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## Todos

Motivate the following:

• Perfectoid spaces
• Unsorted/tilting and untilts
• Galois representations (might have notes elsewhere..?)
• Langlands, period!
• absolute Galois group, particularly $$G({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }/{\mathbb{Q}})$$ (see notes elsewhere)
• Relations to stable homotopy : an isomorphism of Lubin-Tate, and the Drinfeld tower?

## Motivation / Summary

Some notes on the Fargues-Fontaine curve $${X_\mathrm{FF}}$$, “the fundamental curve of p-adic Hodge theory”.

What’s the point? There’s supposed to be a “curve” $${X_\mathrm{FF}}$$ over $${ {\mathbb{Q}}_p }$$ where local Langlands for $${ {\mathbb{Q}}_p }$$ should be encoded as geometric Langlands on $${X_\mathrm{FF}}$$, which glues together important period rings from p-adic Hodge theory. Stems from conjectures of Grothendieck wanting to related de Rham cohomology to Unsorted/etale cohomology, and a similar theorem proved by Faltings in the 80s.

Why care: this is a hot topic right now because of a conjecture related to local Langlands : supposed to give a way to go from the Galois side to the automorphic side.

Important object: For $$G$$ a reductive algebraic group over a local field, $${\mathsf{Bun}}_G$$ the moduli stack of $$G{\hbox{-}}$$torsors over a family of $${X_\mathrm{FF}}$$ curves.

A useful overall analogy: it’s like the Riemann sphere $${\mathbb{CP}}^1 \coloneqq{\mathbb{P}}^1({\mathbb{C}})$$, and in fact the adic version is a $$p{\hbox{-}}$$adic Riemann surface. The full ring of meromorphic functions on $${\mathbb{P}}^1({\mathbb{C}})$$ is $${\mathbb{C}}\qty{z}$$, but $${\mathbb{C}}[z]$$ captures most of the data away from $$\infty$$. $${\mathbb{C}}[z]$$ as a $${\mathbb{C}}{\hbox{-}}$$algebra consists of regular (polynomial) functions on $${\mathbb{P}}^1({\mathbb{C}})$$ with a pole at $$\infty$$ of order equal to the degree of the polynomial. View the $${\mathbb{Z}}{\hbox{-}}$$algebra $${\mathbb{Z}}$$ as the regular functions on $$P$$ the set of primes (finite places), with a point at $$\infty$$ (infinite place) given by the usual valuation $${\left\lvert {{-}} \right\rvert}$$. Make this more algebro-geometric by replacing $${\mathbb{Z}}$$ with either $${ {\mathbb{Z}}_{\widehat{p}} }$$ or $${ {\mathbb{Q}}_p }$$ (so $$p{\hbox{-}}$$adic things) and looking at $${ {\mathbb{Q}}_p }{\hbox{-}}$$algebras $$B$$ as replacements for regular functions. $${X_\mathrm{FF}}$$ is also supposed to “geometrize” period rings from p-adic Hodge theory. One can also geometrize class field theory, and realize $${ \mathsf{Gal}} (\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{Q}}\mkern-1.5mu}\mkern 1.5mu/{\mathbb{Q}})$$ as a fundamental group.

$${X_\mathrm{FF}}$$ is also roughly a moduli of untilts, whisch allow passing between $${\mathbb{F}}_p$$ and $${ {\mathbb{Q}}_p }$$. A major goal is to go from characteristic zero to characteristic $$p$$ (relatively easy) and then to go back to characteristic zero (relatively hard). The curve is useful because many linear algebraic objects of $$p{\hbox{-}}$$adic theory can be translated into vector bundles over $${X_\mathrm{FF}}$$, and there is Grothendieck splitting type of theorem for those.

### Definitions

• Curve: over $k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }$, an integral separated scheme of finite type over $$k$$ of dimension 1.

• Finite type: on affines, when $$\operatorname{Spec}A\to \operatorname{Spec}B$$ induces $$B\to A$$ making $$A \in \mathsf{B}{\hbox{-}}\mathsf{Mod}$$ finitely generated.

• algebraic curve : $$X$$ of pure Krull dimension 1, or equivalently $$X$$ has an affine open cover $$\left\{{\operatorname{Spec}R_i}\right\}_{i\in I} \rightrightarrows X$$ where each $$\operatorname{Spec}R_i$$ is Krull dimension 1.

• For (irreducible) varieties over $k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }$, just an algebraic variety of dimension 1.
• Complete curve: for algebraic varieties $$X$$, universally closed: the projections $$X\times({-})\to({-})$$ are closed maps when evaluated on any variety.

• Equivalently, $$X\to k$$ is a proper morphism (separated, finite type, and universally closed) - For topological spaces, $$X$$ is compact iff $$X$$ is complete. - Complete (smooth) varieties of dimension 1 are always projective.
• Completely valued field: a field $$k$$ equipped with a valuation $$v$$, where $$k$$ is complete as a metric space with respect to $$v$$.

• Valuation: a group morphism $$(K, \cdot) \to ({\mathbb{R}}_{\geq 0}, +)$$ with $$v(a) = \infty \iff a =0$$ satisfying an ultrametric triangle inequality.

• Almost a metric. Supposed to capture the multiplicity of zeros/poles of a function.
• Valuation ring: defined as $${\mathcal{O}}_K \coloneqq\left\{{x\in K {~\mathrel{\Big\vert}~}v(x) \leq 1}\right\}$$, the unit disc with respect to the valuation.

• Has a unique maximal ideal, the interior of the disc: $${\mathfrak{m}}\coloneqq{\mathcal{O}}_K\setminus\left\{{v(x) = 1}\right\}$$.
• Dedekind scheme: a Noetherian integral scheme of dimension 1 where every local ring is regular.

• Example: $$\operatorname{Spec}R$$ for $$R$$ a Dedekind domain.
• Slogan: any non-generic point is a closed point.
• Tilts: for $$k$$ a field, denoted \begin{align*} k {}^{ \flat } \coloneqq\varprojlim(k\xrightarrow{x\mapsto x^p} k \xrightarrow{x\mapsto x^p} k \cdots) ,\end{align*} realized as sequences $$\left\{{x_n}\right\}$$ where $$x_{n+1}^p = x_n$$, made into a characteristic $$p$$ field with pointwise multiplication and a $$p{\hbox{-}}$$twisted addition law involving limits.

• Idea: an inverse limit of applying Frobenius. Yields a perfect $${\mathbb{F}}_p{\hbox{-}}$$algebra. Equipped with a valuation and a valuation ring.
• Untilts: For $$k$$ a field, a pair $$(K, \iota)$$ where $\iota: k\xrightarrow{\sim} K {}^{ \flat }$ is an isomorphism of fields, plus a condition on valuation rings $${\mathcal{O}}_k$$ and ${\mathcal{O}}_K {}^{ \flat }$.

• formal scheme : topologically ringed spaces $$(X, {\mathcal{O}}_X)$$ where $${\mathcal{O}}_X$$ is a sheaf of topological rings, which is locally a formal spectrum of a Noetherian ring.

• Formal spectrum: $$\operatorname{Spf}R \coloneqq\operatorname{Spf}_I R = \colim_{n} \operatorname{Spec}R/I^n$$, where $$I^n$$ is a system of ideals forming neighborhoods of zero in the sense that if $$U\ni 0$$ then $$I^n \subseteq U$$ for some $$n$$. Take the colimit in $${\mathsf{Top}}\mathsf{RingSp}$$. The structure sheaf of $$\operatorname{Spf}_I R$$ is $$\colim_{n} {\mathcal{O}}_{\operatorname{Spec}R/I^n}$$

• Meant to accommodate formal power series as regular functions.
• Examples: $$I{~\trianglelefteq~}R$$, take the $$I{\hbox{-}}$$adic topology with basic open sets $$r + I^n$$. Then $${ {\left\lvert {\operatorname{Spf}A} \right\rvert} } = { {\left\lvert {\operatorname{Spec}A/I} \right\rvert} }$$, so the underlying point-set spaces are the same.
• Formal disk: an infinitesimal thickening of a point.

• Punctured formal disk: remove the unique global point from a formal disk.

• adic space: for $$X$$ a variety over a nonarchimedean field, e.g. $${ {\mathbb{Q}}_p }$$, the associated analytic space $$X^{\mathrm{an}}$$.

• Periods : the results of integrating an algebraic differential form in $$H_\mathrm{dR}^*$$ over a cycle in singular cohomology $$H_{{\mathrm{sing}}}^*$$. Just a number in $${\mathbb{C}}$$!

• Archive/AWS2019/Witt vectors : complicated construction, similar to $$p{\hbox{-}}$$adic integers. Uniquely characterized as a lift $$W(K)$$ of a perfect $${\mathbb{F}}_p{\hbox{-}}$$algebra $$K$$ to $${ {\mathbb{Z}}_{\widehat{p}} }{\hbox{-}}$$algebra, which becomes $$p{\hbox{-}}$$adically complete and $$p{\hbox{-}}$$torsionfree. Also lifts Frobenius.

• Perfectoid space : ? Frobenius is an isomorphism? Plus other conditions?

• For rings, there is a natural map in $$p{\hbox{-}}$$adic Hodge theory $${\mathbb{A}}_\inf(S) \to S$$, and if $$S$$ is perfectoid this is supposed to look like a 1-parameter deformation (“pro-infinitesimal”).
• If $$\operatorname{ch}S = p$$, then $$S$$ is perfectoid iff $$S$$ is perfect, i.e. Frobenius is an automorphism.
• If $$K$$ is a perfectoid field, then $$G(\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu/K) \cong G(\mkern 1.5mu\overline{\mkern-1.5muK {}^{ \flat } \mkern-1.5mu}\mkern 1.5mu/K {}^{ \flat } )$$, so we can study absolute Galois groups here (result of “almost purity” theorems).
• Unsorted/mixed characteristic : a ring $$R$$ with an ideal $$I{~\trianglelefteq~}R$$ with $$\operatorname{ch}R = 0$$ but $$\operatorname{ch}R/I = p > 0$$. The motivating examples: $${\mathbb{Z}}$$, or $${\mathcal{O}}_K$$ for $$K$$ a number field, $${ {\mathbb{Z}}_{\widehat{p}} }$$.

### Main Results

• Why care about tilts/untilts: for $$k$$ a field and $$X\coloneqq\operatorname{Spec}{\mathbb{Q}}$$ as a scheme, characteristic zero tilts of $$k$$ are a good replacement for $$X(k)$$ which makes $$X\times X$$ nontrivial.

If $k = { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }$ is a completely valued characteristic $$p$$ field, then there exists a Dedekind scheme $$X \to \operatorname{Spec}{ {\mathbb{Q}}_p }$$ whose closed points $$x$$ correspond to (isomorphism classes) of characteristic 0 untilts of $$k$$, modulo the action of the Frobenius $$\varphi(x) = x^p$$.

• So moduli of untilts of a perfectoid field, behaves like open disc in $${\mathbb{C}}$$. The unique characteristic $$p$$ untilt is like $$z=0 \in {\mathbb{C}}$$.

• Why the FF “curve” isn’t a curve: it’s a scheme over $${ {\mathbb{Q}}_p }$$, but not finite-type: specifically, the structure morphism $$X\to \operatorname{Spec}{ {\mathbb{Q}}_p }$$ is not finite type.

• Why not finite type? Since if $$x\in X$$ is a closed point then the residue field $\kappa(x) _{/ {{ {\mathbb{Q}}_p }}}$ is not a finite extension.
• Incidentally, $$\kappa(x)$$ is exactly the untilt of $$k$$ corresponding to $$x$$.
• Nice properties of $${X_\mathrm{FF}}$$ coinciding with curves:

• $$\sum \deg_f(x) = 0$$ for every rational function $$f$$ on $${X_\mathrm{FF}}$$, similar to being projective/complete as a scheme/variety.
• Note: very similar to a theorem from complex analysis: sum of orders of poles and zeros equals zero for a meromorphic function on $${\mathbb{P}}^1({\mathbb{C}})$$.
• Line bundles are classified by degree
• $$H^1(C, {\mathcal{O}}_C) = 0$$, similar to having genus $$g=0$$.
• Usually it’s like $$2g = \operatorname{rank}H^1$$
• The structure morphism $$X\to \operatorname{Spec}{ {\mathbb{Q}}_p }$$ has simply connected fibers
• Useful heuristics:

## Constructing the curve

Notation:

• $$B_{\varepsilon}\coloneqq B^{\varphi = 1}_{\mathrm{crys}}$$.

• $${\mathbb{A}}_\inf(K) \coloneqq W(K {}^{ \flat } )$$, i.e. just Archive/AWS2019/Witt vectors of the tilt.

• Supposed to interpolate between the $$\operatorname{ch}0$$ geometry of $$K$$ and $$\operatorname{ch}p$$ geometry of $K {}^{ \flat }$.

• Importantly, even if $$K$$ doesn’t have Frobenius, $K {}^{ \flat }$ does! This will yield an action of Frobenius on cohomology – this is probably used to set up trace formulas.

### Method 1: Schematically, Proj Construction

• Punchline: \begin{align*} {X_\mathrm{FF}}\approx \mathop{\mathrm{Proj}}\qty{\bigoplus_{n\geq 0} B^{\varphi = p^n}} \in {\mathsf{Sch}} .\end{align*}

• $$B^{\varphi = p^n}$$ are the elements in $$B$$ where $$\varphi(x) = p^n x$$ where $$\varphi$$ is the Frobenius.

• $$B \coloneqq\colim_{I\subseteq [0, 1] \subset {\mathbb{R}}} B_{I}$$ is a $$p{\hbox{-}}$$adic Frechet space

• Supposed to look like holomorphic functions of $$p$$.
• For $$I\coloneqq[a, b]$$, $$B_{[a, b]}$$ is the completion of $W({\mathcal{O}}_C {}^{ \flat } ) { \left[ \scriptstyle {{1\over p}, {1\over {\left\lvert {\pi } \right\rvert}}} \right] }$ with respect to Gauss norms, where $$W$$ denotes the Witt vectors, and we’ve localized at $$p$$ and $$\pi$$ a pseudo-uniformizer.

• Alternatively using the Riemann sphere analogy: $${\mathbb{P}}^1({\mathbb{C}})$$ can be recovered as a proj. Write $${\operatorname{Fil}}_k {\mathbb{C}}[z] \coloneqq\left\{{f\in {\mathbb{C}}[z] {~\mathrel{\Big\vert}~}\deg f \leq k}\right\}$$ to be the $$k$$th filtered piece using a filtration by degree.

• The claim is that $${\mathbb{P}}_1({\mathbb{C}}) \cong \mathop{\mathrm{Proj}}\qty{\bigoplus_{k\geq 0} {\operatorname{Fil}}_k {\mathbb{C}}[z]}$$.

• Somehow this uses that $${\mathbb{C}}[z_0, z_1] \xrightarrow{\sim} \bigoplus_{k\geq 0} {\operatorname{Fil}}_k {\mathbb{C}}[z]$$, no clue.
• By analogy, maybe $${X_\mathrm{FF}}= \mathop{\mathrm{Proj}}\qty{ \bigoplus_{k\geq 0} {\operatorname{Fil}}_k B_{\varepsilon}}$$ where $${\operatorname{Fil}}_k B_{\varepsilon}\coloneqq\left\{{b\in B_{\varepsilon}{~\mathrel{\Big\vert}~}v(b) \geq k}\right\}$$, where we take a valuation $$v$$ coming from $$B_\mathrm{dR}$$.

• There is a subalgebra $$B_{\mathrm{crys}}^+ \subseteq B_{\mathrm{crys}}$$ where $$B_{\mathrm{crys}}^+{ \left[ { \scriptstyle \frac{1}{t} } \right] } \cong B_{{\mathrm{crys}}}$$ for some $$t$$, plus some other properties.

• Can then obtain \begin{align*} {X_\mathrm{FF}}= \mathop{\mathrm{Proj}}\qty{\bigoplus _{k\geq 0} \qty{B_{\mathrm{crys}}^+}^{\varphi = p^k}} .\end{align*}

#### Informal Description

Glue together period rings : an affine scheme to a formal disk along a formal punctured disc, so like \begin{align*} {X_\mathrm{FF}}= \operatorname{Spec}B_{{\mathrm{crys}}}^{\varphi = 1} { \displaystyle\coprod_{\operatorname{Spec}B_{\mathrm{dR}}} } \operatorname{Spec}B_{\mathrm{dR}}^+ .\end{align*}

• $$\operatorname{Spec}B_{{\mathrm{crys}}}^{\varphi = 1}$$ is an affine scheme.

• $$\operatorname{Spec}B_{\mathrm{dR}^+}$$ is a formal disk.

• $$\operatorname{Spec}B_{\mathrm{dR}}$$ is a formal punctured disk.

• $$B_\mathrm{dR}$$ is a period ring, cooked up to relate $$H^*_\mathrm{dR}(Y)$$ to $$H^*_{\text{ét}}(Y_{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}; { {\mathbb{Q}}_p })$$ when $$Y$$ is the generic fiber of a scheme over $$W(k)$$ and $$Y_{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu} \coloneqq Y { \underset{\scriptscriptstyle {\operatorname{Spec}K} }{\times} } \operatorname{Spec}\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu$$ (see Crystalline comparison conjecture).

• $$B_{\mathrm{dR}}$$ is a complete DVR, a field, and a (very large) $${ {\mathbb{Q}}_p }{\hbox{-}}$$algebra with residue characteristic zero, and $$B_{{\mathrm{crys}}} \leq B_{\mathrm{dR}}$$ is a subring.
• $$B_\mathrm{dR}^+$$ is its valuation ring (or place), i.e. a subring such that for every element $$x$$ of $$\operatorname{ff}(B_{\mathrm{dR}})$$, either $$x \in B_{\mathrm{dR}}^+$$ or $$x^{-1}\in B_{\mathrm{dR}}^+$$.

• This is supposed to be like a ring of integers $${\mathcal{O}}_K$$ for a number field $$K$$. It is non-canonically isomorphic to the $$p{\hbox{-}}$$adic Laurent field $${\mathbb{C}}_p((t))$$.
• So it’s like the one-point compactification of $$\operatorname{Spec}$$ of the period ring of $$p{\hbox{-}}$$adic Hodge theory.

• This is supposed to look like building the Riemann sphere $${\mathbb{P}}^1({\mathbb{C}})$$ as $${\mathbb{C}}[z]$$ (regular away from $$\infty)$$ glued to $${\mathbb{C}}\qty{1\over z}$$ (germs of meromorphic functions at $$\infty$$).

• $$B_{{\mathrm{crys}}}^{\varphi = 1}$$ is supposed to look like $${\mathbb{C}}[z]$$. Classically, $${\mathbb{C}}\qty{z}$$ is the full field of meromorphic functions on $${\mathbb{CP}}^1$$, and ${\mathbb{C}} { \left[ \scriptstyle {z} \right] }$ are regular functions on $${\mathbb{CP}}^1\setminus\left\{{\infty}\right\}$$.

• And $$B_\mathrm{dR}$$ is like $${\mathbb{C}}\qty{1\over z}$$, germs of meromorphic functions at infinity.

• Taking $${\mathbb{CP}}^1$$ and puncturing at a point yields a disc, so $${\mathbb{CP}}^1\setminus\left\{{\infty}\right\}, {\mathbb{CP}}^1\setminus\left\{{0}\right\}$$ are discs. Their overlap is $${\mathbb{CP}}^1\setminus\left\{{\infty, 0}\right\} \cong {\mathbb{D}}\setminus\left\{{0}\right\}$$, a disc punctured at $$z=0$$.

??

??

## Other Random Notes

• Much like $${\mathbb{P}}^1({\mathbb{C}})$$, we have a Grothendieck splitting principle : every vector bundle $${\mathcal{E}}\to {X_\mathrm{FF}}$$ splits uniquely as $${\mathcal{E}}\cong \bigoplus_{k=1}^m {\mathcal{O}}_{{X_\mathrm{FF}}}(\lambda _k)$$ where $$\lambda_k \in {\mathbb{Q}}$$ are weakly decreasing and $${\mathcal{O}}_{{X_\mathrm{FF}}}(\lambda)$$ is a (somewhat complicated) rational twist.

• Proof uses p-divisible groups, can be used to answer questions in p-adic Hodge theory (e.g. about Galois representations).
• Needs Scholze’s theory of diamonds: Weil’s proof of RH for curves uses the surface $$X{ \underset{\scriptscriptstyle {{\mathbb{F}}_q} }{\times} } X$$ for $$X$$ a curve (this surface is also used in shtuka theory in geometric Langlands), need a similar object in arithmetic geometry that should morally look like $${\mathbb{Z}}\otimes_{{\mathbb{F}}_1} {\mathbb{Z}}$$.

• $$H^1({X_\mathrm{FF}}; {\mathcal{O}}_{{X_\mathrm{FF}}}) = { {\mathbb{Q}}_p }$$, which can apparently be computed using Čech cohomology and the fundamental exact sequence: \begin{align*} 0 \longrightarrow \mathbb{Q}_{p} \longrightarrow B_{e} \longrightarrow B_{\mathrm{dR}} / B_{\mathrm{dR}}^{+} \longrightarrow 0 .\end{align*}