Orientations and p-Adic Analysis

TL;DR

This paper explores the relationship between orientations, p-adic analysis, and formal groups, establishing new connections in complex K-Theory and demonstrating the p-completion of the Todd genus as an E-infinity map.

Contribution

It applies methods from local analytic number theory to formal groups, providing a new proof of the equivalence of two descriptions of complex K-Theory and analyzing the p-completion of the Todd genus.

Findings

Constructed a theory of integration on formal groups of finite height.

Proved the equivalence of two descriptions for complex K-Theory.

Showed the p-completion of the Todd genus is an E-infinity map.

Abstract

Matthew Ando produced power operations in the Lubin-Tate cohomology theories and was able to classify which complex orientations were compatible with these operations. The methods used by Ando, Hopkins and Rezk to classify orientations of topological modular forms can be applied to complex K-Theory. Using techniques from local analytic number theory, we construct a theory of integration on formal groups of finite height. This calculational device allows us to show the equivalence of the two descriptions for complex K-Theory. As an application we show that the $p$-completion of the Todd genus is an $E_\infty$ map.