On the homotopy type of the spectrum representing elliptic cohomology

Abstract

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