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Quasi-elliptic cohomology is closely related to Tate K-theory. It is constructed as an object both reflecting the geometric nature of elliptic curves and more practicable to study than most elliptic cohomology theories. It can be interpreted by orbifold loop spaces and expressed in terms of equivariant K-theories. We formulate the complete power operation of this theory. Applying that we prove the finite subgroups of Tate curve can be classified by the Tate K-theory of symmetric groups modulo a certain transfer ideal.

In addition, we define twisted quasi-elliptic cohomology. They can be related to a twisted equivariant version Devoto’s elliptic cohomology via a Chern character map. Moreover, we construct twisted Real quasi-elliptic cohomology and the Chern character map in this case. This is joint work with Matthew Spong and Matthew Young.