18.786 - Number Theory II

SYLLABUS

Course Overview

We will spend the first 5-6 lectures completing the proofs of local and global class field theory, drawing heavily on the material presented in lectures 23-28 of 18.785 (available here).

The rest of the course will be focused on modular forms and will use very little material from 18.785. There will be only one problem set related to class field theory, so students who are interested only in modular forms are welcome to skip that problem set (and the associated lectures) if they wish. The plan is to begin with a fairly standard introduction to classical modular forms and their L-functions, drawing primarily from the textbooks of Diamond-Shurman and Miyake, and to then branch out into one of several possible directions, depending on student interest. Possible topics include modular curves, L-functions of abelian varieties, Galois rerpesentations, and Hilbert modular forms.

Prerequisites

The official prerequisite for this course is 18.785, but as noted above, after we finish class field theory we will only need a small subset of the material from 18.785. Students wishing to take 18.786 who have not taken 18.785 are welcome but should contact me to discuss prerequisites.

Text Books

There is no required text; lecture notes will be provided (these will typically be posted a few days after each lecture). Below are a number of standard references that I can recommend. These texts can all be accessed on-line from MIT (see the MIT Libraries web page for information on offisite access).

    A first course in modular forms, F. Diamond and J. Shurman.
    Modular forms, T. Miyake.
    Modular functions and modular forms, J.S. Milne.
    Algebraic Number Theory, J. Neukirch.
    Modular forms: a computational approach, W. Stein.
    A Course in Arithmetic, J.-P. Serre.

Problem Sets

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 noon on the due date (the first problem set will be due Wednesday, September 28). 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.

Grading

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.

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. There is a lot of material to cover, and the pace of the course may be faster than you 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 Student 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.