Mathematics 207b

Topics in Number Theory

Base Change and Galois Deformation

Winter Quarter 2014

Haruzo HIDA

**Meeting Time:** Mondays and Wednesdays 1:00pm to
1:50pm, Fridays 4:00pm-5:50pm in MS 6201 (sometimes, we meet at 1pm on Fridays depending on schedule).

**Office hours:** After class meetings, from 4:00pm (MW)
at my office: MS6308.

## Lecture Starts on Monday January 13th in MS 6201

**Texts:** Lecture notes will be posted: The first set of lecture
notes:

Notes (pdf file, a tentative version now posted).

Grading will be based on student presentation at the end of courses on topics close to the course material).
No final exam is planned.
As a reference book,
we list

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

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

- Basics of Galois deformation theory (and representation theory of pro-finite groups);
- Relation of deformation rings for a given starting representation
restricted to open subgroups;
- Introduction to Galois cohomology;
- ``R=T" theorem, applications and open questions (if time allows).
- Related topics will be touched upon in Math 290B student seminar
on Wednesdays at 4:30pm-5:50pm and possibly
Mondays at 4:30pm-5:50pm both in MS 5117 (students are encouraged to attend).

**Prerequisite:**

Good understanding of commutative and non-commutative algebra,
algebraic number theory and very basics of Galois cohomology.