The topological q-expansion principle
Abstract (summary)
Imitating the classical q-expansion principle we use the elliptic character map to develop the relation between elements in elliptic cohomology and their q-series in K-theory. We show that, under certain exactness conditions, the integrality of elliptic objects is completely controlled by their characters.
As an application, we obtain an interpretation of the cooperations in elliptic cohomology as was conjectured by F. Clarke and K. Johnson. It enables us to give a description of the elliptic based Adams-Novikov spectral sequence in terms of cyclic cohomology of modular forms in several variables, and to set up a higher e-invariant with values in N. Katz's ring of divided congruences.
We show how the topological q-expansion principle can be used to equip elliptic cohomology with orientations which obey various Riemann-Roch formulas. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)