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

Abstract

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.

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Notes

  1. 1.

    The notation is to remind of moments of measures with the Euler factor removed.

  2. 2.

    Uniqueness and existence of \(\Phi _{2k,j}^{(m)}\) follow from the Chinese remainder theorem, using that \(1-p^{2(k+m)-1}\) is a unit mod \(p^{v_p(C_p(k))}\).

  3. 3.

    Strictly speaking, rather a line of slope two than the diagonal.

  4. 4.

    The notation means to suggest moments starting in weight 2m of measures with total mass zero.

  5. 5.

    The maps \(\Psi _m^{(0)}\) will not be additive. Fortunately, this will not be required in any of our applications.

  6. 6.

    This condition is vacuous because \(M_{2k+1}(\mathbb {Q})=\{ 0\}\), but we include it to ease the comparison with Theorem 3.1.1.

  7. 7.

    Recall that there is no torsion-ambiguity for maps to \(\mathrm {KO}\), hence ev\(_*\) factors as indicated.

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Acknowledgements

The unpublished PhD thesis [11] of Christian Nerf contains preliminary observations on the problems addressed here.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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Correspondence to Niko Naumann.

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This work was supported by the Collaborative Research Centre SFB1085, funded by the DFG.

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Naumann, N., Sprang, J. Simultaneous Kummer congruences and \({\mathbb {E}}_\infty \)-orientations of KO and tmf. Math. Z. 292, 151–181 (2019). https://doi.org/10.1007/s00209-018-2156-4

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Mathematics Subject Classification

  • 55P42
  • 55P43
  • 55P50
  • 11A07