Langlands correspondence and Bezrukavnikov's equivalence
Course given at the University of Sydney, first semester 2019.
Abstract: A course in two parts:
1) an attempt to explain what the Langlands program is about from an arithmetical point of view;
2) affine Hecke algebras, Deligne-Langlands conjecture and Bezrukavnikov's equivalence.
Extended abstract and bibliography
Notes from Gus Lehrer's course last semester on algebraic number theory.
Notes by Anna Romanova from lectures 1,2,3,6,7,8,9.
Links below will work as I upload the appropriate notes from lectures.
Lecture 1: Introduction to reciprocity
Number of solutions handout for first lecture.
Lecture 2: Review of algebraic number theory
Degree 5 solutions handout for second lecture.
Lecture 3: Zeta function and L-functions
Excerpt from Mazur-Stein giving Riemann's approximations to the zeta function via more and more roots.
Lecture 4: (Gus' lecture, notes thanks to Bregje Pauwels) Artin L-functions
Lecture 5: (Gus' second lecture, notes thanks to Bregje Pauwels) Brauer's induction theorem
Lecture 6: Overview of the Sato-Tate conjecture
Example sheet of first 5000 primes for two curves.
From slides of a lecture by Ito containing a manuscript of Sato.
Lecture 7: Infinite Galois theory, overview of global class field theory (Artin's point of view).
Lecture 8: Structure of local Galois groups; local class field theory.
Two interesting historical accounts:
K. Conrad: History of class field theory.
K. Miyake: Takagi's Class Field Theory -- From where? and to where?
Lecture 9: Heuristic derivation of local Langlands for GL(2); basic rep theory of p-adic groups.
Lecture 10: Precise statement of local Langlands for GL(2) for p ≠ 2.