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 (brubaker@math.stanford.edu) 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. Course Assistant Ryan Vinroot (vinroot@math.stanford.edu) 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. Course Content 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. Syllabus 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

• Important Dates and Class Holidays
  • Monday, January 19th: NO CLASS (MLK, Jr. Day)
  • Sunday, January 25th: Add deadline
  • Sunday, February 1st: Drop deadline
  • Monday, February 16th: NO CLASS (Presidents' Day)
• Projects will be presented in class during the first week of February. These are 15 minute talks (3 scheduled per day) based on topics from the later chapters of Titchmarsh (Ch. 4 and beyond). Notes from your talk should be given to the class (hand-written notes are perfectly acceptable). You can choose any topic as long as you verify it with me. Here are some suggestions (entries with an "*" indicate that two people may choose this and work as a pair):
  • Eisenstein Series and the Maass-Selberg Formula (an alternate proof of the analytic continuation)* -- Yiannis Sakellaridis
  • The Approximate Functional Equation (Theorem 4.13 in Titchmarsh)* -- Paul-Olivier Dehaye
  • The Lindelof Hypothesis (Ch. 5, Titchmarsh, pp. 95-98) -- Kazim Buyukboduk
  • Growth in t at the Edge of the Critical Strip - Vinogradov's Method (Thm 6.14 in Titchmarsh, lots of technical details here) -- Oren Antebi
  • The Second Moment (in t) (Theorem 7.3 in Titchmarsh) -- Rob Easton
  • Dirichlet's Divisor Problem (Sections 12.1 and 12.2 of Titchmarsh) -- Andy Schultz
  • Hardy's Proof that Infinitely Many Zeros Lie on the Critical Line (Section 10.2 in Titchmarsh) -- B-D Kim
  • Irrationality of Special Values of the Zeta Function (Apery's proof) -- David Lecomte
  • Consequences of the Riemann Hypothesis -- Tyler Lawson
  Because Titchmarsh is the assigned text, I have given references corresponding to it. However, many of the above topics are standard facts about the Riemann zeta function which can be found in Edwards' book and many other places. I'll assign these projects on a first come, first served basis. Email me if you know which one you'd like, or have something else in mind. I'll record names next to the selections once they've been taken.
• Additional information and important class updates can be found here in the future. Examples of this include selected solutions to homework and exams, notes building on in-class discussions, and important changes to the course guidelines. These will ALWAYS be announced in class as well.

Homework

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

Handouts

• A first version of Notes on the Prime Number Theorem
• Notes on the continuation for the Dedekind Zeta Function (last revised: Friday, February 27, 4:30 pm) -- The first 10 pages have been carefully edited. The last 12 pages contain the proper content, but haven't been annotated very well yet.