Operations and Cooperations in Elliptic Cohomology, Part I: Generalized modular forms and the cooperation algebra

This is the first of two interconnected parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra for elliptic cohomology, E``∗E``, while Part II is devoted to the more algebraic theory associated with Hecke algebras and stable operations in elliptic cohomology. We investigate the structure of the stable operation algebra E``∗E`` by first determining the dual cooperation algebra E``∗E``. A major ingredient is our identification of the cooperation algebra E``∗E`` with a ring of generalized modular forms whoses exact determination involves understanding certain integrality conditions; this is closely related to a calculation by N. Katz of the ring of all ‘divided congruences’ amongst modular forms. We relate our present work to previous constructions of Hecke operators in elliptic cohomology. We also show that a well known operator on modular forms used by Ramanujan, Swinnerton-Dyer, Serre and Katz cannot extend to a stable operation. Introduction This paper is in two interelated parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra E``∗E``, while Part II will be devoted to the more algebraic theory associated with Hecke algebras and operations in elliptic cohomology. In our earlier paper [6], we defined operations in the ‘level 1’ version of elliptic cohomology E``( ) which restricted to the classical Hecke operators on the coefficient ring E``∗ (defined to be a ring of modular forms for the full modular group SL2(Z)). In the present paper we investigate the structure of the operation algebra E``E`` by determining the dual cooperation algebra E``∗E``, thus following the pattern established in the case of K-theory; we also describe a category Received August 3, 1994 Mathematics Subject Classification. 55N20, 55N22, 55S25.