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Coefficient rings of Tate formal groups determining Krichever genera. (English. Russian original) Zbl 1348.14111

Proc. Steklov Inst. Math. 292, 37-62 (2016); translation from Tr. Mat. Inst. Steklova 292, 43-68 (2016).
A formal group \(F(u,v)\) over a commutative ring \(R\) with identity is called a Buchstaber formal group if it can be presented in the form \[ F(u,v) = \frac{u^2A(v)-v^2A(u)}{uB(v)-vB(u)} \] where \(A(u) = 1+\sum_{i=1}^{\infty} a_iu^i\) and \(B(u) = 1+\sum_{i=1}^{\infty} b_iu^i\). Let \(\mathcal{F}_B(u,v)\) be the universal Buchstaber formal group over the ring \(\mathcal{R}_B = \mathbb{Z}[a_k, k \neq 2, b_m, m \neq 1]/J\) where \(J\) is the associativity ideal. Let \(\mathcal{F}_2(u,v)\) (resp. \(\mathcal{F}_3(u,v)\)) be the specialization of \(\mathcal{F}_B(u,v)\) at \(\{A(u)=1\}\) (resp. \(\{B(u) = A(u)^2\}\)), and let \(\mathcal{S}_2\) (resp. \(\mathcal{S}_3\)) be its coefficient ring.
Let \(\mathcal{F}_T(u,v)\) be the Tate formal group over the ring \(\mathbb{Z}[\mu_1,\mu_2,\mu_3,\mu_4,\mu_6]\). Let \(\mathcal{R}_2\) (resp. \(\mathcal{R}_3\)) be the coefficient ring of the specialization \(\{\mu_k=0, k=1,3,6\}\) (resp. \(\{\mu_2 = -\mu_1^2, \mu_4 = -\mu_1\mu_3, 3\mu_6 = -\mu_3^2\}\)) of \(\mathcal{F}_T(u,v)\).
The main result of this paper is the following theorem:
Theorem. For \(k=2,3\), the ring \(\mathcal{S}_k\) is torsion-free and \(\mathcal{F}_k(u,v)\) is determined by two-parametric elliptic Hirzebruch genera of level \(k\). There exists a classifying homomophisms \(h_k:\mathcal{S}_k \to \mathcal{R}_k\) that is an isomoprhism.
Reviewer: Xiao Xiao (Utica)

MSC:

14L05 Formal groups, \(p\)-divisible groups
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