Rational -equivariant elliptic cohomology

We give a functorial construction of a rational S 1 -equivariant cohomology theory from an elliptic curve A equipped with suitable coordinate data. The elliptic curve may be recovered from the cohomology theory; indeed, the value of the cohomology theory on the compactification of an S 1 -representation is given by the sheaf cohomology of a suitable line bundle on A. This suggests the construction: by considering functions on the elliptic curve with specified poles one may write down the representing S 1 -spectrum in the authorʹs algebraic model of rational S 1 -spectra [Greenlees, Mem. Am. Math. Soc. 661 (1999) xii +289pp.]. nstruction extends to give an equivalence of categories between the homotopy category of module S 1 -spectra over the representing spectrum and a derived category of sheaves of modules over the structure sheaf of A.