Abstract
The purpose of this paper is to give the algebraic topological background for elliptic cohomology theory and to pose some number theoretic problems suggested by these concepts. Algebraic topologists study functors from the category of spaces to various algebraic categories. In particular there are functors to the category of graded rings called multiplicative generalized cohomology theories. (All rings are assumed to be commutative and unital. Graded rings are commutative subject to the usual sign conventions, i.e., odd-dimensional elements anticommute with each other.) These functors satisfy all of the Eilenberg-Steenrod axioms but the dimension axiom. In other words they have the same formal properties as ordinary cohomology except that the cohomology of a point may be more complicated. For a discussion of these axioms the interested reader should consult [S] or [ES]; for generalized cohomology theories a good reference is Part III of [A1]. Among these cohomology theories the following examples are mentioned elsewhere in this volume:
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