Mathematics 207a, Number Theory

## Non abelian class number formulas and adjoint Selmer groups

Winter Quarter 2019

Haruzo HIDA

**Meeting Time:** Wednesdays 2:00pm to
2:50pm and Fridays 1:00pm to 2:50pm in MS 6201. **(Note the lecture is 1 hour on Wednesdays and 2 hour with 10 minutes break on Fridays)**

**Although this is a topic course,
the content is elementary and good for first year students.**

Here is a link to an overview of the course [pdf].

**Office hours:** Fridays 3pm-3:50pm
at my office: MS6308.

## Lecture Starts on Wednesday January 9th at 2pm in MS 6201

**Lecture notes:**

Notes (pdf file, a tentative version).

[Hawaii AMS special session lecture] (summary of the course).

Grading will be based on student presentation either in the seminar course Math 290b.2.19w or in the class meeting at the end of courses on topics (close to the course material)
chosen by each student.
No final exam is planned.
As reference books,
we suggest

[ICF] L. C. Washington, "*Introduction to cyclotomic fields*", Springer Graduate texts in Mathematics,
(any edition),

[LFE] H. Hida, "Elementary Theory of L-functions
and Eisenstein Series",
LMSST **26**, Cambridge University Press, Cambridge, 1993,
[MFG] H. Hida, "Modular forms and Galois cohomology", Cambridge University Press, Cambridge, 2000.
**Topics:** In this course, assuming basic knowledge
of complex analysis,
we describe basics of elliptic modular forms *f*.
We hope to cover the following four topics:

How to get the non-abelian ``class number" formula;
Properties of Galois representations *Ad(rho_f)* and *rho_f* of modular forms *f*;
Definitions of *Sel(Ad(rho_f))*;
the cyclicity question about when *Sel(Ad(rho_f))* is cyclic.

**Prerequisite:**

Good understanding of commutative algebra and
algebraic number theory.