Mathematics 207B

Topics in Number Theory

Image of modular Galois representations.

Winter Quarter 2013

Haruzo HIDA

**Meeting Time: Regularly Mondays and Wednesdays 12:00 noon to
1:50 pm in MS 5233** and some Fridays 12:00 noon to 12:50 pm in MS 5233
(Friday meetings will be announced in the class before the meeting day).

## Lecture Starts on Monday January 7th in MS 5233

**Office hours:** From 3:00pm (M)
at my office: MS6308.
**Texts:** Lecture notes will be posted. The introduction of the notes:

Notes (face page, pdf file, posted)
Notes (pdf file, expanded to cover up to the middle part covering lectures for 5-6 weeks, posted);
Reference (a list of references, posted)

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:

- Lie theory of p-adic Lie groups (algebraic theory and p-adic theory),
- How to to show irreducibility in the p-adic case (e.g. [GME] Section 4.3),
- How to measure the image via Lie theory in the p-adic case (e.g. [GME] Section 4.3),
- All of the above in the Lambda-adic cases.

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.