Rational S^1-equivariant elliptic cohomology
  For each elliptic curve A over the rational numbers we construct a 2-periodic S^1-equivariant cohomology theory E whose cohomology ring is the sheaf cohomology of A; the homology of the sphere of the representation z^n is the cohomology of the divisor A(n) of points with order dividing n. The construction proceeds by using the algebraic models of the author's AMS Memoir ``Rational S^1 equivariant homotopy theory.'' and is natural and explicit in terms of sheaves of functions on A.   This is Version 5.2 of a paper of long genesis (this should be the final version). The following additional topics were first added in the Fourth Edition:   (a) periodicity and differentials treated   (b) dependence on coordinate   (c) relationship with Grojnowksi's construction and, most importantly,   (d) equivalence between a derived category of O_A-modules and a derived category of EA-modules. The Fifth Edition included   (e) the Hasse square and   (f) explanation of how to calculate maps of EA-module spectra. 
J.P.C.Greenlees