Mathematics 207A
Topics in Number Theory
Modular forms and their Galois representations.
Fall Quarter 2012
Haruzo HIDA
Meeting Time: Regularly Mondays and Wednesdays 12:00 noon to
1:50 pm in MS 6118 and some Fridays 12:00 noon to 12:50 pm in MS 6118
(Friday meetings will be announced in the class before the meeting day).
Lecture Starts on Monday October 1st in MS 6118
Office hours: After class meetings, from 2:00pm (M)
at my office: MS6308.
Texts: Lecture notes will be posted. The introduction of the notes:
Entire set of lectures (pdf file, including the first set, reference and the face page)
Grading will be based on student presentation on some Fridays (or in the student participating seminar).
No final exam is planned.
As reference books,
we list
[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:
- analytic/algebraic theory of elliptic modular forms (at the level of my book [LFE]),
- p-adic deformation theory of modular forms via the theory of
p-adic analytic family of classical/p-adic modular forms,
- 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),
- description of the ``big" Galois representation attached to a p-adic families of modular forms
(including its construction assuming the third item).
If we do not reach the last two items within this quarter,
we continue to go in this line in the Winter quarter 2013.
If we finish the objectives listed here within this quarter, Winter 2013 course
will cover slightly more advanced topics.
In addition to above four topics,
- Related topics could be touched upon in Math 290B student seminar
on Wednesdays at 4:30pm-5:50pm in MS 5138
(students are encouraged to attend).
Prerequisite:
Good understanding of the material covered by Math 210 series
(including commutative and non-commutative algebra theory) and
algebraic number theory.