Forms ofK-theory

Morava, Jack

doi:10.1007/bf01214905

Cited by 35 publications

(23 citation statements)

References 14 publications

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“…Rather, there is a canonical short exact sequence of profinite groups (15.1.1) 1 → S 1 → Aut E∞ (E 1 ) → Gal → 1 and the choice of formal group G m over F p gives rise to a splitting. More generally, Morava [Mor89] studied forms of K-theory: p-complete ring spectra K such that there exists an isomorphism of multiplicative cohomology theories The E ∞ -form K α associated to the cohomology class α is given as the homotopy fixed points of the new Galois action on E 1 induced by the inclusion ι α :…”

Section: The Product Decompositionmentioning

confidence: 99%

Topological automorphic forms

Behrens¹,

Lawson²

2010

Memoirs of the AMS

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Introduction ix 0.1. Background and motivation ix 0.2. Subject matter of this book xvii 0.3. Organization of this book xxiii 0.4. Acknowledgments xxv Chapter 1. p-divisible groups 1.1. Definitions 1.2. Classification Chapter 2. The Honda-Tate classification 2.1. Abelian varieties over finite fields 2.2. Abelian varieties over F p Chapter 3. Tate modules and level structures 3.1. Tate modules of abelian varieties 3.2. Virtual subgroups and quasi-isogenies 3.3. Level structures 3.4. The Tate representation 3.5. Homomorphisms of abelian schemes Chapter 4. Polarizations 4.1. Polarizations 4.2. The Rosati involution 4.3. The Weil pairing 4.4. Polarizations of B-linear abelian varieties 4.5. Induced polarizations 4.6. Classification of weak polarizations Chapter 5. Forms and involutions 5.1. Hermitian forms 5.2. Unitary and similitude groups 5.3. Classification of forms Chapter 6. Shimura varieties of type U (1, n − 1) 6.1. Motivation 6.2. Initial data 6.3. Statement of the moduli problem 6.4. Equivalence of the moduli problems 6.5. Moduli problems with level structure 6.6. Shimura stacks v vi CONTENTS Chapter 7. Deformation theory 7.1. Deformations of p-divisible groups 7.2. Serre-Tate theory 7.3. Deformation theory of points of Sh Chapter 8. Topological automorphic forms 8.1. The generalized Hopkins-Miller theorem 8.2. The descent spectral sequence 8.3. Application to Shimura stacks Chapter 9. Relationship to automorphic forms 9.1. Alternate description of Sh(K p ) 9.2. Description of Sh(K p ) F 9.3. Description of Sh(K p ) C 9.4. Automorphic forms Chapter 10. Smooth G-spectra 10.1. Smooth G-sets 10.2. The category of simplicial smooth G-sets 10.3. The category of smooth G-spectra 10.4. Smooth homotopy fixed points 10.5. Restriction, induction, and coinduction 10.6. Descent from compact open subgroups 10.7. Transfer maps and the Burnside category Chapter 11. Operations on TAF 11.1. The E ∞ -action of GU (A p,∞ ) 11.2. Hecke operators Chapter 12. Buildings 12.1. Terminology 12.2. The buildings for GL and SL 12.3. The buildings for U and GU Chapter 13. Hypercohomology of adele groups 13.1. Definition of Q GU and Q U 13.2. The semi-cosimplicial resolution Chapter 14. K(n)-local theory 14.1. Endomorphisms of mod p points 14.2. Approximation results 14.3. The height n locus of Sh(K p ) 14.4. K(n)-local TAF 14.5. K(n)-local Q U Chapter 15. Example: chromatic level 1 15.1. Unit groups and the K(1)-local sphere 15.2. Topological automorphic forms in chromatic filtration 1 Bibliography Index

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Section: The Product Decompositionmentioning

confidence: 99%

Topological automorphic forms

Behrens¹,

Lawson²

2010

Memoirs of the AMS

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“…Reducing modulo 2, the first summand vanishes. Reducing further to Morava Ktheory [46], taking 2 n -th powers is an automorphism of the theory, and u is invertible. Consequently, by completeness, the same conclusion must hold for EO (2): j E (2x) for a bundle x is just an image of j E (x) under an automorphism of the theory, hence cannot vanish if j E (x) does not vanish.…”

Section: Eo(2)-orientable We Have An Eo(2)-orientation Class [X] Eo(mentioning

confidence: 99%

M-theory, type IIA superstrings, and elliptic cohomology

Kříž¹,

Sati²

2004

Advances in Theoretical and Mathematical Physics

166

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The topological part of the M-theory partition function was shown by Witten to be encoded in the index of an E 8 bundle in eleven dimensions. This partition function is, however, not automatically anomalyfree. We observe here that the vanishing W 7 = 0 of the DiaconescuMoore-Witten anomaly [1] in IIA and compactified M-theory partition function is equivalent to orientability of spacetime with respect to (complex-oriented) elliptic cohomology. Motivated by this, we define an elliptic cohomology correction to the IIA partition function, and propose its relationship to interaction between 2-and 5-branes in the M-theory limit.e-print archive: http://lanl.arXiv.org/abs/hep-th/0404013

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“…The first elliptic cohomology theory was discovered by Morava in 1973 (see [Mor89]): the elliptic curve is the Tate elliptic curve, Tate. Its formal group is the multiplicative group, so it is a form of K-theory.…”

Section: Theorem C (Theorem 62) the Elliptic Cohomology Theory Assomentioning

confidence: 99%