Computing the Krichever genus

TL;DR

This paper explores the properties of the Krichever genus, establishing its relation to formal group laws and describing its coefficient ring as a quotient of the Lazard ring, with implications for complex bordism.

Contribution

It proves the relation between the Krichever-H"ohn genus and the formal group law via the isomorphism class and constructs elements in the Lazard ring to define the universal Krichever formal group law.

Findings

Krichever-H"ohn genus equals the composition of a and a^{-1} on the rational complex bordism ring.

Coefficient ring of the universal Krichever formal group law is a quotient of the Lazard ring.

Constructed elements in the Lazard ring define the universal Krichever formal group law.

Abstract

Let $\psi$ denote the genus that corresponds to the formal group law having invariant differential $\omega(t)$ equal to $\sqrt{1+p_1t+p_2t^2+p_3t^3+p_4t^4}$ and let $\kappa$ classify the formal group law strictly isomorphic to the universal formal group law under strict isomorphism $x\CP(x)$. We prove that on the rational complex bordism ring the Krichever-H\"ohn genus $\phi_{KH}$ is the composition $\psi\circ \kappa^{-1}$. We construct certain elements $A_{ij}$ in the Lazard ring and give an alternative definition of the universal Krichever formal group law. We conclude that the coefficient ring of the universal Krichever formal group law is the quotient of the Lazard ring by the ideal generated by all $A_{ij}$, $i,j\geq 3$.