## 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.## References

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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