Volume 271, 2001
On the Adams
for elliptic cohomology
ABSTRACT. We investigate the E2-term of Adams spectral sequence based on
elliptic homology. The main results describe this E2-term from a 'chromatic'
At a prime
3, the Bousfield class of EU is the same as that of
K(O) V K(1) V K(2). Using delicate facts due to Katz (which also play a
major role in work on the structure EU. EU 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
This approach is also of use in
connection with the more general case associated to E(n) for n 2.
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
Heeke operators and the dual cooperation algebras are arithmetically interesting.
Moreover, their reductions modulo a prime
reflect the theory of 'ordinary' reduc-
tions of elliptic curves and the theory of mod
modular forms due to Swinnerton-
Dyer 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
has put the
structure of the cooperation algebra
on a firm footing, and attention has
turned to applications in stable homotopy theory, particularly the Adams spectral
1991 Mathematics Subject Classification. 55N20, 55N22, 55T15 (llFll).
Key words and phrases. Elliptic cohomology, Adams spectral sequence, modular form, Hopf
This paper is dedicated to Peter May on his 60th birthday with thanks for all his mathematical
inspiration and helpfulness over many years.
2001 American Mattwmatical Society