On the Adams E2-term for Elliptic Cohomology

We investigate the E2-term of Adams spectral sequence based on elliptic homology. The main results describe this E2-term from a ‘chromatic’ perspective. At a prime p > 3, the Bousfield class of Ell is the same as that of K(0)∨K(1)∨ K(2). Using delicate facts due to Katz (which also play a major rôle in work on the structure Ell∗Ell by Clarke & Johnson, the author and Laures) as well as our description of supersingular elliptic cohomology in terms of K(2)-theory, we show that the E2-term is chromatically of length 2 and totally determined by the 0, 1 and 2 columns of the usual chromatic spectral sequence for BP . We apply our results to recover results of [7,13] and indeed extend them to completely determine this Adams E2-term. In the Appendix we reprove Katz’s result and a generalisation which allows a similar analysis of the chromatic spectral sequence for the E2-term of the Adams spectral sequence based on E(2). This approach is also of use in connection with the more general case associated to E(n) for n > 2. Introduction. Although the main driving force behind the development of elliptic cohomology undoubtedly lies in its geometric significance, it also has considerable interest to algebraic topologists. Versions of elliptic cohomology defined using Landweber’s Exact Functor Theorem turn out to have a rich enough internal structure to make their analysis worthwhile. For example, their stable operation algebras contain Hecke operators and the dual cooperation algebras are arithmetically interesting. Moreover, their reductions modulo a prime p reflect the theory of ‘ordinary’ reductions of elliptic curves and the theory of mod p modular forms due to SwinnertonDyer and Serre, as well as the theory of supersingular reductions. Such reductions and localisations are related to v1 and v2-periodicity in the associated cohomology theories, which suggests that a geometrically defined version of elliptic cohomology should have the potential to capture both types of periodicity. Work of Clarke & Johnson [7], the author [4] and Laures [13] has put the structure of the cooperation algebra Ell∗Ell on a firm footing, and attention has turned to applications in stable homotopy theory, particularly the Adams spectral sequence based on elliptic (co)homology. It is clear from these papers that interesting connections exist between algebraic or homotopy theoretic questions about the E2-term 1991 Mathematics Subject Classification. 55N20, 55N22, 55T15 (11F11).