A proof for Ando's theorem on norm-coherent coordinates via the Coleman norm operator

Abstract

Ando established an algebraic criterion for when a complex orientation for a Morava E-theory is an $H_\infty$-map. The criterion relates such an orientation to a specific property of the formal group associated to the E-theory, namely, a norm coherence condition on its coordinate. On the other hand, Coleman constructed a norm operator for interpolating division values in local fields, which depends on a Lubin--Tate formal group law. These formal group laws are important tools in explicit local class field theory. In this article, we give a conceptual proof for Ando's theorem using the Coleman norm operator via the bridge of formal group laws between topology and arithmetic.