Simultaneous Kummer congruences and $\mathbb{E}_\infty$-orientations of KO and tmf

TL;DR

This paper proves the existence of many $E_Infinity$-String orientations of KO and tmf, linking them to classical Kummer congruences and Iwasawa theory, extending known genus results.

Contribution

It establishes uncountably many $E_Infinity$-String orientations of KO and tmf, connecting their obstructions to Iwasawa-theoretic conditions and classical Kummer congruences.

Findings

Uncountably many $E_Infinity$-String orientations of KO and tmf.

Obstruction to lifting orientations characterized by Iwasawa theory.

Classical Kummer congruences are central to these results.

Abstract

Building on results of M. Ando, M.J. Hopkins and C. Rezk, we show the existence of uncountably many $\mathbb{E}_\infty$-String orientations of real K-theory KO and of topological modular forms tmf, generalizing the $\hat{A}$- (resp. the Witten) genus. Furthermore, the obstruction to lifting an $\mathbb{E}_\infty$-String orientations from KO to tmf is identified with a classical Iwasawa-theoretic condition. The common key to all these results is a precise understanding of the classical Kummer congruences, imposed for all primes simultaneously. This result is of independent arithmetic interest.