TOKYO INSTITUTE OF TECHNOLOGY

Course description and aims

I will give lectures on the global Langlands correspondence. It is a conjectural correspondence between the following two objects: one is algebraic; the number of solutions mod p of systems of polynomial equations with integer coefficients for each prime number p,　and the other is analytic; automorphic forms. The famous Shimura-Taniyama conjecture on a correspondence between elliptic curves and modular forms, which was a key to solve Fermat's last theorem, can be regarded as a part of the global Langlands correspondence. In the former part of this lecture, first I will give various interesting examples of the global Langlands correspondence. After that, I will give a precise formulation of the global Langlands correspondence in terms of Galois representations and automorphic representations. In the latter part, we focus on the problem attaching automorphic representation to Galois representations (the problem on automorphy of Galois representations). First I will survey the Taylor-Wiles method, which was used to prove Fermat's last theorem. After that, I will explain the Calegari-Geraghty method, which was recently introduced in order to obtain automorphy of Galois representations over imaginary quadratic fields (or more generally, CM fields).
The global Langlands correspondence is a very interesting research topic, since it is related to many problems in number theory. In the former part of this lecture, I would like to explain how it is interesting. I also would like to stress the importance of representation theory; in fact, we need Galois representations and automorphic representations to formulate the global Langlands correspondence precisely. In the latter part, I would like to give the key idea of the proof of Fermat's last theorem, and explain how the idea is developing.

Student learning outcomes

- Learn various examples of the global Langlands correspondence for GL(2).
- Understand a formulation of the global Langlands correspondence for GL(n) in terms of Galois representations and automorphic representations.
- Understand the statement of automorphy lifting theorems.
- Learn the Taylor-Wiles method.
- Understand the reason why Taylor-Wiles method doesn't work over imaginary quadratic fields, and learn the Calegari-Geraghty method, which overcomes the difficulty.

Keywords

Langlands correspondence, automorphic representations, Galois representations, etale cohomology, universal deformation ring, locally symmetric space, Taylor-Wiles method, Calegari-Geraghty method

Competencies that will be developed

Class flow

This is a standard lecture course.

Course schedule/Required learning

Textbook(s)

None required

Reference books, course materials, etc.

Will be announced in the class.

Assessment criteria and methods

Assignments (100%).

Related courses

• MTH.A401 ： Advanced topics in Algebra A
• MTH.A402 ： Advanced topics in Algebra B
• MTH.A501 ： Advanced topics in Algebra E
• MTH.A502 ： Advanced topics in Algebra F

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Basic knowledge on algebra is expected.