SYLLABUS

Course Overview

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 its algebraic and analytic aspects. These connections lie at the heart of many of recent breakthoughs and current areas of research, including the modularity theorem, the Sato-Tate theorem, the Riemann hypothesis the Birch and Swinnerton-Dyer conjecture, and the Langlands program.

Having said that, number theory is, at its core, the study of numbers. Our starting point is thus the integer ring Z, its field of fractions Q, and the various completions and algebraic extensions of Q. This means we will initially cover 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, and Dirichlet's unit theorm. We will spend roughly the first half of the semester on these topics, and then move on to some closely related analytic topics, including zeta functions and L-functions, the prime number theorem, primes in arithmetic progressions, the analytic class number formula, and the Chebotarev density theorem. We will also present the main theorems of local and global class field theory, but we will only have time to cover a few parts of the proofs.

The lecture notes and problem sets from an earlier version of the course are available on the OpenCourseWare website. We will follow a similar path, but there will be some significant changes.

Prerequisites

Die Mathematik ist die Königin der Wissenschaften und die Zahlentheorie ist die Königin der Mathematik -- Carl Friedrich Gauss

As suggested by this quote, number theory is supported by many subfields of mathematics, and we will not hesitate to call upon them as needed. In most cases these supporting subjects will play a minor role, but you should be aware that at various points in the course we will make reference to material from many other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry. When this happens, I will include in 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 prerequsites. In past years, 18.112 (Complex Analysis) was also a formal prerequisite. This is no longer the case, but if you have never studied complex analysis you will need to be prepared to do some extra reading when we come to zeta functions and L-functions.

Undergraduates considering this coures should be sure to also read the Undergraduates section below.

Text Books

There is no required text; lecture notes will be provided. 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. All but one of them can accessed online from MIT (see the MIT Libraries web page for information on offisite access).

    Algebraic Number Theory, J.W.S. Cassels and A. Frohlich. (errata).
    Multiplicative Number Theory, H. Davenport.
    Algebraic Number Theory, J.S. Milne.
    Class Field Theory, J.S. Milne.
    Algebraic Number Theory, S. Lang.
    An Invitation to Arithmetic Geometry, D. Lorenzini. (errata).
    Introduction to Modern Number Theory,Yu. I. Manin and A. A. Panchishkin.
    Algebraic Number Theory, J. Neukirch.
    Number Theory in Function Fields, M. Rosen.
    A Course in Arithmetic, J.-P. Serre.
    Local Fields, J.-P. Serre.

As noted above, commutative algebra is a corequisite for this course. For those who want/need to brush up on their commutative algebra, the course notes for the 2013 edition of 18.705 are available online.

    A Term of Commutative Algebra, A. Altman and S. Kleiman (errata).

I can also recommend the following texts, according to taste (Atiyah-MacDonald is an examplar of brevity, while Eisenbud is wonderfully discursive; Matsumura, my personal favorite, is somewhere in between, and I also recommend Milne's primer.):

    Commutative Algebra, M.F. Atiyah and I.G. MacDonald.
    Commutative Algebra with a View Toward Algebraic Geometry, D. Eisenbud.
    Commutative Ring Theory, H. Matsumura.
    A Primer of Commutative Algebra, J.S. Milne.

Problem Sets

Weekly problem sets will be posted on line. Solutions are to be prepared in typeset form (typically via latex) and submitted electronically as a pdf file via Gradescope by midnight on the due date (the first problem set will be due Friday September 10). Collaboration (with humans and LLMs) is permitted/encouraged, but you must write up your own solutions and explicitly identify any collaborators (including LLMs such as chat-GPT, Claude, or Gemini, if applicable), and/or give the name of your pset group on pset partners, as well as any resources you consulted that are not listed above. If there are none, write Sources consulted: none at the top of your submission.

Grading

Your grade will be determined by your average problem set score, after dropping your lowest/missing score (60 percent), your best score on two in-class quizzes (Oct 16 and Nov 13) (10 percent), and a 3-hour final exam (30 percent).

Undergraduates

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. The pace of this course may be faster than you are accustomed to, and you must be prepared to do any extra reading necessary to acquaint yourself with background material that is unfamiliar to you. It is essential to stay caught up with the problem sets; if you fall behind in a given week it is better to skip that problem set and move on to the next one. This is one reason why late problem sets will not be accepted (you can skip one without penalty). There will also be no makeup quizzes (again, you can skip one without penalty).

Disability Accomodations

Please contact Disability and Access Services as early in the term as possible, if you have not already done so. If you already have an accommodation letter, please be sure to submit a copy to Mathematics Academic Services. Even if you do not plan to use any accommodations, if there is anything I can do to facilitate your learning, please let me know.