Fargues-Fontaine Reading Notes

Fargues-Fontaine Reading Notes

Useful references

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 \(G({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }/{\mathbb{Q}})\) (see notes elsewhere)
  • Relations to stable homotopy : an isomorphism of Lubin-Tate, and the Drinfeld tower?

Motivation / Summary

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 \({ {\mathbb{Q}}_p }\) 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:

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

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 \(z=0\).

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

Method 2: As an adic space

??

Method 3: As a diamond

??

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*}

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