14:45
Tags: #lie-theory #arithmetic-geometry/Langlands
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Cartan subgroup
- Centralizer of a maximal torus.
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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.
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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 \(P\).
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local fields
- Complete with respect to a topology induced by \(v\) a discrete valuation with \(\kappa\) finite.
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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 \(x\in R \implies x = u\pi^k\)
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global field : algebraic number fields, function fields of algebraic curves over finite fields (so finite extensions of \({ \mathbf{F} }_q { \left( {(} \right) } t))\).
- For a 1-dim variety: \(\operatorname{ff}k[X]\), the fraction field of the coordinate ring.
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Note the closed point of \(\operatorname{Spec}{ {\mathbf{Z}}_{\widehat{p}} }\) is \({ \mathbf{F} }_p\) and the generic point is \({ {\mathbf{Q}}_p }\).
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nonarchimedean field
- Existence of infinitesimals, i.e. for a \({\mathbf{Z}}{\hbox{-}}\)module with a linear order, \(x\) is infinitesimal with respect to \(y\) if \(nx < y\) for all \(n\)
- E.g. \({\mathbf{R}} { \left( {(} \right) } x)\) or \({\mathbf{Q}} { \left( {(} \right) } x)\), \(1/x\) is infinitesimal. Or \({ {\mathbf{Q}}_p }\).
- Nonarchimedean local fields are totally disconnected.
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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.
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Iwahori subgroup
- Subgroup of an algebraic group over a nonarchimedean local field, analogous to a Borel.
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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 \(G\coloneqq{ \mathsf{Gal}} (K_\infty {}_{/ {K}} ) = { {\mathbf{Z}}_{\widehat{p}} }\). 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 \({ \operatorname{cl}} (K)[p]\).
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Main conjecture of Iwasawa theory : two methods of defining \(p{\hbox{-}}\)adic \(L{\hbox{-}}\)functions should coincide. Proved by Mazur/Wiles for \({\mathbf{Q}}\), all totally real number fields by Wiles.
- One defining method: interpolate special values.
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Actual definition of Dirichlet characters:
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Fundamental lemma in The Langlands Program
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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?
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Relates orbital integrals on a reductive group over a local fields, to “stable” orbital integrals on its endoscopic groups.
22:17
Tags: #arithmetic-geometry/Langlands
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Define: geometric fiber
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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.
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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.
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Central extension
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Fiber functor
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Algebraic fundamental group
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Certain groups that become isomorphic after field extensions have related automorphic representations.
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Langlands dual: \({\mathcal{L}}(G)\) controls ${}{G}{\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}}{/ {{\mathbf{C}}}} $.
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A connected reductive algebraic group over a separably closed field \(k\) is uniquely determined by its root datum.
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Langlands dual : take root datum, dualize datum, take associated group.
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Langlands’ strategy for proving local and global conjectures: Arthur-Selberg trace formula.
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Equivalence of orbital integrals can somehow be related to Springer fiber??
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Starting point for Langlands: Artin reciprocity, generalizing quadratic reciprocity.
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Chebotarev density theorem is a generalization of Unsorted/Dirichlet’s theorem on primes in arithmetic progressions.
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“The Langlands conjectures associate an automorphic representation of the adelic group \(\operatorname{GL}_n({\mathbf{A}}_{/ {{\mathbf{Q}}}} )\) 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”
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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
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${\mathbf{A}}_{/ {{\mathbf{Q}}}} $: keeps track of all of the completions of \({\mathbf{Q}}\) simultaneously.
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Reciprocity conjecture: a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an \(L\)-group
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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.
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2018: Lafforgue established global Langlands for automorphic forms to Galois representations for connected reductive groups over global function fields
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purity : happens in a specific codimension