A modular description of the K(2)-local sphere at the prime 3

Abstract

Using degree N isogenies of elliptic curves, we produce a spectrum Q(N). This spectrum is built out of spectra related to tmf. At p=3 we show that the K(2)-local sphere is built out of Q(2) and its K(2)-local Spanier-Whitehead dual. This gives a conceptual reinterpretation a resolution of Goerss, Henn, Mahowald, and Rezk.

arXiv comment72 pages, 1 figure, to appear in Topology. Typos in Diagrams (2.11.2) and (2.11.3) fixed, and 2 lemmas added in Section 2.11 to prove that Diagram (2.11.4) exists

References in corpus (1)

• Galois extensions of structured ring spectra

Cited by in corpus (8)

• Congruences between modular forms given by the divided beta family in homotopy theory
• On the homotopy of Q(3) and Q(5) at the prime 2
• Detecting exotic spheres in low dimensions using coker J
• Brauer groups for commutative $S$-algebras
• Higher real K-theories and topological automorphic forms
• The 3-local tmf homology of BSigma_3
• Isogenies of elliptic curves and the Morava stabilizer group
• Buildings, elliptic curves, and the K(2)-local sphere