2021-09-14

14:45

Tags: #representation-theory #subjects/arithmetic-geometry/Langlands

  • 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:

attachments/2021-09-14_15-27-44.png

  • 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

Tags: #subjects/arithmetic-geometry/Langlands

  • 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

    attachments/2021-09-14_22-29-30.png

  • 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

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