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

Specter, Joel

doi:10.1090/tran/7057

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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