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, Ell_*Ell, 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 Ell*Ell by first determining the dual cooperation algebra Ell_*Ell. A major ingredient is our identification of the cooperation algebra Ell_*Ell 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.

The author acknowledges the support of the Science and Engineering Research Council, the Max-Planck-Institut für Mathematik, Glasgow University, Johns Hopkins University, Manchester University and Osaka Prefecture whilst parts of this work were undertaken.