{ "course": "GL-GLC", "planned_lessons": 15, "written_drafts": 15, "whole_course_complete": false, "original_text_license": "CC0-1.0", "lessons": [ { "id": "GL-GLC-01", "source": "src/from-automorphic-functions-to-automorphic-sheaves.md", "proof_obligations": [ "full arithmetic Langlands and excursion theorems", "function–sheaf trace formalism and complete earlier dependency chains" ] }, { "id": "GL-GLC-02", "source": "src/the-moduli-stack-of-bundles.md", "proof_obligations": [ "matching full Lie/flag/torsor/cohomology foundations", "remaining recursive local-algebra foundations" ] }, { "id": "GL-GLC-03", "source": "src/sheaves-and-d-modules-on-bun-g.md", "proof_obligations": [ "geometric truncatability", "global normalized half root", "nilpotent regularity and suitable-open preservation", "derived tempered Satake", "general Riemann–Hilbert", "categorical descent/base-change foundations" ] }, { "id": "GL-GLC-04", "source": "src/hecke-functors-and-hecke-eigensheaves.md", "proof_obligations": [ "full classical and derived Satake theorems and Hecke descent foundations", "trace and closed-place coefficient formalism" ] }, { "id": "GL-GLC-05", "source": "src/geometric-class-field-theory.md", "proof_obligations": [ "full curve Picard/Poincaré and symmetric quotient geometry", "complete coherent/étale/module-descent foundation chains", "recursive algebra/scheme and adic foundations of the Noetherian finite-cover comparison", "recursive categorical platform for the constructed affine ! and geometric projection proofs", "relative Beilinson–Drinfeld collision and lax-unit factorization" ] }, { "id": "GL-GLC-06", "source": "src/the-gl-1-case-as-an-equivalence-of-categories.md", "proof_obligations": [ "recursive Picard/Hilbert, group projectivity, finite-isogeny and coherent foundations of the constructed dual and universal connection", "recursive local-algebra, ambient Ext and ordinary Picard foundations of the written trace proof", "categorical descent/base change and complete earlier foundation chains", "recursive affine/smooth-descent, coherent-duality, regular-ring and DG-category foundations of the torus zero-support proof", "recursive represented Picard/symmetric-product, perfect cohomology, trace duality, regular-local/Cohen–Macaulay and projective dimension/proper-finiteness foundations of the theta proof" ] }, { "id": "GL-GLC-07", "source": "src/the-unramified-correspondence-for-gl-n.md", "proof_obligations": [ "complete general GL_n vanishing and geometric foundations of the written conditional symmetric-group/rank-induction/adjunction deductions", "fixed-degree finite-diagram quotient realizations, subquotient and ULA properties, and restriction-limit assembly", "full torsion-stack family construction, flag smallness and Laumon/Springer IC identity beyond the written rank-one pullback", "actual stack six-operation, duality, perversity and inverse-limit category foundations", "Picard representability and compact-support line-bundle trace/localization foundations", "Whittaker construction, cleanness, Euler constancy, global descent, Hecke compatibility and cuspidality", "transverse inertia, purity and projective-fibre local-system descent foundations", "finite-field adelic, residue, trace, arithmetic Langlands and L-function proof chains", "characteristic-zero de Rham realization and complete recursive mathematical foundations" ] }, { "id": "GL-GLC-08", "source": "src/opers-critical-level-and-the-beilinson-drinfeld-construction.md", "proof_obligations": [ "recursive root/pinning, split-unipotent, faithful adjoint-group and ordinary bundle-descent foundations; recursive Serre/highest-weight and finite-algebra foundations of invariant polynomiality; full derived oper construction", "non-type-A basic quantum lifts and non-type-A coordinate-equivariant vacuum-center/oper comparison; non-type-A completed punctured-disc center and oper-Poisson comparison with exact dual-form normalization; derived/global Hamiltonian reduction; full chiral/factorization, Satake and derived-family compatibility; geometric/Satake/global central-convention comparison", "half-root, uniformization, localization, nonzero specialization, holonomicity and fusion Hecke proofs", "filtered BD quantization and full-field/reductive-center comparisons", "critical FLE, factorization, Ran, determinant, convergence and IndCoh star/shriek foundations", "recursive fixed-curve Picard/coherent/local-algebra/Ext, stack algebraization and optional analytic comparison" ] }, { "id": "GL-GLC-09", "source": "src/the-spectral-side-local-systems-singular-support-and-indcoh-nilp.md", "proof_obligations": [ "global derived LocSys representability, family perfect cotangent complexes, quasi-smooth atlases and Sing=Arth", "general reductive zero-singular-support IndCoh/QCoh theorem beyond the explicit torus model; recursive affine/smooth-descent, coherent-duality, regular-ring and DG-category foundations of the torus comparison", "general reductive spectral Eisenstein construction, properness, quasi-smoothness, support bounds and exact normalized Levi compatibility", "reducible-locus generation, irreducible zero support, categorical localization, Eisenstein transitivity and semisimple-rank induction", "genuine global nilpotent nonzero support object and strictness, distinct from the explicit local exterior model", "global compact generation and general reductive nilpotence-convention comparison", "Whittaker normalization and qualified classical reducibility/singularity statement", "complete earlier curve, local algebra, stable DG-category and derived stack foundations" ] }, { "id": "GL-GLC-10", "title": "The categorical conjecture and its variants", "source": "src/the-categorical-conjecture-and-its-variants.md", "reader": "GL-GLC-10.html", "proof_obligations": [ "full recursive stable/DG-category, monoidal, bar, adjoint, Yoneda and Ind foundations", "actual nonunital Ran localization, unital/action comparison, Satake and kernel vanishing for all connected reductive groups", "critical/Kac–Moody enhanced Hecke eigen-property, proper Eisenstein enhancement and componentwise cuspidal/chiral generation", "actual vacuum Poincare, exponential/half-twist normalization, compactness, rigidity and complete coefficient identity", "compact eventually-coconnective bound, coarsening equivalence and continuous linear Langlands lift", "global stack point/fibre and restricted-variation base-change comparisons; derived-stack geometric detection after all field extensions", "tempered quotient, compact projection and exact coherent-image theorems with normalized Eisenstein compatibility for G and all Levis", "enhanced torus Fourier equivalence including Cartan inversion, actual vacuum and zero-singular-support theorem", "restricted regular-singular Riemann–Hilbert, Betti specified-functor comparisons and return to original coefficient/base fields", "actual irreducible fibre and normalized multiplicity/perversity consequences, with automorphisms and exceptional hypotheses retained", "finite-field categorical Frobenius trace and its accessible local-term compatibility" ] }, { "id": "GL-GLC-11", "title": "Constructing the Langlands functor (GLC I)", "source": "src/constructing-the-langlands-functor.md", "reader": "GL-GLC-11.html", "proof_obligations": [ "full recursive stable/DG-category, adjoint, coherent Yoneda/action and Ind foundations", "actual normalized half-root, exponential/character duality, vacuum compactness and complete natural Whittaker identity", "actual Ran localization, derived Satake, kernel vanishing, spectral rigidity and perfect-action amplitude", "Lin kernel comparison, nilpotent kernel pairing, geometric Casselman–Shalika and tempered Hecke exactness", "restricted spectral exact/conservative probes, enhanced coefficient/action and all-field base-change comparisons", "truncatability, compact-Dmodule Cousin and affine !-amplitude geometry, global all-field !-fibre criterion", "actual t-exact bounded coarsening equivalence and ordinary-fibre upper detection/projection bounds", "enhanced torus Fourier–Mukai–Laumon/Cartan normalization and zero-singular-support theorem", "nonzero constant full-image example and its bounded source/coarse-vanishing assertions", "Betti quotient-exponential, projector, finite vacuum construction, compact-image and spectral-action theorems", "restricted regular-singular Riemann–Hilbert, field comparisons, geometric detection and tempered/coherent-image/Eisenstein comparisons for all Levis", "eigensheaf regularity/perversity and irreducible fibre/pushforward comparisons; microstalk, perfect restriction, spectral component and scheme-theoretic cycle geometry", "full GLC equivalence and complete earlier mathematical foundation chains" ] }, { "id": "GL-GLC-12", "title": "Kac-Moody localization and the fundamental local equivalence (GLC II)", "source": "src/kac-moody-localization-and-the-fundamental-local-equivalence.md", "reader": "GL-GLC-12.html", "proof_obligations": [ "full critical Kac–Moody, arc-action, coordinate, t-structure and factorization-category constructions", "full Feigin–Frenkel duality, centre and birth-of-opers construction with central and factorization compatibility", "actual affine Skryabin, derived Satake, geometric Casselman–Shalika and pointwise tempered reconstruction", "actual compact preservation, fibrewise-to-factorization comparison, unitality, convergence and collisions", "infinite-type star/shriek IndCoh, placidity, duality, ambient module-category and base-change foundations", "critical localization, infinitesimal Hecke descent, normalized half twist and constant determinant lines", "global monodromy-free oper scalar extension, coherent Hecke property, spectral fibres and localized nonvanishing", "actual local-to-global unitalization and factorization-homology pairing identities", "spectral localization/global-sections full faithfulness, spectral Poincaré zero-support and automorphic tempered images", "full torus unbounded critical FLE, categorical arc equivariance, infinite-dimensional descent and coordinate compatibility", "derived regular-central-character comparison, actual Frenkel–Gaitsgory category and reductive Lie foundations", "unrestricted external fusion comparison with valid hypotheses; no use of the unqualified printed B.11.14 assertion", "complete stable/DG-category, adjoint, coherent Yoneda, localization, derived tensor–Hom, Ind and recursive earlier foundations", "full global GLC equivalence and all original fifteen-lesson mathematical corrections" ] }, { "id": "GL-GLC-13", "title": "Eisenstein series and the reduction to the cuspidal part (GLC III)", "source": "src/eisenstein-series-and-the-reduction-to-the-cuspidal-part.md", "reader": "GL-GLC-13.html", "proof_obligations": [ "actual bundle/local-system stacks, normalized half twists, Drinfeld compactifications, D-module and IndCoh operations, nilpotent-support and geometric adjunctions", "semi-infinite Satake, Jacquet, critical FLE compatibility and coherent local-to-global parabolic comparisons", "full spectral generation including the P=G quasi-coherent contribution, and the geometric extension producing the left adjoint", "miraculous duality, Chevalley involution, determinant normalizations and geometric compactness of the left adjoint", "global linear-kernel representation, tempered compatibility, compact preservation, formal-completion comparison and coherent full faithfulness proving A_G is quasi-coherent", "faithful quasi-coherent embeddings with coherent action and unit compatibility, and geometric conservativity of L_G", "irreducible-open geometry, zero-nilpotent-support comparison, proper-parabolic generation of reducible support and exact support localization", "both compactified constant-term comparisons and the actual line-bundle and supported-unit detection arguments under all proper-Levi conjectures", "root-datum structure of reductive groups, full torus GLC and the irreducible algebra-unit theorem", "extension-stack geometry, any existence assertion beyond the chosen one-form hypothesis, and the recursive algebraic foundations of the linked earlier curve proofs", "stable/DG categories, coherent adjunctions and monads, Yoneda, derived tensor and module categories, localization and recursive earlier mathematical foundations", "the full original fifteen-lesson GLC proof and correction scope" ] }, { "id": "GL-GLC-14", "title": "Ambidexterity, opers and multiplicity one (GLC IV and V)", "source": "src/ambidexterity-opers-and-multiplicity-one.md", "reader": "GL-GLC-14.html", "proof_obligations": [ "actual bundle and de Rham local-system stacks, half twists, D-module and IndCoh operations, nilpotent support and coherent geometric adjunctions", "full proper-Levi and torus conjectures, Eisenstein generation, and the supported-unit comparison for A_G", "Whittaker square, Lin cuspidal duality, Casselman–Shalika/Chevalley compatibility, miraculous duality and determinant-line comparisons", "localization square, critical FLE, common cuspidal support, spectral star/shriek comparisons and the geometric premises of the displayed twist/shift cancellation", "FR nonvanishing and conservativity, Beraldo Ramanujan/tempered comparison, and GL_n full-faithfulness and eigensheaf existence", "generic oper geometry, pseudo-proper presentations, base change, rational maps and coherent unital-Ran/iQCoh comparisons", "classical smoothness of U_G, compatible fibre self-duality and generic-to-global finite étale monodromy", "fundamental groups of de Rham stack components, central character-line classification, local-system topology comparisons and finite-cover descent", "spectral locally complete-intersection/CM theorem, stable-bundle and reducible-local-system codimension estimates with the genus-two A1 exceptions", "vacuum component nonvanishing, scalar H0 endomorphisms, vanishing under nontrivial central translation and two-categorical Fourier–Mukai compatibility", "almost-isogeny, centre, product and low-genus reductions and the classical-group generic-oper theorem", "stable/enriched DG category foundations, Yoneda, coherent adjunctions and duality, derived tensor/Hom and local cohomology, compactness/perfection and smooth descent", "recursive earlier commutative-algebra and rank-one geometric foundations and all retained original fifteen-lesson proof/correction scope" ] }, { "id": "GL-GLC-15", "title": "The proof as a map", "source": "src/the-proof-as-a-map.md", "reader": "GL-GLC-15.html", "proof_obligations": [ "Actual bundle and derived local-system stacks, half twists, D-module/IndCoh operations, nilpotent singular support and coherent geometric adjunctions.", "Whittaker-normalized construction, spectral action, compact eventually-coconnective bounds, corrected critical FLE/factorization and localization comparisons.", "Eisenstein generation, actual global algebra monad with its specified unit, exact open-supported localization and the proper-Levi supported-unit theorem.", "Actual FR nonvanishing/tempered comparisons and conservativity; normalized ambidexterity, generic-opers/Ran geometry, compactness and finite-monodromy bundle theorem.", "Fundamental groups, central-character line decomposition, spectral component topology, CM and codimension estimates, supported-section vanishing with the stated genus and type exceptions.", "Every-component vacuum nonvanishing, vacuum Hom dimensions and twisted vanishings, gerbe/central-Hecke comparisons, and positivity distinct from the upper dimension bound.", "Full categorical torus equivalence, reductive root structure, product/Kunneth comparisons, actual almost-isogeny base change and the separate low-genus/type-A1 results.", "Full/restricted/tempered and de Rham/Betti comparisons, field extensions, compact coherent-image and regular-singular Riemann–Hilbert theorems.", "Actual spectral/eigen-fibre comparison, nonzero fibres and faithful forgetting, irreducible fibre theorem, normalized coefficient, regularity/perversity/residual-gerbe/Peter–Weyl and characteristic-cycle inputs.", "Subsequent positive-characteristic specialization, component/trace theorems, factorization-crystal descent, arithmetic vacuum/orbit comparisons; explicitly conjectural orbit, analytic and local/ramified claims retain their status.", "Stable/DG-category, coherent adjunction, monoidal action, relative tensor, Ind, derived-module, local cohomology and recursive earlier curve/algebra/stack foundations.", "Full geometric Langlands proof, all group and genus reductions, and complete proofs of the mathematical foundations used throughout the course." ] } ], "status": "Draft. Self-checked by the writing AI.", "reference_note": "References credit the human mathematical sources; citations do not replace proofs." }