Section
Langlands Letter
A historical annotated reading of Langlands' 1967 letter to Weil, cross-linked to modern Langlands vocabulary.
Core idea
An annotated reading of Robert Langlands' 1967 letter to André Weil, which launched the Langlands program. This collection preserves the language and logical route of the source while linking bundled historical pages to the separate modern Langlands foundations.
The letter predates local packets and their refinements, Arthur parameters, stabilized trace formulas, excursion operators, categorical geometric Langlands, and Fargues--Scholze local geometrization. It should therefore be read as a foundational historical document, not as a complete statement of the current program.
Modern companion paths
- Local Langlands and packets
- Global reciprocity
- Trace formula and endoscopy
- Function-field parameterization
- Local geometrization
- Geometric Langlands
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Definitions
Click any term to expand its definition inline.
Algebraic Groups
- Split reductive algebraic group
- Simply connected semisimple group
- Maximal torus and weight lattice
- Roots, Weyl group, and dominant weights
- Coroots and the weight–coroot pairing
- Root lattice, weight lattice, and isogeny forms
- Chevalley basis
- Chevalley lattice and integral model
- Pinning and pinned automorphisms
- Semisimple element and conjugacy class
- Characters separate semisimple conjugacy classes
- Borel–Mostow normalizer representative
Langlands Duality
- Langlands dual group
- Dual lattice
- L-group and Satake parameter
- Langlands functoriality and L-homomorphisms
- Contragredient (dual) representation
Number Fields and Local Fields
- Global field
- Number field
- Global function field
- Place of a global field
- Completion at a place
- Local field
- Nonarchimedean local field
- p-adic field
- Galois extension and Galois group
- Unramified extension of a p-adic field
- Unramified prime and Frobenius element
- Choosing embeddings Q̄ → Q̄ₚ
Adeles and Automorphic Forms
- Adeles and restricted products
- Ideles, Hecke characters, and Artin reciprocity
- Automorphic form and Hecke eigenvalues
- Eisenstein series on a reductive group
Hecke Algebras and L-functions
- Spherical Hecke algebra and Satake isomorphism
- Maximal compact and hyperspecial subgroup
- Group algebra of a lattice
- Euler product and local L-factor