Let $K$ be a finite extension of $\mathbb{Q}\_p$ and let $G$ be a 1-dimensional formal group defined over $\mathcal{O}\_K$. Consider the field $K\_\infty$ obtained by adjoining to $K$ all the solutions to $[p]^n(x) = 0$ with $x \in \mathbb{C}\_p$ with $|x|<1$ and $n\geq1$. (These torsion points depend on a choice of coordinate for the formal group, but the field $K\_\infty$ does not). Equivalently, if $V(G)$ is the Tate module of $G$, considered as a $\mathrm{Gal}(\overline{K}/K)$-representation, the field $K\_\infty$ is the subfield of $\overline{K}$ which corresponds to its kernel via Galois theory.


Consider the $p$-adic completion $\widehat{K}\_\infty$ of $K\_\infty$. **My Question is**: is $\widehat{K}\_\infty$ perfectoid?


At the very least, I know that $K\_\infty$ must be infinitely ramified (the Tate module of $G$ has a nonzero Hodge-Tate weight). Also, the answer is yes when $G$ is a Lubin-Tate formal group.


Thanks!



Is the completion of the field generated by torsion points of a 1-dimensional formal group perfectoid?
Yes it is. The extension $K\_\infty/K$ is deeply ramified. Now apply proposition 6.6.6 of Gabber-Ramero's "Almost ring theory".


