Global group laws and equivariant bordism rings

We prove that the homotopical $A$-equivariant complex bordism ring is isomorphic to the $A$-equivariant Lazard ring for every abelian compact Lie group $A$, settling a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian $2$-groups. This generalizes classical theorems of Quillen on the connection between non-equivariant bordism rings and formal group laws, and extends the case $A=C_2$ due to Hanke--Wiemeler. We work in the framework of global homotopy theory, which is essential for our proof. Using this framework, we also give an algebraic characterization of the collection of equivariant complex bordism rings as the universal contravariant functor from abelian compact Lie groups to commutative rings that is equipped with a coordinate. More generally, the ring of $n$-fold cooperations of equivariant complex bordism is shown to be universal among such functors equipped with a strict $n$-tuple of coordinates.