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Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $G_K=\mathrm{Gal}(\overline{K}/K)$ be its absolute Galois group. There are the Fargues-Fontaine analytic curves $Y_{FF}$ and $X_{FF}$ associated to $\mathbb{C}_p$, and they each have a $G_K$-action.

Now suppose I take a connected open affinoid $U$ in either $Y_{FF}$ or $X_{FF}$, and suppose $U$ is $G_K$-stable. Do we know that the fixed functions are exactly $K_0$, for $K_0$ the maximal unramified extension of $\mathbb{Q}_p$ that is contained in $K$? Namely, do we know that $H^0(U,\mathcal{O}_{Y_{FF}})^{G_K}=K_0$ (respectively $H^0(U,\mathcal{O}_{X_{FF}})^{G_K}=K_0$)?

The answer is yes for the $U = Y_I$ for $I$ a compact interval, this is proposition 10.2.7 in the book of Fargues-Fontaine. In other words this is the equality $B_I^{G_K} = K_0$ for the rings usually considered in the theory. In fact, it's even true that $\mathrm{Frac}{B_I}^{G_K} = K_0$.

So for general $U$, is this still true? And if not, what can be said about these $G_K$-invariants?

Thanks!

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  • $\begingroup$ By the way, I think that probably by the same argument of Fargues and Fontaine, one can show that the answer is yes if $U$ contains the point at infinity or one of its Frobenius translates in the case that $U \subset Y_{FF}$ $\endgroup$
    – xlord
    Commented Feb 16, 2022 at 21:39

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