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

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.

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