Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [Ber04] Laurent Berger, An introduction to the theory of $ p$-adic representations, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 255-292 (English, with English and French summaries). MR 2023292
  • [Ber14a] Laurent Berger, Iterated extensions and relative Lubin-Tate groups, Ann. Math. Qué. 40 (2016), no. 1, 17-28 (English, with English and French summaries). MR 3512521, https://doi.org/10.1007/s40316-015-0052-4
  • [Ber14b] Laurent Berger, Lifting the field of norms, J. Éc. polytech. Math. 1 (2014), 29-38 (English, with English and French summaries). MR 3322781, https://doi.org/10.5802/jep.2
  • [Fon94] Jean-Marc Fontaine, Le corps des périodes $ p$-adiques, Astérisque 223 (1994), 59-111 (French), with an appendix by Pierre Colmez; Périodes $ p$-adiques (Bures-sur-Yvette, 1988). MR 1293971
  • [Haz09] Michiel Hazewinkel, Witt vectors. I, Handbook of algebra. Vol. 6, Handb. Algebr., vol. 6, Elsevier/North-Holland, Amsterdam, 2009, pp. 319-472. MR 2553661, https://doi.org/10.1016/S1570-7954(08)00207-6
  • [KR09] Mark Kisin and Wei Ren, Galois representations and Lubin-Tate groups, Doc. Math. 14 (2009), 441-461. MR 2565906
  • [Kob77] Neal Koblitz, $ p$-adic numbers, $ p$-adic analysis, and zeta-functions, Graduate Texts in Mathematics, Vol. 58, Springer-Verlag, New York-Heidelberg, 1977. MR 0466081
  • [LMS02] François Laubie, Abbas Movahhedi, and Alain Salinier, Systèmes dynamiques non archimédiens et corps des normes, Compositio Math. 132 (2002), no. 1, 57-98 (French, with English summary). MR 1914256, https://doi.org/10.1023/A:1016009331800
  • [Li] Hua-Chieh Li, On heights of $ p$-adic dynamical systems, Proc. Amer. Math. Soc. 130 (2002), no. 2, 379-386. MR 1862116, https://doi.org/10.1090/S0002-9939-01-06166-4
  • [Li02] Hua-Chieh Li, $ p$-typical dynamical systems and formal groups, Compositio Math. 130 (2002), no. 1, 75-88. MR 1883692, https://doi.org/10.1023/A:1013792029235
  • [Lub94] Jonathan Lubin, Non-Archimedean dynamical systems, Compositio Math. 94 (1994), no. 3, 321-346. MR 1310863
  • [LT65] Jonathan Lubin and John Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1965), 380-387. MR 0172878, https://doi.org/10.2307/1970622
  • [Sar10] Ghassan Sarkis, Height-one commuting power series over $ \mathbb{Z}_p$, Bull. Lond. Math. Soc. 42 (2010), no. 3, 381-387. MR 2651931, https://doi.org/10.1112/blms/bdp130
  • [Sar05] Ghassan Sarkis, On lifting commutative dynamical systems, J. Algebra 293 (2005), no. 1, 130-154. MR 2173969, https://doi.org/10.1016/j.jalgebra.2005.08.007
  • [SS13] Ghassan Sarkis and Joel Specter, Galois extensions of height-one commuting dynamical systems, J. Théor. Nombres Bordeaux 25 (2013), no. 1, 163-178 (English, with English and French summaries). MR 3063836
  • [SW13] Peter Scholze and Jared Weinstein, Moduli of $ p$-divisible groups, Camb. J. Math. 1 (2013), no. 2, 145-237. MR 3272049, https://doi.org/10.4310/CJM.2013.v1.n2.a1
  • [W] Jean-Pierre Wintenberger, Automorphismes des corps locaux de caractéristique $ p$, J. Théor. Nombres Bordeaux 16 (2004), no. 2, 429-456 (French, with English and French summaries). MR 2143563

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11S20, 11S31, 11S82, 14L05, 13F25, 14F30

Retrieve articles in all journals with MSC (2010): 11S20, 11S31, 11S82, 14L05, 13F25, 14F30


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

American Mathematical Society