group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
An (even) 2-periodic cohomology theory or just periodic cohomology theory for short is an (even) multiplicative cohomology theory $E$ with a Bott element $\beta \in E^2({*})$ which is invertible (under multiplication in the cohomology ring of the point) so that multiplication by it induces an isomorphism
Via the Brown representability theorem this corresponds to a periodic ring spectrum.
Compare with the notion of weakly periodic cohomology theory.
More generally one considers $2n$-periodic cohomology theories
For $E$ an E-∞ ring representing a periodic cohomology (a periodic ring spectrum) double suspension/looping on any $E$-∞-module $N$ is equivalent to the identity
This equivalence ought to be coherent to yield a $\mathbb{Z}/2\mathbb{Z}$ ∞-action on the (∞,1)-category of (∞,1)-modules $E Mod$ (MO discussion).
There is an analogue of the Landweber exact functor theorem for even 2-periodic cohomology theories, with MU replaced by MP (Hovey-Strickland 99, theorem 2.8, Lurie lecture 18, prop. 11).
elliptic cohmology?
The concept of even 2-periodic multiplicative cohomology theories originates with
The analogue of the Landweber exact functor theorem for even 2-periodic cohomology is discussed in
Mark Hovey, Neil Strickland, theorem 2.8 of Morava K-theories and localisation Mem. Amer. Math. Soc., 139(666):viii+100, 1999.
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 , Lecture 18 Even periodic cohomology theories (pdf)
Review includes
section 5.1 by Markus Land in these “TMF seminar” notes: pdf
Akhil Mathew, Lennart Meier, section 2.1 of Affineness and chromatic homotopy theory, J. Topol. 8 (2015), no. 2, 476–528 (arXiv:1311.0514)