In mathematics, **elliptic cohomology** is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.

Historically, elliptic cohomology arose from the study of elliptic genera. It was known by Atiyah and Hirzebruch that if acts smoothly and non-trivially on a spin manifold, then the index of the Dirac operator vanishes. In 1983, Witten conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant. This led to certain other problems concerning -actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera. In turn, Witten related these to (conjectural) index theory on free loop spaces. Elliptic cohomology, invented in its original form by Landweber, Stong and Ravenel in the late 1980s, was introduced to clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differential operators on free loop spaces. In some sense it can be seen as an approximation to the K-theory of the free loop space.

Call a cohomology theory even periodic if for i odd and there is an invertible element . These theories possess a complex orientation, which gives a formal group law. A particularly rich source for formal group laws are elliptic curves. A cohomology theory with

is called *elliptic* if it is even periodic and its formal group law is isomorphic to a formal group law of an elliptic curve over . The usual construction of such elliptic cohomology theories uses the Landweber exact functor theorem. If the formal group law of is Landweber exact, one can define an elliptic cohomology theory (on finite complexes) by

Franke has identified the condition needed to fulfill Landweber exactness:

- needs to be flat over
- There is no irreducible component of , where the fiber is supersingular for every

These conditions can be checked in many cases related to elliptic genera. Moreover, the conditions are fulfilled in the universal case in the sense that the map from the moduli stack of elliptic curves to the moduli stack of formal groups

is flat. This gives then a presheaf of cohomology theories

over the site of affine schemes flat over the moduli stack of elliptic curves. The desire to get a universal elliptic cohomology theory by taking global sections has led to the construction of the topological modular forms ^{ [1] }^{pg 20}

as the homotopy limit of this presheaf over the previous site.

In mathematics, **complex geometry** is the study of complex manifolds, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

In mathematics, a **sheaf** is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.

The **Riemann–Roch theorem** is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus *g*, in a way that can be carried over into purely algebraic settings.

In mathematics, in particular algebraic geometry, a **moduli space** is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces are formal moduli.

In mathematics, the **étale cohomology groups** of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct **ℓ-adic cohomology**, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

In mathematics, an **algebraic stack** is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin.

In mathematics, a **gerbe** is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.

In mathematics, **complex cobordism** is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.

In mathematics, specifically in symplectic topology and algebraic geometry, **Gromov–Witten** (**GW**) **invariants** are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten.

In algebraic geometry, a **noetherian scheme** is a scheme that admits a finite covering by open affine subsets , noetherian rings. More generally, a scheme is **locally noetherian** if it is covered by spectra of noetherian rings. Thus, a scheme is noetherian if and only if it is locally noetherian and quasi-compact. As with noetherian rings, the concept is named after Emmy Noether.

In algebraic geometry, a **moduli space of** (**algebraic**) **curves** is a geometric space whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding **moduli problem** and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.

In mathematics, **topological modular forms (tmf)** is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer *n* there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space *X*, one obtains an abelian group structure on the set of homotopy classes of continuous maps from *X* to . One feature that distinguishes tmf is the fact that its coefficient ring, (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring.

In mathematics a **stack** or **2-sheaf** is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.

In algebraic topology, a branch of mathematics, the **Čech-to-derived functor spectral sequence** is a spectral sequence that relates Čech cohomology of a sheaf and sheaf cohomology.

In mathematics, the **Landweber exact functor theorem**, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal group law. The Landweber exact functor theorem can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.

**Peter Steven Landweber** is an American mathematician working in algebraic topology.

In mathematics, **chromatic homotopy theory** is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories it studies include: complex K-theory, elliptic cohomology, Morava K-theory and tmf.

In algebraic geometry, the **moduli stack of formal group laws** is a stack classifying formal group laws and isomorphisms between them. It is denoted by . It is a "geometric “object" that underlies the chromatic approach to the stable homotopy theory, a branch of algebraic topology.

In mathematics, the **moduli stack of elliptic curves**, denoted as or , is an algebraic stack over classifying elliptic curves. Note that it is a special case of the Moduli stack of algebraic curves . In particular its points with values in some field correspond to elliptic curves over the field, and more generally morphisms from a scheme to it correspond to elliptic curves over . The construction of this space spans over a century because of the various generalizations of elliptic curves as the field has developed. All of these generalizations are contained in .

In algebraic geometry, a **presheaf with transfers** is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additive functor from the category of finite correspondences to the category of abelian groups.

- Franke, Jens (1992), "On the construction of elliptic cohomology",
*Mathematische Nachrichten*,**158**(1): 43–65, doi:10.1002/mana.19921580104 . - Landweber, Peter S. (1988), "Elliptic genera: An introductory overview", in Landweber, P. S. (ed.),
*Elliptic Curves and Modular Forms in Algebraic Topology*, Lecture Notes in Mathematics,**1326**, Berlin: Springer, pp. 1–10, ISBN 3-540-19490-8 . - Landweber, Peter S. (1988), "Elliptic cohomology and modular forms", in Landweber, P. S. (ed.),
*Elliptic Curves and Modular Forms in Algebraic Topology*, Lecture Notes in Mathematics,**1326**, Berlin: Springer, pp. 55–68, ISBN 3-540-19490-8 . - Landweber, P. S.; Ravenel, D. & Stong, R. (1995), "Periodic cohomology theories defined by elliptic curves", in Cenkl, M. & Miller, H. (eds.),
*The Čech Centennial 1993*, Contemp. Math.,**181**, Boston: Amer. Math. Soc., pp. 317–338, ISBN 0-8218-0296-8 . - Lurie, Jacob (2009), "A Survey of Elliptic Cohomology", in Baas, Nils; Friedlander, Eric M.; Jahren, Björn; et al. (eds.),
*Algebraic Topology: The Abel Symposium 2007*, Berlin: Springer, pp. 219–277, doi:10.1007/978-3-642-01200-6, hdl: 2158/373831 , ISBN 978-3-642-01199-3 .

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