Topics in Number Theory
Winter Quarter 2017
Modular forms and their Galois representations.
Meeting Time: Regularly Mondays and Wednesdays 2:00 pm to
2:50 pm in MS 5148 and Fridays either 2:00 pm to 2:50pm or 4:00 pm to 5:30 pm in MS 5148
(Friday meeting time will be announced in the class before the meeting day).
Lecture Starts on Monday January 9th in MS 5148
Office hours: Before class meetings, from 1:00pm (MW)
at my office: MS6308.
Texts: Tentative lecture notes has been posted:
Grading will be based on student presentation in the last week of teaching (or in the student participating seminar).
No final exam is planned.
Notes (pdf file).
An old lecture notes on elliptic curves and modular forms (describes how to compute the equation of a given elliptic curve):
Notes no.2 (pdf file)
As reference books,
[LFE]:"Elementary Theory of L-functions and Eisenstein Series,"
LMSST 26, Cambridge University Press, Cambridge, 1993, Chapters 5, 7
[GME]:"Geometric Modular Forms and Elliptic Curves," (World Scientific) First (2000) or Second edition (2011), Chapters 1, 2 and 4
[ALR]: Serre's book: "Abelian l-Adic Representations
and Elliptic Curves," any edition, for example,
Research Notes in Mathematics 7, A K Peters, 1998
Topics: I would touch the following topics in this course:
If we do not reach the last item within this quarter,
we continue to go in this line in the Spring quarter 2017.
If we finish the objectives listed here within this quarter, Spring 2017 course
will cover slightly more advanced topics.
- analytic/algebraic theory of elliptic modular forms (at the level of my book [LFE]),
- description of Galois representation attached to modular forms (not the construction in [GME]
that requires good knowledge, out of the scope of this course,
of functorial algebraic/arithmetic geometry of Grothendieck),
- p-adic deformation theory of modular forms via the theory of
p-adic analytic family of classical/p-adic modular forms,
- description of the ``big" Galois representation attached to a p-adic families of modular forms
(including its construction assuming the second item).
In addition to above four topics,
- Related topics could be touched upon in Math 290B student seminar
on Wednesdays at 4:00pm-5:30pm in MS 5118
(all students are encouraged to attend).
Good understanding of the material covered by Math 210 series
(including commutative and non-commutative algebra theory) and
algebraic number theory.