On the homotopy of Q(3) and Q(5) at the prime 2

Abstract

We study modular approximations Q(l), l = 3,5, of the K(2)-local sphere at the prime 2 that arise from l-power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with Q(l) and record Hill, Hopkins, and Ravenel's computation of the homotopy groups of TMF_0(5). Using these tools and formulas of Mahowald and Rezk for Q(3) we determine the image of Shimomura's 2-primary divided beta-family in the Adams-Novikov spectral sequences for Q(3) and Q(5). Finally, we use low-dimensional computations of the homotopy of Q(3) and Q(5) to explore the role of these spectra as approximations to the K(2)-local sphere.

62 pages, 10 figures; v5: a sentence on p22 clarified. Final version, to appear in AGT

References in corpus (2)

• A modular description of the K(2)-local sphere at the prime 3
• The homotopy groups of the spectrum Tmf

Cited by in corpus (10)

• The $C_2$-spectrum $Tm{f}_1(3)$ and its invertible modules
• Topological resolutions in K(2)-local homotopy theory at the prime 2
• Detecting exotic spheres in low dimensions using coker J
• On the ring of cooperations for 2-primary connective topological modular forms
• The slice spectral sequence of a $C_4$-equivariant height-4 Lubin-Tate theory
• Invertible $K(2)$-Local $E$-Modules in $C_4$-Spectra
• (Topological) modular forms with level structures: decompositions and duality
• Elliptic cohomology is unique up to homotopy
• Hecke operators on topological modular forms
• The Tate spectrum of the higher real $K$-theories at height $n=p-1$