TOKYO INSTITUTE OF TECHNOLOGY

Course description and aims

The purpose of a series of lectures is to study the modularity of a Galois representation and its related conjecture called Serre conjecture. We also learn recent works toward a generalization of Serre conjecture. Galois representations in question are a variety of linear representations which takes the values in several fields as the field of complex numbers, finite fields, and local fields. By using Galois theory with complex linear representations of finite groups give examples of Galois representations. Geometric objects such as algebraic varieties give Galois representations over l-adic fields via etale cohomology. On the other hand automorphic forms also give Galois representations which are obtained by using (arithmetic) geometries and a trace formula. The modularity problem for a given Galois representation asks if there exists an automorphic form which gives rise to the same Galois representation. It related to Shimura-Taniyama conjecture proved by Wiles which plays an important role for Fermat's last theorem. After his results there is much progress in this area.
A standard way to attach the modularity problem is twofold so that firstly we prove the modularity problem for Galois representations over finite fields and then next we lift its modularity to a representation over a field of characteristic zero. The former is called Serre conjecture which is proved only in the case of GL(2) over the field of rational numbers. It is still standing as an open problem in how to precisely formulate Serre conjecture in general. In a series of lectures we first learn basics of Galois representations and automorphic forms. After that the original version of Serre cojecture for GL(2) over the field of rational numbers and its proof due to Khare-Wintenberger would be explained. Further we learn its known generalizations and focus on difficulties came up there. To learn basics of Galois representations and automorphic forms would be regarded as a part of our purpose.

Student learning outcomes

Students would learn basics of Galois representations and what kinds of representations correspond to geometric or analytic objects. They also learn principles and background behind several theories. To be more precise, the followings will be organized:

1. We learn the definition of Galois representations and its constructions
2. We learn the basics of automorphic forms
3. We understand the contents of modularity problems
4. We understand the contents of Serre conjecture

Keywords

Galois representations, automorphic forms, modularity problems, and Serre conjecture

Competencies that will be developed

Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Required learning

Textbook(s)

None required.

Reference books, course materials, etc.

Summer school for number theory 2009 l-adic Galois representations and number theory for Galois deformations (available at the website)
Recent progress on R=T ~ Sato-Tate conjecture and Serre conjecture ~ (available at the website)

Assessment criteria and methods

Assignments (100%).

Related courses

• MTH.A301 ： Algebra I
• MTH.A302 ： Algebra II
• MTH.A331 ： Algebra III
• MTH.A201 ： Introduction to Algebra I
• MTH.A202 ： Introduction to Algebra II
• MTH.A203 ： Introduction to Algebra III
• MTH.A204 ： Introduction to Algebra IV

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

To have basic knowledge in algebra and number theory.