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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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