Global Langlands conjectures and functoriality

Draft course, written by AI. Statements whose proofs are still missing are identified explicitly in the lessons.

The global Langlands programme connects Galois representations with automorphic representations. The course develops reductive groups and L-groups, the Satake isomorphism, automorphic L-functions, functoriality and reciprocity, symmetric powers and base change, modularity and its consequences, the function-field correspondence, and endoscopy and the classification of classical-group representations. The available draft lessons include written proofs of local analytic foundations, deformation-ring constructions and global cuspidal restriction. Each lesson identifies the additional proofs needed for its full subject.

1. Reductive groups, root data and the L-group
2. The Satake isomorphism for unramified groups and unramified L-factors
3. Automorphic representations and automorphic L-functions
4. L-functions for GL_n: Godement–Jacquet and Rankin–Selberg
5. The principle of functoriality
11. Modularity lifting: the Taylor–Wiles method in outline
14. The function-field case: Drinfeld and L. Lafforgue
15. Endoscopy and the classification of automorphic representations of classical groups: an outlook

Original text CC0 1.0 unless the lesson states other terms.