1 2021-09-14

1.1 14:45

Terms:

1.2 22:17

1.3 23:59

2 2021-09-12

2.1 \({\mathsf{FI}}{\hbox{-}}\)modules (23:45)

Reference: Church-Ellenberg-Farb

3 2021-08-05

3.1 Classical / Analytic Moduli Theory

Reference: see https://www.math.purdue.edu/~arapura/preprints/shimura2.pdf

3.2 Moduli as Stacks

\begin{align*} \text { ord }_{x}(f):= \begin{cases}0 & \text { if } \mathrm{f} \text { is holomorphic and non-zero at } x \\ k & \text { if } \mathrm{f} \text { has a zero of order } k \text { at } x \\ -k & \text { if } \mathrm{f} \text { has a pole of order } k \text { at } x\end{cases} .\end{align*}

To pick back up: https://repositorio.uniandes.edu.co/bitstream/handle/1992/43725/u830743.pdf?sequence=1

4 2021-07-25

5 Fargues-Fontaine Stuff

Useful references:

5.1 Motivation / Summary

Some notes on the Fargues-Fontaine curve \({X_\mathrm{FF}}\), the fundamental curve of \(p{\hbox{-}}\)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 \(p{\hbox{-}}\)adic Hodge theory. Stems from conjectures of Grothendieck wanting to related de Rham cohomology to étale cohomology, and a similar theorem proved by Faltings in the 80s. 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: \({\mathsf{Bun}}_G\), the moduli stack of \(G{\hbox{-}}\)bundles for \(G\) a reductive algebraic group over a local field over a family of \({X_\mathrm{FF}}\) curves.

A useful overall analogy: it’s like the Riemann sphere \({\mathbb{CP}}^1 \mathrel{\vcenter{:}}={\mathbb{P}}^1({\mathbb{C}})\), 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 \({ {\mathbb{Z}}_p }\) 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 \(p{\hbox{-}}\)adic Hodge theory. One can also geometrize class field theory somehow, and realize \({ \mathsf{Gal}} (\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{Q}}\mkern-1.5mu}\mkern 1.5mu/{\mathbb{Q}})\) as some kind of fundamental group.

\({X_\mathrm{FF}}\) is also roughly a moduli of “untilts,” which 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.

Todos: motivate

5.1.1 Definitions

5.1.2 Main Results

5.2 Constructing the curve

Notation:

5.2.1 Method 1: Schematically, Proj Construction

5.2.1.1 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*}