Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 151–181 | Cite as

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

  • Niko NaumannEmail author
  • Johannes Sprang


Building on results of Ando, Hopkins and 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.

Mathematics Subject Classification

55P42 55P43 55P50 11A07 



The unpublished PhD thesis [11] of Christian Nerf contains preliminary observations on the problems addressed here. The second author would like to thank Christian Nerf. It was his PhD thesis which drew his attention to the set of problems studied in this paper. He is also grateful for remarks and comments by Uli Bunke, Thomas Nikolaus and Michael Völkl. Last but not least, he would like to thank Guido Kings for introducing him to the beautiful subject of p-adic interpolation. The first author thanks Charles Rezk for suggesting Theorem 3.4.1. We also thank an anonymous referee for helpful comments improving the readability.


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Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany

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