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arXiv · 1501.04611

The crystalline period of a height one $p$-adic dynamical system over $\mathbf{Z}_p$

Joel Specter
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Abstract

Let $f$ be a continuous ring endomorphism of $\mathbf{Z}_p[[x]]/\mathbf{Z}_p$ 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 $\mathbf{Z}_p[[x]]$ 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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Joel Specter. 2015-01-19. The crystalline period of a height one $p$-adic dynamical system over $\mathbf{Z}_p$. https://arxiv.org/abs/1501.04611

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