On the homotopy type of the spectrum representing elliptic cohomology

Abstract:

In this paper we analyse the homotopy type at primes $p > 3$ of the ring spectrum $E\ell \ell$ representing a version of elliptic cohomology whose coefficient ring $E\ell {\ell _ * }$ agrees with the ring of modular forms for $S{L_2}(\mathbb {Z})$. For any prime (=maximal) graded ideal $\mathcal {P} \triangleleft E\ell {\ell _*}$ containing the Eisenstein function ${E_{p - 1}}$ as well as $p$, we show that there is a morphism of ring spectra \[ \widehat {E(2)} \to (E\ell \ell )_{\hat {\mathcal {P}}}\] and a corresponding splitting \[ (E\ell \ell )_{\hat {\mathcal {P}}} \simeq \bigvee \limits _i {\Sigma ^{2\theta (i)}}\widehat {E(2)}\] of algebra spectra over $\widehat {E(2)}$ (the ${I_2}$-adic completion of $E(2)$); here $(\;)_{\hat {\mathcal {P}}}$ denotes the $\mathcal {P}$-adic completion of the spectrum $E\ell \ell$. Moreover, there is a multiplicative reduction ${(E\ell \ell /\mathcal {P})^ * }(\;)$ and we similarly show that there is a splitting of $K(2)$ algebra spectra \[ E\ell \ell /\mathcal {P} \simeq \bigvee \limits _i {\Sigma ^{2\theta ’(i)}}K(2).\] In each case the indexing $i$ ranges over a finite set.