Math 248B - Algebraic Number Theory - Winter 2004

| General information | Syllabus | Announcements & Dates | Homework | Handouts |

General information

Meeting Time Mon., Wed., Fri., 10:00 - 10:50 (may be negotiated otherwise)
Location Building 380 (Math), room 381-T
Professor Ben Brubaker (
Office: 382-F (2nd floor, Math building 380)
Office Phone: 3-4507
Office Hours: Monday 2-3:30, Wednesday 2-3:30, or by appointment.
Ryan Vinroot (
Office: 380-G (Basement, Math building)
Office hours: By appointment only.
Textbooks Multiplicative Number Theory, by Harold Davenport.

The Theory of the Riemann Zeta Function, by E.C. Titchmarsh.

We will review the basics of class field theory in the first week through the lens of reciprocity. Then we'll head to Davenport, covering the first 8 chapters, essential and classical analytic number theory from which all else follows. We'll break and do several weeks on the zeta function. Classical stuff again, but often shoved under the rug. We'll end with a potpourri of topics, including Siegel's theorem and proofs of algebraic facts using analytic information. We conclude by discussing automorphic forms in general and their role in solving fundamental questions in algebraic number theory. It will be sweet.
Prerequisites There are essentially no prerequisites for this course. If you took Rubin's 248A in the fall, or know the equivalent information, you'll have a better view from which to understand the arc of the course, but its not at all necessary in understanding the day to day material.


Date Material covered Relevant Reading

Wed., 1/7

Reciprocity Laws and Class Field Theory I

Cassels/Frohlich, Lang, "Alg. Num. Thy."

Fri., 1/9

Reciprocity Laws and Class Field Theory II

Cassels/Frohlich, Lang, "Alg. Num. Thy."

Mon., 1/12

Primes in Arithmetic Progression (prime modulus)

Davenport, Ch. 1, Lang, "Alg. Num. Thy."

Wed., 1/14

Gauss Sums and Cyclotomy

Davenport Ch. 2,3

Fri., 1/16

Primes in Arithmetic Progression (general case)

Davenport, Ch. 4

Mon., 1/19

No Class (MLK)

Wed., 1/21

Primitive Characters and Quadratic Forms

Cox, "Primes of the Form x^2 + ny^2," Davenport, Ch. 5

Fri., 1/23

Dirichlet's Class Number Formula

Davenport, Ch. 6

Mon., 1/26

    "Two Weeks of Zeta" Festival Begins!
Introduction, Proofs of Functional Equation, Facts about Gamma

Titchmarsh, "Theory of the Riemann Zeta Function," Chs. 1 & 2

Wed., 1/28

Infinite Products, Zero-free regions

Titchmarsh, Ch. 3

Fri., 1/30

The Prime Number Theorem

Titchmarsh, Ch. 3

    Mon., 2/2 -
Mon., 2/9

Week of Individual Projects Begin (see "Announcements").

Titchmarsh, Chs. 4-15

Wed., 2/11

Wrap-Up of Zeta Topics, Even Moments -> Lindelof

Titchmarsh, Patterson

Fri., 2/13

Siegel's Theorem

Titchmarsh, Course Notes (available soon in handouts section)

Mon., 2/16

NO CLASS (Presidents' Day)

Wed., 2/18

Finish Proof of Siegel's Theorem

Titchmarsh, Notes

Fri., 2/20

Begin Discussing Zeta Functions on Number Fields

Class Notes

Mon., 2/23

Begin Hecke's proof of Dedekind zeta functional equation

Class Notes

Wed., 2/25

Theta Functions

Class Notes

Fri., 2/27

Finish Hecke's proof of Dedekind zeta functional equation

Class Notes

Final 2-3 Weeks

Hecke L-functions, Basics of Automorphic Forms

Class Notes, Bump Ch. 1

Announcements & Dates


To receive credit for the course, you are expected to complete problem sets and do a 10-minute presentation on a topic of your choice concerning the zeta function. Homework exercises will often come about naturally as unfinished business during lecture. Eventually, these and other questions will be collected and written down, and then you'll have some time to work on them. There will be roughly 3 or 4 of these problem sets over the course of the quarter.

Assignment Due date Relevant Reading
#1 Homework 1 Problems Wednesday, February 18 Davenport, Ch. 1-6, Titchmarsh, Ch. 2, Cox, Chs. 1,2