## 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.

### MSC:

 55Q45 Stable homotopy of spheres 11F11 Holomorphic modular forms of integral weight 55Q51 $$v_n$$-periodicity 58J26 Elliptic genera

### Citations:

Zbl 1205.55012; Zbl 0924.55004; Zbl 1222.55013; Zbl 1134.55010
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