Computations of orbits for the Lubin–Tate ring

Abstract

We take a direct approach to computing the orbits for the action of the automorphism group \(\mathbb {G}_2\) of the Honda formal group law of height 2 on the associated Lubin–Tate rings \(R_2\). We prove that \((R_2/p)_{\mathbb {G}_2} \cong \mathbb {F}_p\). The result is new for \(p=2\) and \(p=3\). For primes \(p\ge 5\), the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.

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Notes

  1. 1.

    If one is willing to work with the formal group law of a super-singular elliptic curve rather than the Honda formal group law, an analogue of Theorem 2.3 follows from Section 6 of [1] where the results were obtained directly. The analogue of Theorem 2.1 also holds in this case, the proof being completely analogous to the one provided below.

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Acknowledgements

We thank some of the usual suspects for useful conversations: Tobias Barthel, Mark Behrens, Paul Goerss, Hans-Werner Henn, Mike Hopkins, Niko Naumann and Vesna Stojanoska. We also thank the referee and the editors their input.

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Correspondence to Agnès Beaudry.

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This material is based on work supported by the CU Boulder Department of Mathematics in the context of its internal Research For Undergraduates program. This material is also based upon work supported by the National Science Foundation under Grant no. DMS-1725563.

Communicated by Craig Westerland.

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Beaudry, A., Downey, N., McCranie, C. et al. Computations of orbits for the Lubin–Tate ring. J. Homotopy Relat. Struct. 14, 691–718 (2019). https://doi.org/10.1007/s40062-018-00228-7

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Keywords

  • Honda formal group law
  • Lubin-Tate ring
  • Morava E-theory
  • Morava stabilizer group
  • Chromatic Vanishing Conjecture