## Fall 2020   Topics in Number Theory: Fermat's Last Theorem

### Class information

• Instructor: Liang Xiao (θζ’)
• Lectures: Monday 1-2, Wednesday 7-8 (even week)
• Lecture room: Lecture Building #3, Room 206 (δΈζ206)
• Office Hours: Friday 10-11am at BICMR 75101-1 (in the BICMR, Jingchun Yuan)
• Email: lxiao at bicmr.pku.edu.cn

The focus of this course will be on understanding the Langlands correspondence for modular forms and the Taylor-Wiles method. The course consists of five parts:
• We review basic facts on structures of Galois group of local and global number fields, and discuss Galois cohomology and duality theorems.
• We discuss deformation of Galois representations, local and global.
• We shift to discuss modular forms and associated Galois representations.
• We prove a modularity lifting theorem, through which we deduce Fermat's Last Theorem.
• An certainly incomplete list of references:

• Homeworks: 50%, every two-weeks starting week 4, in total 6 times, with lowest grade dropped
• Take-home final exam: 50%, exams will be available on January 7, you may consult notes or any reference, but you need to complete the final exam on your own, and return the exam no later than January 23.

### Homework policy

• Homework problems are posted on the course webpage, and are usually due on the Wednesday of odd weeks. You are welcome and encouraged to work with other students on the problems, but you should write up your homework independently.

All the lecture videos are hosted at bilibili website.

### Course Schedule

The course schedule is subject to change.  Tentative schedule Lecture Date Content 1 9/21 Introduction and Background in Number Theory I: l-adic representations of local Galois group, structure of local Galois group. (notes, video) 2 9/28 Background in Number Theory II: Grothendieck's l-adic monodromy theorem, Weil--Deligne representations, higher ramification groups, local class field theory in terms of Artin maps. (notes, video) 3 9/30 Background in Number Theory III: Global class field theory, Galois cohomology. (notes, video) a comment on Artin map. a short note on congruent subgroups of units in number field. Happy National's Day and Mid-Autumn Festival! 4 10/12 Background in Number Theory IV: Local duality theorems for Galois cohomology. (notes, video) more details on the proof of local Tate duality. a short note on first Galois cohomology and extensions of representations. 5 10/14 Background in Number Theory V: Global duality theorems for Galois cohomology (notes, video, Homework 1 due, Solution) 6 10/19 Overflow material from duality theorems (video) 7 10/26 Galois deformation I: Framed deformation, computation of tangent space. (notes, video, Homework 2 due) 8 10/28 Galois deformation II: Quotient by free actions, existence of deformation ring. (notes, will not cover first two pages, video) 9 11/2 Galois deformation III: Relations of deformation rings, and some examples. (notes, video) 10 11/9 Galois deformation IV: Relative deformation problem, Galois deformation with local conditions, more examples. (notes, video) 11 11/11 Galois deformation IV: Taylor-Wiles primes and Galois patching. (notes, video, Homework 3 due) 12 11/16 Galois deformation V: Conditioned local Galois deformation (notes for lectures 12 and 13, video) 13 11/23 Galois deformation VI: Conditioned local Galois deformation at l≠p (LX away, watch the lecture video here.) 14 11/25 p-adic Hodge theory I: A quick overview (LX away, watch the lecture video here, notes for lectures 14 and 15.) 15 11/30 p-adic Hodge theory II: Fontaine-Laffaille deformation (LX called sick, watch the lecture video here) 16 12/7 Galois representations appearing in geometry (notes, video, Homework 4 due) 17 12/9 Modular forms I: Jacobian of curves (notes, video). 18 12/14 Modular forms II: Galois representations attached to modular forms (notes, video) 19 12/21 Modular forms III: Geometric modular forms (notes, video). 20 12/23 Modular forms IV: Eichler-Shimura relations (notes, video, Homework 5 due) 21 12/28 Modularity lifting theorem I: preparation (notes, video) 22 1/4 Modularity lifting theorem II: patching argument.(notes, video) 23 1/6 Modularity lifting theorem III: Wiles' 3-5 trick and finish of the proof (notes, video, Homework 6 due) Take-home Final Exam