group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
An elliptic spectrum is a spectrum which represents an elliptic cohomology theory.
For $E$ a ring spectrum, write $E^\bullet(\ast)$ for its coefficient ring and generally $E^\bullet(X)$ for its generalized cohomology ring over any $X$.
An elliptic spectrum is a triple consisting of
an elliptic curve $A$ over the coefficient ring $E^\bullet(\ast)$;
an even periodic ring spectrum $E$;
an equivalence
between the algebraic spectrum of the $E$-cohomology ring over the classifying space for complex line bundles (see at complex oriented cohomology theory) and the formal Picard group $Pic_A^0$ of $A$.
This is due to (Ando-Hopkins-Strickland01, def. 1.2). See for instance also (Gepner 05, def. 15).
Originally (and still in many or even most references), def. 1 is stated with the formal Picard group $Pic_A^0$ replaced by the formal completion $\hat A$ of $A$ at its neutral element.
These two versions of the definition in itself are equivalent, since elliptic curves are self-dual abelian varieties equipped with a canonical isomorphism $A\simeq Pic_X^0$exhibited by the Poincaré line bundle.
But for the development of the theory, notably for application to equivariant elliptic cohomology, for the relation of elliptic cohomology to loop group representations etc., it is crucial to understand that $E^\bullet(B U(1))$ is the space of sections of a line bundle over a (formal) moduli space of line bundles on the elliptic curve, instead of on the elliptic curve itself.
Indeed, generally for $G$ a compact Lie group, then $E^\bullet(B G)$ is the space of sections of the WZW model-line bundle on the (formal) moduli space of flat connections on $G$-principal bundles over the elliptic curve. This is the central statement at equivariant elliptic cohomology. As the appearance of the WZW model here shows, this is also crucial for understanding the role of elliptic spectra in quantum field theory/string theory, see at equivariant elliptic cohomology – Interpretation in Quantum field theory/String theory for more on this.
Moreover, understanding $Spec E^\bullet(BU(1))$ as being about moduli of line bundles on the elliptic curve is crucial for understanding the generalization of the concept of elliptic spectra, for instance to K3-spectra. This is indicated in the following table
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
The concept of elliptic spectrum was introduced in
A brief review is for instance in of
Survey includes
Charles Rezk, Elliptic cohomology and elliptic curves, Felix Klein Lectures, Bonn 2015 (web)