Contemporary Mathematics

Volume 271, 2001

On the Adams

~-term

for elliptic cohomology

Andrew Baker

ABSTRACT. 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 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

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

Heeke operators and the dual cooperation algebras are arithmetically interesting.

Moreover, their reductions modulo a prime

p

reflect the theory of 'ordinary' reduc-

tions of elliptic curves and the theory of mod

p

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

&

Johnson

[7],

the author

[4]

and Laures

[13]

has put the

structure of the cooperation algebra

Eff.Efl!

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

algebroid.

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

http://dx.doi.org/10.1090/conm/271/04347