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