On the Homotopy Type of the Spectrum Representing Elliptic Cohomology

In this paper we analyse the homotopy type at primes p > 3 of the ring spectrum E`` representing a version of elliptic cohomology whose coefficient ring E``∗ agrees with the ring of modular forms for SL2(Z). For any prime (=maximal) graded ideal P / E``∗ containing the Eisenstein function Ep−1 as well as p, we show that there is a morphism of ring spectra [ E(2) −→ E`` b P and a corresponding splitting E`` b P '_ i Σ2θ(i)[ E(2) of algebra spectra over [ E(2) (the I2-adic completion of E(2)); here ( )b P denotes the P-adic completion of the spectrum E``. Moreover, there is a multiplicative reduction (E``/P)∗( ) and we similarly show that there is a splitting of K(2) algebra spectra