Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Prerequisites
This is the first semester of a one year graduate course in number theory. There is no official prerequisite.
Corequisite
Description
Historically, number theory has often been separated into algebraic and analytic tracks, but we will not make such a sharp distinction. Indeed, one of the central themes of modern number theory is the intimate connection between various algebraic and analytic objects; these connections lie at the heart of many of recent breakthroughs and current programs of research, including the modularity theorem, the Sato-Tate theorem, the conjecture of Birch and Swinnerton-Dyer, and the Langlands program.
Having said that, number theory is, after all, the study of numbers, and our starting point is the ring Z, its field of fractions Q, and the various completions and algebraic extensions of Q. This means we will start with many of the standard topics in algebraic number theory, including: Dedekind domains, decomposition of prime ideals, local fields, ramification, the discriminant and different, ideal class groups, Dirichlet's unit theorm, and the ring of adeles and group of ideles. We will spend at least the first half of the semester on these topics, but then move on to some closely related analytic topics, including zeta functions and L-functions, the analytic class number formula, and Chebotarev density theorem. I also plan to cover at least the statement of the main theorems of local and global class field theory.
A unifying principle of twentieth-century number theory is the observation that, for any finite field F, the finite extensions of the function field F(t) have much in common with the finite extensions of Q; in particular, the subset of integral elements in such an extension form a ring that is a Dedekind domain. By working in the more general context of the fraction field of a Dedekind domain, we obtain theorems that apply to both cases; most of the results we will prove for number fields have direct analogs in the function field setting that we will make explicit as we go along.
More on Prerequisites
Die Mathematik ist die Königin der Wissenschaften und die Zahlentheorie ist die Königin der Mathematik—Carl Friedrich Gauss
[Translation: "Mathematics is the queen of the sciences and number theory is the queen of mathematics."]
As suggested by this quote, number theory (Zahlentheorie) is supported by many subfields of mathematics, and we will not hesitate to call upon them when needed. In most cases these supporting subjects will play a minor role, and it does not make sense to require you to spend an entire semester with them before meeting the queen (Königin). But you should be aware that at various points in the course we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry. When this happens, I will add to the lecture notes a quick review of any terminology and theorems we need that fall outside of the official corequisite for this course, which is 18.705 Commutative Algebra. Note that 18.705 transitively includes 18.100 Real Analysis, as well as 18.701 Algebra I and 18.702 Algebra II, as prerequisites. In past years, 18.112 Complex Analysis was also a formal prerequisite, but this is no longer the case. However, if you have never worked with functions of a complex variable before you will need to be prepared to do some extra reading when we come to zeta functions (e.g., the notion of an analytic continuation).
For graduate students in Mathematics, I expect that none of this will be an issue, but undergraduates and students from other departments may need to spend some time acquainting (or reacquainting) themselves with supporting material as it arises.
Motivated undergraduate students with adequate preparation are welcome to register for this course, but should do so with the understanding that it is a graduate level course aimed at students who are planning to do research in number theory or a closely related field. I expect students taking this course to be amply motivated and to take personal responsibility for mastering the material—this means keeping up with the lectures and doing whatever outside study may be necessary to fill in any gaps in your background.
Textbook and Notes
There is no required text; lecture notes will be provided.
I encourage you to take notes in class that include definitions and statements of lemmas and theorems, but only a high level summary of the proofs (many of which I will only sketch in class in any case). After class, you should attempt to fill in the proofs on your own. This is a great way to learn and will help you absorb the material much more effectively than a purely passive approach. You can then consult the lecture notes I will provide and / or any of the texts below to fill in gaps and to compare your approach with mine.
Number theory is a vast subject, and it is good to see it from many different perspectives. Below are a number of standard references that I can recommend. If you purchase the 2010 reprint of classic text of Cassels and Frohlich, it corrects most of the errata noted below. The text by Manin and Panchishkin is more of an encyclopedia than a textbook; it gives a panoramic view of number theory that necessarily omits a lot of details but gives a good indication of the scope of the subject.
Suggested Texts
Cassels, John W. S., and Albrecht Fröhlich. Algebraic Number Theory. London Mathematical Society, 2010. ISBN: 9780950273426. [On Algebraic Number Theory, Held from September 1st to September 17th 1965, in the University of Sussex, Brighton]; Including a List of Errata. (
errata (PDF))
Davenport, Harold, and Hugh L. Montgomery. Multiplicative Number Theory. Springer, 2000. ISBN: 9780387950976.
Milne, J. S. Algebraic Number Theory. (PDF - 1.2MB) 2014. Available at J. S. Milne Mathematics Site.
———. Class Field Theory. (PDF - 1.9MB) 2013. Available at J. S. Milne Mathematics Site.
Lang, Serge. Algebraic Number Theory. Springer, 2013. ISBN: 9781461269229.
Manin, Yuri Ivanovic, and Alexei A. Panchishkin. Introduction to Modern Number Theory Fundamental Problems, Ideas and Theories. Springer, 2010. ISBN: 9783642057977.
Neukirch, Jürgen. Algebraic Number Theory. Springer, 2010. ISBN: 9783642084737.
Serre, Jean-Pierre. A Course in Arithmetic. World Publishing Corporation, 2009. ISBN: 9787510005350.
———. Local Fields. Springer, 2013. ISBN: 9781475756753. [Preview with Google Books]
As noted above, commutative algebra is a corequisite for this course. We won't need anything heavy, but for those who want / need to brush up on their commutative algebra, the course notes from last year's version of 18.705 Commutative Algebra are available online.
A Term of Commutative Algebra, Lecture notes by Alan Altman and Steven Kleiman. Available online at The Worldwide Center of Mathematics (WCoM) (errata).
I can also recommend the following texts, according to taste (Atiyah-MacDonald is an exemplar of brevity, while Eisenbud is wonderfully discursive; Matsumura, my personal favorite, is somewhere in between):
Atiyah, Michael Francis, and I. G. Macdonald. Notes on Commutative Algebra. Mathematical Institute, 1966.
Eisenbud, David. Commutative Algebra with a View Toward Algebraic Geometry. Springer, 2008.
Matsumura, Hideyuki. Commutative Ring Theory. Cambridge University Press, 1989. ISBN: 9780521367646. [Preview with Google Books]
Problem Sets & Exams
There will be weekly problem sets. Solutions are to be prepared in typeset form (typically via latex) and submitted electronically as PDF files by 5pm on the due date. Collaboration is permitted / encouraged, but you must write up your own solutions and explicitly identify your collaborators, as well as any resources you consulted that are not listed above. If there are none, please indicate this by writing Sources consulted: None at the top of your submission.
There are no exams and no final.
Grading
Your grade will be determined by your performance on the problem sets; your lowest score will be ignored (this means you can afford to skip a problem set without penalty).