Rational S^1-equivariant elliptic cohomology

TL;DR

This paper constructs a 2-periodic S^1-equivariant elliptic cohomology theory for elliptic curves over rationals, linking sheaf cohomology of the curve with equivariant cohomology, and explores its algebraic and computational aspects.

Contribution

It provides an explicit, natural construction of elliptic cohomology theories associated with elliptic curves over rationals, connecting sheaf cohomology with equivariant homotopy theory.

Findings

Constructed a 2-periodic S^1-equivariant cohomology theory for elliptic curves.

Established an explicit link between sheaf cohomology and equivariant cohomology.

Explored algebraic models and computational methods for the theory.

Abstract

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.