Mathematics 205a

Number Theory

Galois Deformation, Modular Lifting and R=T Theorems

Fall Quarter 2015

Haruzo HIDA

**Meeting Time:The first meeting is tentatively on Monday September 28 in MS 5148**

Mondays and Wednesdays 1:00pm to
1:50pm in MS 5148 and on Fridays, either 1:00pm to 1:50pm in MS 5148 or 4:00pm to 5:50pm in MS 5117
(the room and time for the Friday meeting following each Monday will
be announced earlier in the week)

**Office hours:** Before class meetings, from 12:00noon to 12:50pm (MW)
in my office: MS6308.

**Texts:** Lecture notes are now posted:

Notes (pdf file, now posted);
Reference (a list of references, a tentative version posted)

Grading will be based on student presentation in the last two weeks of teaching.
No final exam is planned.
As reference books,
we list

"Modular forms and Galois cohomology" (Cambridge studies in advanced mathematics 69)

"Geometric Modular Forms and Elliptic Curves" (World Scientific)

**Topics:** I would touch the following topics in this course:

- Basics of Galois deformation theory (and representation theory of pro-finite groups);
- Sketch of proofs of different R=T theorems.

- Related topics will be touched upon in Math 290B student seminar
on Mondays (MS5117) or Wednesdays (MS5138) at 4:00pm-5:50pm (students are encouraged to attend; meeting day depends on the week).

**Prerequisite:**

Good understanding of commutative and non-commutative algebra,
algebraic number theory, basic arithmetic geometry
(old course notes on arithmetic modular forms/elliptic curves [pdf]) and basic Galois cohomology
(old course notes on Galois cohomology [pdf]).