A Survey of Elliptic Cohomology
By Jacob Lurie. 2009
135 citations recorded by the source corpus.
This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in Sect. 1 with an overview of the classical theory of elliptic cohomology. In Sect. 2 we review the theory of E∞-ring spectra and introduce the language of derived algebraic geometry.We apply this theory in Sect. 3, where we introduce the notion of an oriented group scheme and describe connection between oriented group schemes and equivariant cohomology theories. In Sect. 4 we sketch a proof of our main result, which relates the classical theory of elliptic cohomology to the classification of oriented elliptic curves. In Sect. 5 we discuss various applications of these ideas, many of which rely upon a special feature of elliptic cohomology which we call 2-equivariance. The theory that we are going to describe lies at the intersection of homotopy theory and algebraic geometry. We have tried to make our exposition accessible to those who are not specialists in algebraic topology; however, we do assume the reader is familiar with the language of algebraic geometry, particularly with the theory of elliptic curves. In order to keep our account readable, we will gloss overmany details, particularly where the use of higher category theory is required. A more comprehensive account of the material described here, with complete definitions and proofs, will be given in [1].
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/3a49d2f5272b5a287f15761355ebcf92c823b35f
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Citations in the corpus
- The C2–spectrum Tmf1(3) and its invertible modules (2015, background): “As a last point, we mention that Lurie defines in [40] a sheaf of E1–ring spectra on the n–torsion of the universal elliptic curve over Mell whose global sections provide the value of Cn –equivariant TMF at a point; if we invert n, this can be analyzed in terms of the TMF1.”
- Affineness and chromatic homotopy theory (2013, methodology): “A construction of Mell is detailed in [8], and another is sketched in [34].”
- Commutativity conditions for truncated Brown–Peterson spectra of height 2 (2011, background): “These in turn can be produced from families of elliptic curves (leading to topological modular forms [38]) and, more generally, from suitable abelian varieties (leading to topological automorphic forms [8]).”
- Derived algebraic geometry (2014, background): “4]), and has found an impressive application to the study of topological modular forms (see [92]).”
- Moduli Problems for Ring Spectra (2011, methodology): “One can use this enhancement to give a moduli-theoretic reformulation of the Goerss-Hopkins-Miller theory of topological modular forms; we refer the reader to [21] for are more detailed discussion.”
- Topological modular forms with level structure (2013, background): “We being by recalling the following (see [2,6,34]).”
- Nilpotence and descent in equivariant stable homotopy theory (2015, methodology): “[11]), we will need to use the existence of models of certain E∞-algebras in SpG (namely, equivariant real and complex K-theory), even though an ∞-categorical treatment (and new construction) is to appear in Lurie’s forthcoming work on elliptic cohomology (see [43] for a survey).”
- Elliptic stable envelopes (2016, background): “2 Equivariant elliptic cohomology, developed in [24, 26, 27, 31, 41, 55] and other papers, defines a functor”
- Derived Algebraic Geometry (2014, background): “4]), and has found an impressive application to the study of topological modular forms (see [92]).”