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Transactions of the American Mathematical Society

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

 
 

 

The crystalline period of a height one $ p$-adic dynamical system


Author: Joel Specter
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
Published electronically: December 29, 2017
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Abstract: Let $ f$ be a continuous ring endomorphism of $ \mathbf {Z}_p\llbracket x\rrbracket /\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\llbracket x\rrbracket $ 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
Email: jspecter@jhu.edu

DOI: https://doi.org/10.1090/tran/7057
Keywords: $p$-adic dynamical system, $p$-adic Hodge theory, Lubin-Tate group
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.
Article copyright: © Copyright 2017 American Mathematical Society

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