Topics in Number Theory
Winter Quarter 2016
Hilbert modular forms and their Galois representations
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:
Grading will be based on student presentation in the last two weeks of teaching.
No final exam is planned.
Overview (pdf file, an overview of the course);
Notes (pdf file, a tentative version of the lecture notes)
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).
Good understanding of commutative and non-commutative algebra,
algebraic number theory, basic arithmetic geometry.