Buchstaber, Ochanine, Krichever, and Witten Genera

Abstract

We introduce a new class of one-dimensional commutative formal group laws with formal inverse $\bar u=-u$ whose modulus square construction yields Buchstaber's family of polynomials, and prove that the formal group law $F_{\mathrm{Bc}}(u,v)$ is universal for this class over commutative $\mathbb{Z}[1/2]$-algebras. This class is related to, but does not coincide with, the family of formal group laws associated with the Krichever genus. We compute the values of the corresponding Hirzebruch genus $\mathrm{Bc}$ on theta divisors and complex projective spaces, describe its relation to the Ochanine, Krichever, and Witten genera, and show how this construction gives examples not arising from Hirzebruch's elliptic genera of level $n$. We construct a complex-oriented multiplicative cohomology theory $\mathrm{Buc}^*$ such that the complex orientation $\mathrm{U}^*\to\mathrm{Buc}^*$ induces the genus $\mathrm{Bc}$ on coefficient rings, and prove that its localization at the discriminant is naturally isomorphic to the Landweber-exact elliptic cohomology theory $\mathrm{U}^*(-)\otimes_{\Omega_{\mathrm{U}}}\mathbb{Z}[1/2,a_1,a_2,a_3,\Delta^{-1}]$. Finally, we prove that $\mathbb{Z}[a_1,12a_2,360a_3]$ is the smallest subring of $\mathbb{Q}[a_1,a_2,a_3]$ over which the Buchstaber exponential $f_{\mathrm{Bc}}(u)$ is a Hurwitz series. As a corollary, we prove the Hurwitz-integrality statement predicted by Bunkova's coefficient-divisibility conjecture and show that $\mathbb{Z}[g_2/2,6g_3]$ is the minimal Hurwitz coefficient ring of the Weierstrass sigma-function.