ELLIPTIC COHOMOLOGY , p - ADIC MODULAR FORMS AND ATKIN ’ S OPERATOR

We construct a p-adic version of Elliptic Cohomology whose coefficient ring agrees with Serre’s ring of p-adic modular forms. We then construct a stable operation Ûp in this theory agreeing with Atkin’s operator Up on p-adic modular forms. Throughout the paper we assume given a fixed prime p ≥ 5. We begin as in [2] by considering the universal Weierstrass cubic (for Z(p) algebras) Ell/R∗: Ell:Y 2 = 4X − g2X − g3 where R∗ = Z(p)[g2, g3] is the graded ring for which |gn| = 4n. We can also assign gradings 4, 6 to X, Y respectively. Now the discriminant ∆Ell = g 2 − 27g 3 is non-zero and hence Ell is an elliptic curve over R∗. Thus we can define an abelian group structure on Ell considered as a projective varietysee [5], [11]. This has the unique point at infinity O = [0, 1, 0] as its zero. We can take the local parameter T = − Y and then the group law on Ell yields a formal group law (commutative and 1 dimensional) F over R∗. This is explained in detail in for example [11]. Associated to this is an invariant differential ω Ell = dT ∂ ∂Y F E``(T, 0) = dX Y which can also be written as ω Ell = d log E`` (T ). The formal group law F is classified by a unique homomorphism φ: L∗ −→ R∗ where L∗ is Lazard’s universal ring (given its natural grading). But topologists are aware that L∗ is isomorphic to MU∗, the coefficient ring of complex (co)bordism MU∗( ), and moreover the natural orientation for complex line bundles in this theory has associated to it a universal formal group law F . This is all explained in for example [1]. Thus we obtain a genus φE``: MU∗ −→ R∗. The ring R∗ can be identified with a ring of modular forms for SL2(Z) which are holomorphic at i∞ as explained in [2]. Under this identification we have R∗ ∼= S(Z(p))∗ where g2 ←→ 1 124 and g3 ←→ − 1 216 E6 and E2n denotes the weight 2n Eisenstein function. We will use this identification without further comment. Now Elliptic Cohomology is usually defined by first localising R∗ at the multiplicative set generated by ∆Ell which makes R∗[∆−1 Ell] universal for elliptic curves. However, this is not necessary if we only worry about the formal group law (in fact such a Weierstrass cubic is always non-singular at O). We define a functor on the category of finite CW complexes by E``[1]∗( ) = R∗(V1)⊗MU∗MU( ) where V1 ∈ R2(p−1) is the image of the Eisenstein function Ep−1 in R∗. ‡ We would like to thank the SERC for support whilst this research was carried out.