Building on results of M. Ando, M.J. Hopkins and C. Rezk, we show the
existence of uncountably many E∞-String orientations of real
K-theory KO and of topological modular forms tmf, generalizing the A^-
(resp. the Witten) genus. Furthermore, the obstruction to lifting an
E∞-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.Comment: final versio