Hecke Algebras Acting on Elliptic Cohomology

Introduction. In our earlier papers [2,3,4,5,6], we investigated stable operations and cooperations in elliptic cohomology and its variants, relating these to known operations on rings of modular forms. The purpose of this article is to give an introduction to these stable operation algebras, in particular explaining the connections with Hecke algebras and Morava stabilizer algebras; further details will appear in [7]. In [9] we make use of some of the Hecke operations described here to calculate parts of the Adams E2-term based on elliptic homology. In work currently in progress we describe the operation algebra for the supersingular elliptic cohomology of §4 in terms of isogenies of supersingular elliptic curves over finite fields and use this to study the v2-periodic part of elliptic cohomology Adams E2-term which was determined in [8]. I would like to thank Jack Morava for providing the original stimulus for this paper and also to offer felicitations and thanks to Mark Mahowald, who always knows a good operation.