# Fontaine-Fargues curve and period rings and untilt

When I read the paper "THE FARGUES–FONTAINE CURVE AND DIAMONDS" of Matthew Morrow, I have a question on page 11.

Question: The arthur said that the de Rham and crystalline period rings implicitly depended on having chosen an untilt of $$F$$, where $$F=C_p^b$$ is the tilt of the p-adic complex field $$C_p$$. I can't understand this. If the author just defines the $$\infty$$ point on Fontaine-Fargues curve to be the class $$[C_p]$$, why the author mention the period rings at the bottom of page11 when he defines the $$\infty$$ point?

Definition: The definitions of the de Rham and crystalline period rings I know are $$B_{dR}:=Frac(lim W(R)[\frac{1}{p}]/{(ker\theta)}^n])$$, $$B_{cris}:=Frac(A_{cris}[\frac{1}{p}])$$, $$A_{cris}:=limA^0_{cris}/p^nA^0_{cris}$$, and $$A^0_{cris}$$ is just the sub $$W(R)$$-module of $$W(R)[\frac{1}{p}]$$ generated by the $$\frac{\xi^n}{n!}$$ where $$n$$ takes all positive integers. These definitions come from Fontaine's readable book Theory of p-adic Galois Representations.

• Doesn't the map $\theta$ (and hence $\xi$) depend on a choice of untilt? Also, I think $A^0_{cris}$ should be a subring of $B_{dR}$, not $W(R)[1/p]$, right? – Daniel Litt Jul 6 '19 at 15:07
• @DanielLitt Thanks for your answer. The map $\theta$ is defined the natural extension of $W(R)\rightarrow O_C:(x_0,x_1,...,x_n,...)\rightarrow \sum p^nx_n^{(n)}$ where $x_n=(x_n^{(m)})$ and $x_n^{(m)}\in O_C$. And $A_{cris}^0\subseteq W(R)[\frac{1}{p}]\subseteq B_{dR}^+\subseteq B_{dR}$. The untilt of a perfectoid field $K$ is a pair $(L,r)$, where $L$ is a perfectoid field and $r:L^b\cong K$ is an isomorphism. But I still can't see why $\theta$ determines such a pair $(L,r)$. – user141691 Jul 6 '19 at 15:54
• $C$ is a choice of untilt! – Daniel Litt Jul 6 '19 at 17:32

As Daniel Litt says, the choice of $$C$$ is actually an untilt. The "classical" approach to period rings, which you might have in mind, was to start with a certain complete, algebraically closed field $$C_p$$, then to construct $$R$$ out of the quotient $$O_{C_p}/p$$ and out of its Witt vectors to construct the period rings.
The content of Morrow's Proposition 5.1 is that $$R$$ may arise from many other fields $$C$$ other than from your initially chosen $$C_p$$ and all these form, modulo $$\varphi^\mathbb{Z}$$, all the equivalence classes of untilts. But to produce $$\theta$$ you need a target, hence you need to pick one of these choices. By Theorem 2.3 ibid. this choice is a point on The Curve, call it $$\infty$$: the construction of the line bundle depends on this point. Morrow's remark that a choice has been made means the following: suppose $$B_e$$ could be constructed independently of the choice of an untilt. Then the construction in (6) would give the same result irrespectively of the chosen point, providing you with a canonical line bundle on it, or equivalently with a prefered point on $$\mathbb{P}^1$$, which is absurd. On the other hand, the two choices match with each other: a point on $$\mathbb{P}^1$$ (rather, on $$X^{FF}$$) call it $$\infty$$ and an untilt, for instance $$C_p$$.
• Thanks for your detailed answer. When I study p-adic Hodge, I haven't thought that we can replace $C_p$ with other complete algebraic closed non-archimedean field in the constructions of these period rings. One more question: Does the construction in (6) you mentioned mean the two hypotheses $Pic(X)\cong Z$ and $H^1(X,O_X(k))=0$ for all $k\geq0$ on page 11? – user141691 Jul 7 '19 at 13:45
• Yea, especially in the isomorphism $\mathrm{Pic}(X)\cong\mathbb{Z}$. – Filippo Alberto Edoardo Jul 8 '19 at 16:12