Rational S1-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 author's algebraic model of rational S-1-spectra [Greenlees, Mem. Am. Math. Soc. 661 (1999) xii +289pp.]. The construction 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. (c) 2005 Elsevier Ltd. All rights reserved.