Mathematics 207b

Topics in Number Theory

Hilbert modular forms and their Galois representations

Winter Quarter 2016

Haruzo HIDA

## Meeting Time:The first meeting is on Wednesday January 6 at 1:00pm in MS 6201

Mondays and Wednesdays 1:00pm to
1:50pm in MS 6201 and on Fridays, sometimes from 1:00pm to 1:50pm in MS 6201 and often from 4:00pm to 5:50pm in MS 5138
(the room and time for the Friday meeting following each Monday will
be announced earlier in the week)

**Office hours:** On Mondays (starting Jan.11), after class meetings for 2pm-3pm, and on Wednesdays (starting Jan. 13), before class meetings, from 12:00noon to 12:50pm
in my office: MS6308.
**Texts:** Tentative lecture notes are posted:

Overview (pdf file, an overview of the course);
Notes (pdf file, a tentative version of the lecture notes)

Grading will be based on student presentation in the last two weeks of teaching.
No final exam is planned.
Reference books are listed in the first page of the lecture notes.

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

- Basics of analytic/algebraic theory of Hilbert/quaternion automorphic forms,
- Relation between Quaternionic automorphic forms and Hilbert modular forms (quaternionic automorphic forms
are indispensable in construction of the Galois representation though we do not go into details of construction),
- Description of Galois representation attached to modular forms,
- Description of the ``big" Galois representation attached to a
*p*-adic families of modular forms
(if time allows).
Since this is a topic course, for some of the topics, we just give the results without detailed proofs.

**Seminar:** Related topics will be touched upon in Math 290B student seminar
on Mondays in MS 5118 or Wednesdays in MS 5138 from 4:30pm to 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.