---
date: 2021-09-14
tags: [ web/quick-notes ]
---


# 2021-09-14

## 14:45

Tags:
#lie-theory #arithmetic-geometry/Langlands

- [Cartan](Cartan) subgroup
  - Centralizer of a maximal torus.
- [Borel](Borel.md) subgroup
  - Maximal connected solvable subgroup
  - Why care: critical to structure theory of simple [reductive algebraic group](reductive%20algebraic%20group).
    Uses pairs $(B, N)$ where $N = N_G(T)$ is the normalizer of a maximal torus.
- [Parabolic](Parabolic) subgroup
  - Literally any $P\leq G$ such that $B \subseteq P \subseteq G$
  - $G/P$ is a complete variety, so all projections $X\times (\wait) \to (\wait)$ are closed maps.
  - $G/B$ is the largest complete variety since $B \subseteq P$ for all $P$.
- [local fields](local%20fields)
  - Complete with respect to a topology induced by $v$ a discrete valuation with $\kappa$ finite.
- [valuation](valuation) : For $v: k \to G\union\ts{\infty}$ and $G\in \Ab$ [totally ordered](totally%20ordered).
  - [value group](value%20group) : $\im v$
  - [valuation ring](valuation%20ring.md) : $R_v \da \ts{v(x) \geq 0}$
  - Prime/maximal ideal: $\mfm_v \da \ts{v(x)>0}$
  - Residue field $\kappa_v \da R_v/\mfm_v$
  - [places](place) : $\ts{v}/\sim$ where $v_2\sim v_1 \iff v_2 = \phi \circ v_1$.
  - [uniformizer](uniformizer) : for $R$ a DVR, a generator $\pi$ for the unique maximal ideal, so $R\units \gens{\pi} = R$ and $x\in R \implies x = u\pi^k$

- [global field](global%20field.md) : algebraic number fields, function fields of [algebraic curves](algebraic%20curve.md) over finite fields (so finite extensions of $\FF_q\functionfield(t))$.
  - For a 1-dim variety: $\ff k[X]$, the fraction field of the coordinate ring.
- Note the closed point of $\spec \ZZpadic$ is $\FF_p$ and the generic point is $\QQpadic$.

- [nonarchimedean field](nonarchimedean%20field)
  - Existence of infinitesimals, i.e. for a $\ZZ\dash$module with a linear order, $x$ is infinitesimal with respect to $y$ if $nx < y$ for all $n$
  - E.g. $\RR\functionfield(x)$ or $\QQ\functionfield(x)$, $1/x$ is infinitesimal.
    Or $\QQpadic$.
  - Nonarchimedean local fields are totally disconnected.
- [proper morphism](proper%20morphism.md)
  - Separated, finite type, universally closed (so for $X\to Y$, all projections $X\fiberprod{Y}Z\to Z)$ are closed maps).
  - For spaces: preimages of compact subspaces are compact.
  - For locally compact Hausdorff spaces: continuous and closed with compact fibers.
- [Iwahori](Iwahori) subgroup
  - Subgroup of an algebraic group over a nonarchimedean local field, analogous to a Borel.

- Fun fact: $p\dash$torsion in an [Unsorted/class group](Unsorted/class%20group.md) was the main obstruction to a direct proof of FLT.
  Observed by Kummer.

  - Motivates defining $K_\infty \da \colim_n L(\mu_{p^{n+1}})$, using $\Gal(K_n{}\slice K) = C_{p^n}$ so $G\da \Gal(K_\infty {}\slice K) = \ZZpadic$.
  Set $I_n = \cl(K_n)[p]$ to be the $p\dash$torsion in the ideal class group of $K_n$, form $I\da \colim_n I_n$ using norm maps to get module structure, recover info about $\cl(K)[p]$.

- Main conjecture of [Iwasawa theory](Iwasawa%20theory.md) : two methods of defining $p\dash$adic $L\dash$functions should coincide.
  Proved by Mazur/Wiles for $\QQ$, all totally real number fields by Wiles.
  - One defining method: interpolate special values.

- Actual definition of Dirichlet characters:

![](attachments/2021-09-14_15-27-44.png)

- Fundamental lemma in [The Langlands Program](The%20Langlands%20Program)
  - Relates orbital integrals on a [reductive group](reductive%20group) over a [local fields](local%20fields), to "stable" orbital integrals on its endoscopic groups.
    - Endoscope: $H\leq G$ a quasi-split group whose Langlands dual $H\dual$ is the connected component of $C_{G\dual}(x)$ for $x\in G\dual$ some semisimple element.
  - Want to get at [automorphic forms](automorphic%20forms) and the arithmetic of [Shimura varieties](Shimura%20varieties)
  - Some "stabilized" version of the [Grothendieck-Lefschetz Trace Formula](Grothendieck-Lefschetz%20Trace%20Formula)?

## 22:17

Tags:
#arithmetic-geometry/Langlands 

- Define: [geometric fiber](geometric%20fiber.md)
- [Reductive](Reductive), [semisimple](semisimple), simply connected, etc for $G\in\Grp\Sch\slice S$: affine and smooth over $S$, where geometric fibers are reductive. s.s., etc [algebraic groups](algebraic%20groups).
- [etale morphism](etale%20morphism.md)
  - For $f \in \Mor_\Sch(X, Y)$ [finite type](finite%20type.md) and $X, Y$ locally [Noetherian](Noetherian.md), $f$ is [etale](Unsorted/etale.md) at $y\in Y$ if $f^*: \OO_{f(y)} \to \OO_y$ is flat and $\OO_{f(y)}/\mfm_{f(y)} \to \OO_{f(y)}/ f^*(\mfm_{f(y)} \OO_y)$ is a finite [separable extension](separable%20extension.md).
- Central extension
- Fiber functor

- Algebraic fundamental group

  ![](attachments/2021-09-14_22-29-30.png)

- Certain groups that become isomorphic after field extensions have related [automorphic representations](automorphic%20representations).
- [Langlands dual](Langlands%20dual): $\mcl(G)$ controls $\mods{G}$ somehow, arises as an extension $\Gal(k^s \slice k) \to \mcl(G) \to H$ where $H \in \Lie\Grp\slice \CC$.
- A connected reductive algebraic group over a separably closed field $k$ is uniquely determined by its [root datum](root%20datum).
- [Langlands dual](Langlands%20dual) : take root datum, dualize datum, take associated group.
- Langlands' strategy for proving local and global conjectures: [Arthur-Selberg trace formula](Arthur-Selberg%20trace%20formula).
- Equivalence of orbital integrals can somehow be related to [Springer fiber](Springer%20fiber)??
- Starting point for Langlands: [Artin reciprocity](Artin%20reciprocity), generalizing [quadratic reciprocity](quadratic%20reciprocity.md).
- [Chebotarev density](Chebotarev%20density.md) theorem is a generalization of [Unsorted/Dirichlet's theorem on primes in arithmetic progressions](Unsorted/Dirichlet's%20theorem%20on%20primes%20in%20arithmetic%20progressions.md).
- "The [Langlands conjectures](Langlands%20conjectures) associate an automorphic representation of the [adelic](adelic) group $\GL_n(\AA\slice \QQ)$ to every $n\dash$dimensional irreducible representation of the Galois group, which is a [cuspidal representation](cuspidal%20representation) if the Galois representation is irreducible, such that the [Artin L function](Artin%20L%20function.md) of the Galois representation is the same as the [automorphic L function](automorphic%20L%20function.md) of the automorphic representation"
- Serre's modularity conjecture: an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form
- $\AA\slice \QQ$: keeps track of all of the completions of $\QQ$ simultaneously.
- Reciprocity conjecture: a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an $L$-group
- [geometric Langlands](geometric%20Langlands.md) : relates [l-adic representations](l-adic%20representations.md) of the [étale fundamental group](étale%20fundamental%20group) of an [algebraic curve](algebraic%20curve.md) to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles over the curve.
- 2018: Lafforgue established [global Langlands](global%20Langlands) for automorphic forms to [Galois representations](Galois%20representations.md) for connected reductive groups over global function fields
- [purity](purity) : happens in a specific codimension