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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The crystalline period of a height one $p$-adic dynamical system
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by Joel Specter PDF
Trans. Amer. Math. Soc. 370 (2018), 3591-3608 Request permission

Abstract:

Let $f$ be a continuous ring endomorphism of $\mathbf {Z}_p\lBrack x\rBrack /\mathbf {Z}_p$ of degree $p.$ We prove that if $f$ acts on the tangent space at $0$ by a uniformizer and commutes with an automorphism of infinite order, then it is necessarily an endomorphism of a formal group over $\mathbf {Z}_p.$ The proof relies on finding a stable embedding of $\mathbf {Z}_p\lBrack x\rBrack$ in Fontaine’s crystalline period ring with the property that $f$ appears in the monoid of endomorphisms generated by the Galois group of $\mathbf {Q}_p$ and crystalline Frobenius. Our result verifies, over $\mathbf {Z}_p,$ the height one case of a conjecture by Lubin.
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Additional Information
  • Joel Specter
  • Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evans- ton, Illinois 60208
  • Address at time of publication: Department of Mathematics, Johns Hopkins University, 419 Krieger Hall, 3400 N. Charles Street, Baltimore, Maryland 21218
  • MR Author ID: 1022895
  • Email: jspecter@jhu.edu
  • Received by editor(s): April 11, 2016
  • Received by editor(s) in revised form: August 16, 2016
  • Published electronically: December 29, 2017
  • Additional Notes: The author was supported in part by National Science Foundation Grant DMS-1404620 and by a National Science Foundation Graduate Research Fellowship under Grant No. DGE-1324585.
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3591-3608
  • MSC (2010): Primary 11S20, 11S31, 11S82; Secondary 14L05, 13F25, 14F30
  • DOI: https://doi.org/10.1090/tran/7057
  • MathSciNet review: 3766859