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**The beta family at the prime two and modular forms of level three.**
*(English)*
Zbl 1355.55011

For studying the stable homotopy groups \(\pi_*(S^0)_{(p)}\) of spheres localizing at a prime \(p\), the Adams-Novikov spectral sequence \(E_2^{*,*}=\text{Ext}_{BP_*BP}^{*,*}(BP_*,BP_*)\Rightarrow \pi_*(S^0)_{(p)}\) gives us much information. The beta family at a prime \(p\) is a family of generators of the second line \(E_2^{2,*}\) of the spectral sequence. At a prime \(p\geq 5\), M. Behrens [Geom. Topol. 13, No. 1, 319–357 (2009; Zbl 1205.55012)], [G. Laures, Topology 38, No. 2, 387–425 (1999; Zbl 0924.55004)], [M. Behrens and G. Laures, Geom. Topol. Monogr. 16, 9–29 (2009; Zbl 1222.55013)] derived the \(f\)-invariant of the beta family taking values in a group which is closely related to divided congruences of modular forms. By a different approach, J. Hornbostel and N. Naumann [Am. J. Math. 129, No. 5, 1377–1402 (2007; Zbl 1134.55010)] studied the \(f\)-invariant of \(\beta_s\) and \(\beta_{2^ns/2^n}\) at the prime two.

In the paper under review, the author computes the \(f\)-invariant of all members of the beta family at the prime two. Using the Hirzebruch genus taking values in the ring of modular forms for the congruence subgroup \(\Gamma_1(3)\subset \text{SL}(2,\mathbb Z)\) of level three, we obtain an elliptic homology theory \(E_*^{\Gamma_1(3)}\) with a map of coefficient rings \(\varphi: BP_*\to E_*^{\Gamma_1(3)}\). Then, an analogy of the universal Greek letter map is defined on Ext\(^{2,*}(E_*^{\Gamma_1(3)},E_*^{\Gamma_1(3)})\), and the image of the beta family under the induced map \(\varphi_*\) is computed, from which the \(f\)-invariant is obtained. This is done by a straightforward calculation.

In the paper under review, the author computes the \(f\)-invariant of all members of the beta family at the prime two. Using the Hirzebruch genus taking values in the ring of modular forms for the congruence subgroup \(\Gamma_1(3)\subset \text{SL}(2,\mathbb Z)\) of level three, we obtain an elliptic homology theory \(E_*^{\Gamma_1(3)}\) with a map of coefficient rings \(\varphi: BP_*\to E_*^{\Gamma_1(3)}\). Then, an analogy of the universal Greek letter map is defined on Ext\(^{2,*}(E_*^{\Gamma_1(3)},E_*^{\Gamma_1(3)})\), and the image of the beta family under the induced map \(\varphi_*\) is computed, from which the \(f\)-invariant is obtained. This is done by a straightforward calculation.

Reviewer: Katsumi Shimomura (Kochi)