18.785 - Number Theory I

# SYLLABUS

## Course Overview

Historically, number theory has often been separated into algebraic and analytic tracks, but we will not make such a sharp distinction. One of the central themes of modern number theory is the intimate connection between various arithmetic and analytic objects; these connections lie at the heart of many of recent breakthoughs and current programs 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, after all, the study of numbers, thus 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 begin with some 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 at most of 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. I also plan to cover at least the statement of the main theorems of local and global class field theory, as well as a preliminary discussion of Artin representations, Hecke characters, and their associated L-functions.

The lecture notes and problem sets from last year are available on the OpenCourseWare website.

## 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 when 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 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 (e.g., the notion of an analytic continuation).

For graduate students in course 18 (the target audience of this course), none of this should be an issue. But undegraduates and students from other departments may need to spend some time acquainting (or reacquainting) themselves with supporting material as it arises. Undergraduates should also be sure to read the Undergraduates section below.

## Text Books

There is no required text; lecture notes will be provided (these will typically be posted the day after each lecture). You can find the lecture notes for last year's course here. We will cover much of the same material, but there will be some differences in emphasis and a few new topics.

(1) 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.

(2) 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 on-line from MIT (see the MIT Libraries web page for information on offisite access). The classic text of Cassels and Frohlich is not "officially" available in online form, but a bit of googling will lead you to many scanned pdf versions; alternatively, you can purchase the 2010 reprint, which corrects most of the errata noted below and is available on Amazon for around \$40. 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.

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.
Introduction to Modern Theory,Yu. I. Manin and A. A. Panchishkin.
Algebraic Number Theory, J. Neukirch.
A Course in Arithmetic, J.-P. Serre.
Local Fields, J.-P. Serre.

As noted above, commutative algebra is a corequisite for this course. We won't need anything heavy duty, but for those who want/need to brush up on their commutative algebra, the course notes for 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):

Commutative Algebra, M.F. Atiyah and I.G. MacDonald.
Commutative Algebra with a View Toward Algebraic Geometry, D. Eisenbud.
Commutative Ring Theory, H. Matsumura.

## 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 by 5pm on the Monday due date (the first problem set is due September 19). Late problem sets will not be accepted -- your lowest score is dropped, so you can afford to skip one problem set without penalty. Collaboration is permitted/encouraged, but you must write up your own solutions and explicity 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.

Your grade will be determined by your performance on the problem sets; your lowest score will be ignored (this includes any problem set you did not submit). There are no exams and no final.

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. There is a lot of material to cover, and the pace of the course may be faster than you are are used to. This makes it essential to stay caught up; if you fall behind in a given week it is better to simply skip that problem set and move on to the next one (this is one reason why late problem sets will not be accepted). I expect students taking this course to be amply motivated and to take personal responsibility for mastering the material -- this includes doing whatever outside reading may be necessary to fill in any gaps in your background.

## Disability Accomodations

Please contact Kathleen Monagle, Associate Dean in Studen Disability 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 in room 2-110. 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.