Buildings, elliptic curves, and the K (2)-local sphere

We investigate a dense subgroup Γ of the second Morava stabilizer group given by a certain group of quasi-isogenies of a supersingular elliptic curve in characteristic p. The group Γ acts on the Bruhat-Tits building for GL2(ℚ) through its action on the -adic Tate module. This action has finite stabilizers, giving a small resolution for the homotopy fixed point spectrum (EhΓ 2)hGal by spectra of topological modular forms. Here, E2 is a version of Morava E-theory and Gal = Gal().