# 2021-09-14

## 14:45

• Cartan subgroup

• Centralizer of a maximal torus.
• Borel subgroup

• Maximal connected solvable subgroup
• Why care: critical to structure theory of simple reductive algebraic group. Uses pairs $$(B, N)$$ where $$N = N_G(T)$$ is the normalizer of a maximal torus.
• Parabolic subgroup

• Literally any $$P\leq G$$ such that $$B \subseteq P \subseteq G$$
• $$G/P$$ is a complete variety, so all projections $$X\times ({-}) \to ({-})$$ are closed maps.
• $$G/B$$ is the largest complete variety since $$B \subseteq P$$ for all $$P$$.
• local fields

• Complete with respect to a topology induced by $$v$$ a discrete valuation with $$\kappa$$ finite.
• valuation : For $$v: k \to G\cup\left\{{\infty}\right\}$$ and $$G\in {\mathsf{Ab}}$$ totally ordered.

• value group : $$\operatorname{im}v$$
• valuation ring : $$R_v \coloneqq\left\{{v(x) \geq 0}\right\}$$
• Prime/maximal ideal: $${\mathfrak{m}}_v \coloneqq\left\{{v(x)>0}\right\}$$
• Residue field $$\kappa_v \coloneqq R_v/{\mathfrak{m}}_v$$
• places : $$\left\{{v}\right\}/\sim$$ where $$v_2\sim v_1 \iff v_2 = \phi \circ v_1$$.
• uniformizer : for $$R$$ a DVR, a generator $$\pi$$ for the unique maximal ideal, so $$R^{\times}\left\langle{\pi}\right\rangle = R$$ and $$x\in R \implies x = u\pi^k$$
• global field : algebraic number fields, function fields of algebraic curves over finite fields (so finite extensions of $${\mathbb{F}}_q { \left( {(} \right) } t))$$.

• For a 1-dim variety: $$\operatorname{ff}k[X]$$, the fraction field of the coordinate ring.
• Note the closed point of $$\operatorname{Spec}{ {\mathbb{Z}}_{\widehat{p}} }$$ is $${\mathbb{F}}_p$$ and the generic point is $${ {\mathbb{Q}}_p }$$.

• nonarchimedean field

• Existence of infinitesimals, i.e. for a $${\mathbb{Z}}{\hbox{-}}$$module with a linear order, $$x$$ is infinitesimal with respect to $$y$$ if $$nx < y$$ for all $$n$$
• E.g. $${\mathbb{R}} { \left( {(} \right) } x)$$ or $${\mathbb{Q}} { \left( {(} \right) } x)$$, $$1/x$$ is infinitesimal. Or $${ {\mathbb{Q}}_p }$$.
• Nonarchimedean local fields are totally disconnected.
• proper morphism

• Separated, finite type, universally closed (so for $$X\to Y$$, all projections $$X{ \underset{\scriptscriptstyle {Y} }{\times} }Z\to Z)$$ are closed maps).
• For spaces: preimages of compact subspaces are compact.
• For locally compact Hausdorff spaces: continuous and closed with compact fibers.
• Iwahori subgroup

• Subgroup of an algebraic group over a nonarchimedean local field, analogous to a Borel.
• Fun fact: $$p{\hbox{-}}$$torsion in an Unsorted/class group was the main obstruction to a direct proof of FLT. Observed by Kummer.

• Motivates defining $$K_\infty \coloneqq\colim_n L(\mu_{p^{n+1}})$$, using $${ \mathsf{Gal}} (K_n{}_{/ {K}} ) = C_{p^n}$$ so $$G\coloneqq{ \mathsf{Gal}} (K_\infty {}_{/ {K}} ) = { {\mathbb{Z}}_{\widehat{p}} }$$. Set $$I_n = { \operatorname{cl}} (K_n)[p]$$ to be the $$p{\hbox{-}}$$torsion in the ideal class group of $$K_n$$, form $$I\coloneqq\colim_n I_n$$ using norm maps to get module structure, recover info about $${ \operatorname{cl}} (K)[p]$$.
• Main conjecture of Iwasawa theory : two methods of defining $$p{\hbox{-}}$$adic $$L{\hbox{-}}$$functions should coincide. Proved by Mazur/Wiles for $${\mathbb{Q}}$$, all totally real number fields by Wiles.

• One defining method: interpolate special values.
• Actual definition of Dirichlet characters:

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

## 22:17

• Define: geometric fiber

• Reductive, semisimple, simply connected, etc for $G\in{\mathsf{Grp}}{\mathsf{Sch}}_{/ {S}}$: affine and smooth over $$S$$, where geometric fibers are reductive. s.s., etc algebraic groups.

• etale morphism

• For $$f \in \mathop{\mathrm{Mor}}_{\mathsf{Sch}}(X, Y)$$ finite type and $$X, Y$$ locally Noetherian, $$f$$ is etale at $$y\in Y$$ if $$f^*: {\mathcal{O}}_{f(y)} \to {\mathcal{O}}_y$$ is flat and $${\mathcal{O}}_{f(y)}/{\mathfrak{m}}_{f(y)} \to {\mathcal{O}}_{f(y)}/ f^*({\mathfrak{m}}_{f(y)} {\mathcal{O}}_y)$$ is a finite separable extension.
• Central extension

• Fiber functor

• Algebraic fundamental group

• Certain groups that become isomorphic after field extensions have related automorphic representations.

• Langlands dual: $${\mathcal{L}}(G)$$ controls $${\mathsf{G}{\hbox{-}}\mathsf{Mod}}$$ somehow, arises as an extension $${ \mathsf{Gal}} (k^s _{/ {k}} ) \to {\mathcal{L}}(G) \to H$$ where $H \in \mathsf{Lie}{\mathsf{Grp}}_{/ {{\mathbb{C}}}}$.

• A connected reductive algebraic group over a separably closed field $$k$$ is uniquely determined by its root datum.

• Langlands dual : take root datum, dualize datum, take associated group.

• Langlands’ strategy for proving local and global conjectures: Arthur-Selberg trace formula.

• Equivalence of orbital integrals can somehow be related to Springer fiber??

• Starting point for Langlands: Artin reciprocity, generalizing quadratic reciprocity.

• Chebotarev density theorem is a generalization of Unsorted/Dirichlet’s theorem on primes in arithmetic progressions.

• “The Langlands conjectures associate an automorphic representation of the adelic group $$\operatorname{GL}_n({\mathbb{A}}_{/ {{\mathbb{Q}}}} )$$ to every $$n{\hbox{-}}$$dimensional irreducible representation of the Galois group, which is a cuspidal representation if the Galois representation is irreducible, such that the Artin L function of the Galois representation is the same as the automorphic L function 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

• ${\mathbb{A}}_{/ {{\mathbb{Q}}}}$: keeps track of all of the completions of $${\mathbb{Q}}$$ simultaneously.

• Reciprocity conjecture: a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an $$L$$-group

• geometric Langlands : relates l-adic representations of the étale fundamental group of an algebraic curve 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 for automorphic forms to Galois representations for connected reductive groups over global function fields

• purity : happens in a specific codimension

#web/quick-notes #representation-theory #subjects/arithmetic-geometry/Langlands