New York Journal of Mathematics
Volume 1 (1994-1995) 39-74


Andrew Baker

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

Published: January 10, 1995
Keywords: Elliptic cohomology, modular forms, operations and cooperations
Subject: 55N20, 55N22, 55S25

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.

Author information

Department of Mathematics, Glasgow University, Glasgow G12 8QW, Scotland.