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Topological modular forms of level 3. (English) Zbl 1192.55006
This paper presents calculations related to the stable homotopy groups of the spheres localized with respect to the second Morava \(K\)-theory, \(K(2)\), at the prime 2. More precisely, Behrens has conjectured that there be a decomposition of a Galois cover of the \(K(2)\)-local sphere at the prime 2 which involves a spectrum \(Q(3)\) that can be build from the periodic topological modular forms spectrum \(\text{TMF}\) and a variant thereof, \(\text{TMF}(\Gamma_0(3))\), which involves topological modular forms with respect to \(\Gamma_0(3)\)-level structures. The authors compute the homotopy groups of the latter by means of the homotopy fixed point spectral sequence for a 2-fold Galois cover \(\text{TMF}(\Gamma_1(3))\), where the homotopy groups co-incide with the ordinary modular forms with respect to \(\Gamma_1(3)\)-level structures. The differentials are determined by means of the String orientation of \(\text{TMF}\), which allows to detect elements in the image of the \(J\)-homomorphism in the homotopy of \(Q(3)\). In addition, the authors also discuss connective models for \(\text{TMF}(\Gamma_0(3))\).

MSC:
55N34 Elliptic cohomology
55Q45 Stable homotopy of spheres
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