Fargues-Fontaine Reading Notes

Useful references

https://www.mathi.uni-heidelberg.de/fg-sga/docs/Programm_la-courbe.pdf

Seminar outline: - https://www.math.ias.edu/~lurie/205notes/Lecture1-Overview.pdf - https://www.math.ias.edu/~lurie/205notes/Lecture6-Curve.pdf

Todos

Motivate the following:

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

Motivation / Summary

attachments/Pasted%20image%2020220503105503.png

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 ${ {\mathbf{Q}}_p }$ where local Langlands for ${ {\mathbf{Q}}_p }$ 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 ${\mathbf{CP}}^1 \coloneqq{\mathbf{P}}^1({\mathbf{C}})$, and in fact the adic version is a $p{\hbox{-}}$adic Riemann surface. The full ring of meromorphic functions on ${\mathbf{P}}^1({\mathbf{C}})$ is ${\mathbf{C}}\qty{z}$, but ${\mathbf{C}}[z]$ captures most of the data away from $\infty$. ${\mathbf{C}}[z]$ as a ${\mathbf{C}}{\hbox{-}}$algebra consists of regular (polynomial) functions on ${\mathbf{P}}^1({\mathbf{C}})$ with a pole at $\infty$ of order equal to the degree of the polynomial. View the ${\mathbf{Z}}{\hbox{-}}$algebra ${\mathbf{Z}}$ 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 ${\mathbf{Z}}$ with either ${ {\mathbf{Z}}_{\widehat{p}} }$ or ${ {\mathbf{Q}}_p }$ (so $p{\hbox{-}}$adic things) and looking at ${ {\mathbf{Q}}_p }{\hbox{-}}$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 ${ \mathsf{Gal}} (\mkern 1.5mu\overline{\mkern-1.5mu{\mathbf{Q}}\mkern-1.5mu}\mkern 1.5mu/{\mathbf{Q}})$ as a fundamental group.

${X_\mathrm{FF}}$ is also roughly a moduli of untilts, whisch allow passing between ${ \mathbf{F} }_p$ and ${ {\mathbf{Q}}_p }$. 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 $A \in {}_{B}{\mathsf{Mod}}$ 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 $(K, \cdot) \to ({\mathbf{R}}_{\geq 0}, +)$ 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 ${ \mathbf{F} }_p{\hbox{-}}$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. ${ {\mathbf{Q}}_p }$, 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 ${\mathbf{C}}$!
Archive/AWS2019/Witt vectors : complicated construction, similar to $p{\hbox{-}}$adic integers. Uniquely characterized as a lift $W(K)$ of a perfect ${ \mathbf{F} }_p{\hbox{-}}$algebra $K$ to ${ {\mathbf{Z}}_{\widehat{p}} }{\hbox{-}}$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 ${\mathbf{A}}_\inf(S) \to S$, 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: ${\mathbf{Z}}$, or ${\mathcal{O}}_K$ for $K$ a number field, ${ {\mathbf{Z}}_{\widehat{p}} }$.

Main Results

Why care about tilts/untilts: for $k$ a field and $X\coloneqq\operatorname{Spec}{\mathbf{Q}}$ 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 $X \to \operatorname{Spec}{ {\mathbf{Q}}_p }$ 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 ${\mathbf{C}}$. The unique characteristic $p$ untilt is like $z=0 \in {\mathbf{C}}$.
Why the FF “curve” isn’t a curve: it’s a scheme over ${ {\mathbf{Q}}_p }$, but not finite-type: specifically, the structure morphism $X\to \operatorname{Spec}{ {\mathbf{Q}}_p }$ is not finite type.
- Why not finite type? Since if $x\in X$ is a closed point then the residue field $\kappa(x) _{/ {{ {\mathbf{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 ${\mathbf{P}}^1({\mathbf{C}})$.
- 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 $X\to \operatorname{Spec}{ {\mathbf{Q}}_p }$ has simply connected fibers
Useful heuristics:

attachments/2021-07-02_01-42-27.png

Constructing the curve

Notation:

$B_{\varepsilon}\coloneqq B^{\varphi = 1}_{\mathrm{crys}}$.
${\mathbf{A}}_\inf(K) \coloneqq W(K {}^{ \flat } )$, 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.
$B \coloneqq\colim_{I\subseteq [0, 1] \subset {\mathbf{R}}} B_{I}$ 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: ${\mathbf{P}}^1({\mathbf{C}})$ can be recovered as a proj. Write ${\operatorname{Fil}}_k {\mathbf{C}}[z] \coloneqq\left\{{f\in {\mathbf{C}}[z] {~\mathrel{\Big\vert}~}\deg f \leq k}\right\}$ to be the $k$th filtered piece using a filtration by degree.
The claim is that ${\mathbf{P}}_1({\mathbf{C}}) \cong \mathop{\mathrm{Proj}}\qty{\bigoplus_{k\geq 0} {\operatorname{Fil}}_k {\mathbf{C}}[z]}$.
- Somehow this uses that ${\mathbf{C}}[z_0, z_1] \xrightarrow{\sim} \bigoplus_{k\geq 0} {\operatorname{Fil}}_k {\mathbf{C}}[z]$, 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} { \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 $H^*_{\text{ét}}(Y_{\mkern 1.5mu\overline{\mkern-1.5muK\mkern-1.5mu}\mkern 1.5mu}; { {\mathbf{Q}}_p })$ 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) ${ {\mathbf{Q}}_p }{\hbox{-}}$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 ${\mathbf{C}}_p((t))$.
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 ${\mathbf{P}}^1({\mathbf{C}})$ as ${\mathbf{C}}[z]$ (regular away from $\infty)$ glued to ${\mathbf{C}}\qty{1\over z}$ (germs of meromorphic functions at $\infty$).
- $B_{{\mathrm{crys}}}^{\varphi = 1}$ is supposed to look like ${\mathbf{C}}[z]$. Classically, ${\mathbf{C}}\qty{z}$ is the full field of meromorphic functions on ${\mathbf{CP}}^1$, and ${\mathbf{C}} { \left[ \scriptstyle {z} \right] } $ are regular functions on ${\mathbf{CP}}^1\setminus\left\{{\infty}\right\}$.
- And $B_\mathrm{dR}$ is like ${\mathbf{C}}\qty{1\over z}$, germs of meromorphic functions at infinity.
- Taking ${\mathbf{CP}}^1$ and puncturing at a point yields a disc, so ${\mathbf{CP}}^1\setminus\left\{{\infty}\right\}, {\mathbf{CP}}^1\setminus\left\{{0}\right\}$ are discs. Their overlap is ${\mathbf{CP}}^1\setminus\left\{{\infty, 0}\right\} \cong {\mathbb{D}}\setminus\left\{{0}\right\}$, a disc punctured at $z=0$.

attachments/2021-07-03_21-14-26.png

Method 2: As an adic space

Method 3: As a diamond

Other Random Notes

Much like ${\mathbf{P}}^1({\mathbf{C}})$, 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 $\lambda_k \in {\mathbf{Q}}$ 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 $X \underset{\scriptscriptstyle {{ \mathbf{F} }_q} }{\times} X$ 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 ${\mathbf{Z}}\otimes_{{ \mathbf{F} }_1} {\mathbf{Z}}$.
$H^1({X_\mathrm{FF}}; {\mathcal{O}}_{{X_\mathrm{FF}}}) = { {\mathbf{Q}}_p }$, 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*}