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arXiv · 1409.5314

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

J. Sprang·N. Naumann
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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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J. Sprang, N. Naumann. 2019-12-19. Simultaneous Kummer congruences and $\mathbb{E}_\infty$-orientations of KO and tmf. https://doi.org/10.1007/s00209-018-2156-4

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