citation Context ...the Witt ring of k.) DefG ∼ = Spf(B(k, G)) B(k, G) ∼ = W(k)[[u1, . . . , un−1]]. Let � G/B(k, G) denote the universal deformation of G. The following theorem was proven by Goerss, Hopkins, and Miller =-=[GH04]-=-. Theorem 4.2 (Goerss-Hopkins-Miller). Let C be the category of pairs (k, G) where k is a perfect field of characteristic p and G is a formal group of finite height over k. There is a functor C → E∞ r... |

22 | Galois extensions of structured ring spectra. Stably dualizable groups
- Rognes
(Show Context)
Citation Context ...) Consider the finite (Z/p) × Galois extension E h(1+pZp) 1 � (tmf K(2)) K(1). of S K(1) given by the homotopy fixed points of E1-theory with respect to the open subgroup 1 + pZp ⊂ Z × p (see [DH04], =-=[Rog08]-=-). Note that we have (8.3) (E h(1+pZp) 1 ) h(Z/p)× � SK(1). Define spectra (tmf (N, p) K(2)) K(1) := (tmf (N) K(2)) K(1) ∧S K(1) E h(1+pZp) 1 (tmf (p) K(2)) K(1) := (tmf K(2)) K(1) ∧S K(1) E h(1+pZp) ... |

18 | Topological automorphic forms
- Behrens, Lawson
(Show Context)
Citation Context ... moduli problem Mord ell (pk) is rigid (i.e. it takes values in sets, not groupoids). Since p generates the ideal of definition of the formal stack Mord ell (pk), it follows from Proposition 3.5.1 of =-=[BL10]-=- that it suffices to show that the mod p reduction M ord rigid. Let (M ord ell )n Fp ell (pk )Fp denote the moduli stack (over Fp) of ordinary elliptic curves with the structure of an n-jet at the bas... |

17 |
Finite subgroups of formal groups
- Strickland
- 1997
(Show Context)
Citation Context ... of tmf , and I would have gotten it wrong except for the help of Niko Naumann and Charles Rezk.) Let i : Subp( ˜ C) → Spf(B) be the formal scheme of “subgroups of ˜ C” of order p (see, for instance, =-=[Str97]-=-). The formal scheme Subp( ˜ C) is of the form Spf(Γ0(p)( ˜ C)). Observe that because the p-divisible group of ˜ C is entirely formal, we have Γ0(p)( ˜ C) = Γ0(p)( ˜ C ∧ ). Let ˜ Hcan be the universal... |

15 | The sigma orientation is an H∞ map
- Ando, Hopkins, et al.
(Show Context)
Citation Context ...ompatible. Remark 6.3. As a side-effect of our construction of O top K(1) it will be the case that the E∞-structure on the spectrum of sections E = O top K(1) (Spf(R)) is θ-compatible. Remark 6.4. In =-=[AHS04]-=-, the authors define the notion of an H∞-elliptic ring spectrum, which is a stronger notion than that of an elliptic H∞-ring spectrum in that they require a compatibility between the H∞-structure and ... |

15 | Geometric Modular Forms and Elliptic Curves, World Scientific - Hida - 2000 |

14 | Arithmetic moduli of generalized elliptic curves - CONRAD |

13 | The Tate spectrum of vn –periodic complex oriented theories, Mathematische Zeitschrift 222 - Greenlees, Sadofsky - 1996 |

13 | The stack of formal groups in stable homotopy theory
- Naumann
(Show Context)
Citation Context ... formal group of the universal generalized elliptic curve is flat (this can be verified using Serre-Tate theory, see [BL, Lemma 9.1.6]). Thus the spectrum of sections E = O top (R) is Landweber exact =-=[Nau]-=-. Fibrant presheaves of spectra satisfy homotopy descent, and so the values of the presheaf are determined by values on the affine opens using étale descent. Remark 1.5. The spectrum tmf is defined to... |

12 |
Completions in algebra and topology.” Handbook of algebraic topology
- Greenlees, P
(Show Context)
Citation Context ...C. Then (1) E is E(2)-local. (2) Suppose that R is p-complete, and that the classifying map Spf(R) → (Mell)p for C is flat. Then there is an equivalence E K(1) � E[v −1 1 ]p. Proof. Greenlees and May =-=[GM95]-=- proved that there is an equivalence They also showed there is a spectral sequence E E(n) � E[I −1 n+1 ]. (8.1) H s (Spec(R) − Xn, ω ⊗t ) ⇒ π2t−sE[I −1 n+1 ] where Xn = Spec(R/In+1) is the locus of Sp... |

10 | Moduli problems for structured ring spectra (2005) Available at http://hopf.math.purdue.edu Algebraic & Geometric Topology, Volume 9 (2009) theory of modules over operads in symmetric spectra 1679
- Goerss, Hopkins
(Show Context)
Citation Context ...r R, ∼ = −→ � C) = an isomorphism of formal groups. (Note that the existence of η implies that C has ordinary reduction modulo p.) The ring V ∧ ∞ possesses a special structure: it is a θ-algebra (see =-=[GH]-=-). That is, it has actions of operators ψ k , k ∈ Z × p , ψ p , lift of the Frobenius, θ, satisfying ψ p (x) = x p + pθ(x). The operations ψ k and ψ p are ring homomorphisms. The operation θ is determ... |

9 | Arithmetic moduli of elliptic curves. Annals of Mathematics studies, study 108 - Katz, Mazur - 1985 |

8 |
Uniformizing the stacks of abelian sheaves, Number fields and function fields—two parallel worlds
- Hartl
- 2005
(Show Context)
Citation Context ...rtant comments on our use of formal geometry in this paper. (1) The object (Mell)p is a formal Deligne-Mumford stack. We shall use these throughout this paper — we refer the reader to the appendix of =-=[Har05]-=- for some of the basic definitions. Given a formal Deligne-Mumford stack X and a ring R complete with respect to an ideal I, we define the R-points of X by X (R) = lim ←−i X (R/I i ). (2) If R is comp... |

7 | CW simplicial resolutions of spaces with an application to loop spaces - Blanc |

5 |
Replacing homotopy actions by topological actions
- Cooke
- 1978
(Show Context)
Citation Context ...t there is a map of groups (Z/p) × → [tmf (p) ord , tmf (p) ord ] × E∞ . The obstructions to lifting this homotopy action to a point-set action may be identified using the obstruction theory of Cooke =-=[Coo78]-=- (adapted to the topological category of E∞-ring spectra). Namely, the obstructions lie in the group cohomology H s ((Z/p) × , πs−2(E∞(tmf (p) ord , tmf (p) ord ), Id)), s ≥ 3. Since the space E∞(tmf ... |

4 | Tate cohomology and periodic localization of polynomial functors - Kuhn |

4 |
K(1)-local topological modular forms
- Laures
(Show Context)
Citation Context ...ctrum tmf K(1) such that there is an isomorphism of graded reduced θ-algebras (KO ∧ 2 )∗tmf K(1) ∼ = KO∗ ⊗ V ∧ ∞. Remark 7.14. There is another construction of tmf K(1) at p = 2 which is described in =-=[Lau04]-=- (see also [Hop]). The spectrum is explicitly constructed by attaching two K(1)-local E∞-cells to the K(1)-local sphere. Unfortunately, it seems that this approach does not generalize to primes p ≥ 5,... |

1 |
K(1)-local E∞-ring spectra, available from Doug Ravenel’s web-page
- Hopkins
(Show Context)
Citation Context ...h that there is an isomorphism of graded reduced θ-algebras (KO ∧ 2 )∗tmf K(1) ∼ = KO∗ ⊗ V ∧ ∞. Remark 7.14. There is another construction of tmf K(1) at p = 2 which is described in [Lau04] (see also =-=[Hop]-=-). The spectrum is explicitly constructed by attaching two K(1)-local E∞-cells to the K(1)-local sphere. Unfortunately, it seems that this approach does not generalize to primes p ≥ 5, though it does ... |

1 |
modular forms, the Witten genus, and the theorem of the cube
- Topological
- 1994
(Show Context)
Citation Context ...properties of the spectrum of sections of E = O top (Spec(R)) enumerated in Theorem 1.1 make E an elliptic spectrum associated to the generalized elliptic curve C/R in the sense of Hopkins and Miller =-=[Hop95]-=-. Thus Theorem 1.1 gives a functorial collection of E∞-elliptic spectra associated to the collection of generalized elliptic curves whose classifying maps are étale. Remark 1.4. This theorem practical... |

1 | Supplementary notes for math 512, Available at http://www.math.uiuc.edu/∼rezk - Rezk |

1 |
Higher homotopy operations and André-Quillen cohomology
- Blanc, Johnson, et al.
(Show Context)
Citation Context ...→ πs,t(X•; P ) → π ♮ s−2,t+1 (X•; P ) → · · · . These We shall omit P from the notation when P = S0 . We will closely follow the explicit treatment of obstruction theory given by Blanc-Johnson-Turner =-=[BJT]-=-, adapted to our setting. Namely, we will produce a free simplicial resolution W• of the reduced theta algebra A0, and then analyze the obstructions to inductively producing an explicit object X• ∈ sA... |