## Elliptic spectra, the Witten genus and the theorem of the cube (1997)

Venue: | Invent. Math |

Citations: | 69 - 16 self |

### BibTeX

@ARTICLE{Ando97ellipticspectra,,

author = {M. Ando and M. J. Hopkins and N. P. Strickland},

title = {Elliptic spectra, the Witten genus and the theorem of the cube},

journal = {Invent. Math},

year = {1997},

volume = {146},

pages = {10--1007}

}

### Years of Citing Articles

### OpenURL

### Abstract

2. More detailed results 7 2.1. The algebraic geometry of even periodic ring spectra 7

### Citations

302 | On the structure of Hopf algebras - Milnor, Moore - 1965 |

167 |
Stable Homotopy and Generalised Homology
- Adams
- 1974
(Show Context)
Citation Context ...lusion S 0 → P L−1 of the bottom cell. It is equivalent to give a class x ∈ ˜ E 2 (P ) whose restriction to ˜ E 2 (S 2 ) is the suspension of 1 ∈ ˜ E 0 S 0 ; this is the description of maps MU → E in =-=[Ada74]-=-. 2.5. The σ–orientation of an elliptic spectrum.sELLIPTIC SPECTRA 19 2.5.1. Elliptic spectra and the Theorem of the Cube. Let C be a generalized elliptic curve over an affine scheme S. To begin, note... |

130 |
Les schémas de modules de courbes elliptiques
- Deligne, Rapoport
- 1973
(Show Context)
Citation Context ... of schemes and sheaf cohomology. For more information, and proofs of results merely stated here, see [Del75, KM85, Sil94, DR73]. Note, however, that our definition is not quite equivalent to that of =-=[DR73]-=-: their generalized elliptic curves are more generalized than ours, so what we call a generalized elliptic curve is what they would call a stable curve of genus 1 with a specified section in the smoot... |

127 |
p-adic properties of modular schemes and modular forms
- Katz
- 1973
(Show Context)
Citation Context ...he Tate curve CTate, and give an explicit formula for the cubical structure s( � CTate). For further information about the Tate curve, the reader may wish to consult for example [Sil94, Chapter V] or =-=[Kat73]-=-. By way of motivation, let’s work over the complex numbers. Elliptic curves over C can be written in the form C × /(u ∼ qu) for some q with 0 < |q| < 1. This is the Tate parameterization, and as is c... |

124 | modules, and algebras in stable homotopy theory - Rings - 1997 |

121 | The Spectrum of an Equivariant Cohomology Ring - Quillen - 1971 |

106 |
Formal Groups and Applications
- Hazewinkel
- 1978
(Show Context)
Citation Context ...lgebra. For d ≥ 2, the group C 2 d( � Ga, � Ga)(A) is the free A-module on the single generator c(d). Let ⎛ E(t) = exp ⎝ � k≥0 t pk p k ⎞ (3.4) ⎠ (3.6) be the Artin-Hasse exponential (see for example =-=[Haz78]-=-). It is of the form 1 mod (t), and it has coefficients in Z (p). For d ≥ 2, let g2(d, b) ∈ C 2 ( � Ga, Gm)(Q[b]) be the power series � δ g2(d, b) = 2 ×(E(bxd ) −p ) if d is a power of p δ2 ×(E(bxd ))... |

76 |
Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs
- Demazure, Gabriel
- 1970
(Show Context)
Citation Context ... connect the topology to the algebraic geometry, we shall express some facts about even periodic ring spectra in the language of algebraic geometry. 2.1.1. Formal schemes and formal groups. Following =-=[DG70]-=-, we will think of an affine scheme as a representable covariant functor from rings to sets. The functor (co-)represented by a ring A is denoted spec A. The ring (co-)representing a functor X will be ... |

61 | The arithmetic of elliptic curves - Tate - 1974 |

60 | The arithmetic of elliptic curves, Graduate Texts in Mathematics 106 - Silverman - 1992 |

58 | Lectures on p-divisible groups - Demazure - 1972 |

56 |
Elliptic Genera and Quantum Field Theory
- Witten
- 1987
(Show Context)
Citation Context ...KO making the diagram MSU −−−−→ MU −−−−→ K ⏐ � � � � MSpin −−−−→ KO −−−−→ K commute. As above, it is customary to suppress the grading and write ⎛ �A ⎝M; � Symqn( ¯ ⎞ TC) ⎠ , which is formula (27) in =-=[Wit87]-=-. We have proved Proposition 2.67. The invariant n≥0 π∗MSpin → Z[[q]] associated to the σ-orientation on KTate is the Witten genus. 2.8. Modularity. Proposition 2.68. For any element [M] ∈ π2nMU〈6〉, t... |

55 | On the formal group laws of unoriented and complex cobordism theory
- Quillen
- 1969
(Show Context)
Citation Context ...lassifying the tensor product of line bundles makes the scheme PE into a (one-dimensional commutative) formal group over SE. The formal group PE is not quite the same as the one introduced by Quillen =-=[Qui69]-=-. The ring of functions on Quillen’s formal group is E ∗ (P ), while the ring of functions on PE is E 0 (P ). The homogeneous parts of E ∗ (P ) can interpreted as sections of line bundles over PE. For... |

54 |
Manifolds and modular forms
- Hirzebruch, Berger, et al.
- 1992
(Show Context)
Citation Context ...us is a cobordism invariant of Spin-manifolds for which λ = 0, and it takes its values in modular forms (of level 1). It has exhibited a remarkably fecund relationship with geometry (see [Seg88], and =-=[HBJ92]-=-). Rich as it is, the theory of the Witten genus is not as developed as are the invariants described by the index theorem. One thing that is missing is an understanding of the Witten genus of a family... |

53 | Algebraic Geometry, volume 52 of Graduate Texts in Mathematics - Hartshorne - 1977 |

52 | The index of the Dirac operator in loop space - Witten - 1988 |

46 | The Hopf ring for complex cobordism - Ravenel, Wilson - 1977 |

40 | Advanced Topics in the Arithmetic of Elliptic Curves, volume 151 of Graduate Texts in Mathematics - Silverman - 1994 |

31 |
Periodic cohomology theories defined by elliptic curves’, The Čech
- Landweber, Ravenel, et al.
- 1993
(Show Context)
Citation Context ...as to which E to choose, and how, in this language, to express the modular invariance of the Witten genus. One candidate for E, elliptic cohomology, was introduced by Landweber, Ravenel, and Stong in =-=[LRS95]-=-. To keep the technicalities to a minimum, we focus in this paper on the restriction of the Witten genus to stably almost complex manifolds with a trivialization of the Chern classes c1 and c2 of the ... |

31 | Products on MU -modules - Strickland - 1999 |

30 | On modular invariance and rigidity theorems - Liu - 1995 |

27 |
Fonctions thêta et theoreme du cube
- Breen
- 1983
(Show Context)
Citation Context ...bundles) (rigid) s(0, 0, 0) = 1 (symmetry) s(a σ(1), a σ(2), a σ(3)) = s(a1, a2, a3) (cocycle) s(b, c, d)s(a, b + c, d) = s(a + b, c, d)s(a, b, d).s4 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND (See =-=[Bre83]-=-, and Remark 2.42 for comparison of conventions.) Our main result (2.50) asserts that the set of multiplicative maps MU〈6〉 −→ E is naturally in one to one correspondence with the set of cubical struct... |

25 |
Topological modular forms, the witten genus, and the theorem of the cube
- Hopkins
- 1994
(Show Context)
Citation Context ...roduction This paper is part of a series ([HMM98, HM98] and other work in progress) getting at some new aspects of the topological approach to elliptic genera. Most of these results were announced in =-=[Hop95]-=-. In [Och87] Ochanine introduced the elliptic genus—a cobordism invariant of oriented manifolds taking its values in the ring of (level 2) modular forms. He conjectured and proved half of the rigidity... |

23 | Groupes de monodromie en géométrie algébrique. I - Rim - 1972 |

21 |
On the Rigidity Theorems of Witten
- Bott, Taubes
- 1989
(Show Context)
Citation Context ... Wit88] Witten interpreted Ochanine’s invariant in terms of index theory on loop spaces and offered a proof of the rigidity theorem. Witten’s proof was made mathematically rigorous by Bott and Taubes =-=[BT89]-=-, and since then there have been several new proofs of the rigidity theorem [Liu95, Ros98]. In the same papers Witten described a variant of the elliptic genus now known as the Witten genus. There is ... |

20 |
Courbes elliptiques: formulaire d’après
- Deligne
- 1972
(Show Context)
Citation Context ...o modular forms of the indicated weights, and one checks directly from the definitions that c 3 4 − c 2 6 = 1728∆. The proof that MF∗ is precisely Z[c4, c6, ∆]/(1728∆ − c 3 4 + c 2 6) can be found in =-=[Del75]-=- and will not be reproduced here.sELLIPTIC SPECTRA 57 Definition B.8. The q-expansion of a modular form g is the series h(q) ∈ Z[[q ] = ODTate such that g(CTate/DTate) = h(q)d(x/y) k 0. Note that if τ... |

18 |
On the construction of elliptic cohomology
- Franke
- 1992
(Show Context)
Citation Context ...C. This is the classical construction of elliptic cohomology; and gives rise to many examples. In fact, the construction identifies a representing spectrum E up to canonical isomorphism, since Franke =-=[Fra92]-=- and Strickland [Str99a, Proposition 8.43] show that there are no phantom maps between Landweber exact elliptic spectra.s10 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND Example 2.7. In §2.6, we descri... |

14 |
Arithmetic Moduli of Elliptic Curves, volume 108 of Annals of Mathematics Studies
- Katz, Mazur
- 1985
(Show Context)
Citation Context ...isors and line bundles. We will need to understand the relationship between divisors and line bundles in a form which is valid for non-Noetherian schemes. An account of divisors on curves is given in =-=[KM85]-=-, but we need to genera-Lise this slightly to deal with divisors on C ×S C ×S C over S, for example. The issues involved are surely well-known to algebraic geometers, but it seems worthwhile to have a... |

10 | Über die Addition und Multiplication der elliptischen Functionen, Jour. für die reine und angewandte Mathematik 88 (1880 - Frobenius, Stickelberger |

9 | Equivariant elliptic cohomology and rigidity - Rosu - 1999 |

8 | Biextensions of formal groups - Mumford - 1968 |

8 | Formal schemes and formal groups
- Strickland
- 1997
(Show Context)
Citation Context ...whereas the construction X ↦→ X E is only functorial for commutative H-spaces and H-maps. It is in fact possible to carry out this program, at least for k ≤ 3. It relies on the apparatus developed in =-=[Str99a]-=-, and the full strength of the present paper is required even to prove that C3(G) (as defined by a suitable universal property) exists. Details will appear elsewhere. 2.4. The complex-orientable homol... |

7 | Elliptic functions, volume 112 of Graduate Texts in Mathematics - Lang - 1987 |

6 |
Sur les genres multiplicatifs définis par des intégrales elliptiques
- Ochanine
- 1987
(Show Context)
Citation Context ...is paper is part of a series ([HMM98, HM98] and other work in progress) getting at some new aspects of the topological approach to elliptic genera. Most of these results were announced in [Hop95]. In =-=[Och87]-=- Ochanine introduced the elliptic genus—a cobordism invariant of oriented manifolds taking its values in the ring of (level 2) modular forms. He conjectured and proved half of the rigidity theorem—tha... |

6 |
Connective fiberings over
- Singer
- 1968
(Show Context)
Citation Context ...L1) + (1 − L2) − (1 − L1L2) = (1 − L1)(1 − L2). Now let E be an even periodic ring spectrum. Applying E-homology to the map ρk gives a homomorphism E0ρk : E0P k → E0BU〈2k〉. For k ≤ 3, BU〈2k〉 is even (=-=[Sin68]-=- or see §4), and of course the same is true of P , and so we may consider the adjoint ˆρk of E0ρk in E0BU〈2k〉�⊗E 0 P k . Proposition 2.25 then implies the following. Corollary 2.26. The element ˆρk ∈ ... |

5 | Forms of K-theory - Morava - 1989 |

5 |
Elliptic cohomology
- Segal
- 1988
(Show Context)
Citation Context ...he Witten genus is a cobordism invariant of Spin-manifolds for which λ = 0, and it takes its values in modular forms (of level 1). It has exhibited a remarkably fecund relationship with geometry (see =-=[Seg88]-=-, and [HBJ92]). Rich as it is, the theory of the Witten genus is not as developed as are the invariants described by the index theorem. One thing that is missing is an understanding of the Witten genu... |

4 |
modules
- Clifford
- 1964
(Show Context)
Citation Context ...zation of I(0), then δf = f ′ (0)Dx f(x) where Dx is the invariant differential with value dx at 0. The K-theory orientation of complex vector bundles MP → K (2.66) constructed by Atiyah-Bott-Shapiro =-=[ABS64]-=- corresponds to the coordinate 1 − u on the formal completion of Gm = spec Z[u, u −1 ]. The invariant differential is D(1 − u) = − du u , and the restriction of (2.66) to MU → K is classified by the Θ... |

4 |
pairings and Morava K-theory
- Weil
(Show Context)
Citation Context ...the set of cubical structures on L = O(−{0}). We have chosen a computational approach to the proof of this theorem partly because it is elementary, and partly because it leads to a general result. In =-=[AS98]-=-, the first and third authors give a less computational proof of this result (for formal groups of finite height in positive characteristic), using ideas from [Mum65, Gro72, Bre83] on the algebraic ge... |

4 |
A note on the Thom isomorphism
- Mahowald, Ray
- 1981
(Show Context)
Citation Context ...now turn our attention to the Thom spectra MU〈2k〉. We first note that when k ≤ 3, the map BU〈2k〉 → BU〈0〉 = Z × BU is a map of commutative, even H-spaces. The Thom isomorphism theorem as formulated by =-=[MR81]-=- implies that E0MU〈2k〉 is an E0BU〈2k〉-comodule algebra; and a choice of orientation MU〈0〉 → E gives an isomorphism E0MU〈2k〉 ∼ = E0BU〈2k〉 of comodule algebras. In geometric language, this means that th... |

3 |
Abelian varieties, volume 5 of Tata Institute of Fundamental Research Studies in Mathematics
- Mumford
- 1970
(Show Context)
Citation Context ... are constants, the requirement s(0, 0, 0) = 1 determines the section uniquely, and shows that it satisfies the “symmetry” and “cocycle” conditions. In fact the “theorem of the cube” (see for example =-=[Mum70]-=-) shows more generally that any line bundle over any abelian variety has a unique cubical structure. Over the complex numbers, a transcendental formula for f(x, y, z) is σ(x + y + z) σ(z) σ(x + y) σ(x... |

2 |
Primitive elements in the K-theory of BSU
- Adams
- 1976
(Show Context)
Citation Context ...cocycle F2 does not have a product decomposition F2 = � g2(d, bd), d≥2 with g2(d, bd) having leading term of degree d, until one localizes at a prime p. The analogous result for H∗BSU is due to Adams =-=[Ada76]-=-. Fix a prime p. For d ≥ 2, let c(d) ∈ Z[x1, x2] be the polynomial � 1 p c(d) = (xd1 + xd 2 − (x1 + x2) d ) d = ps for some s ≥ 1 xd 1 + xd 2 − (x1 + x2) otherwise The following calculation of C 2 ( �... |

2 | Elliptic curves and stable homotopy theory I. in preparation - Hopkins, Mahowald, et al. - 1998 |

2 | Fibre Bundles, volume 33 of Graduate Texts in Mathematics - Husemoller - 1975 |

1 | Elliptic curves and stable homotopy theory II - Hopkins, Mahowald - 1998 |

1 | Formulae novae in theoria transcendentium ellipticarum fundamentales. Crelle J. für die reine u - Jacobi - 1881 |