 

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 


Abstract
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, SwinnertonDyer,
Serre and Katz cannot extend to a stable operation.


Acknowledgements
The author acknowledges the support of the Science and Engineering Research Council, the MaxPlanckInstitut 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.
andy@@maths.gla.ac.uk

