SYLLABUS
Course Overview
For the 2024 spring semester, 18.786 will be devoted to the topic of modular forms, with a focus on arithmetic aspects including a discussion of modular curves, Galois representations, modular abelian varieties, and their L-functions. We will start in the classical setting (GL2/Q), but time permitting, we may also discuss Hilbert and Bianchi modular forms.
Prerequisites
The official prerequisite for this course is 18.785, but we will only need a small subset of the material covered in 18.785 and students who have not taken 18.785 are welcome to take 18.786. We will assume familiarity with the basics of complex analysis, topology, and algebra; anyone who has taken 18.783 should have all the necessary background.
Text Books
There is no required text for you to purchase, all of the references listed below a can all be accessed on-line from MIT (see the MIT Libraries web page for information on offisite access). We will primarily use Diamond-Shurman and Miyake, but will also occasionally need to refer to some of the other references listed below, as well as articles and notes I will post when we need them.
An introduction to the Langlands program, Bump , Cogdell , Shalit , Gaitsgory , Kowalski , Kudla.
Modular forms: A classical approach, H. Cohen and F. Strömberg.
Algorithms for modular elliptic curves, J.E. Cremona.
Modular forms and modular curves, F. Diamond and J. Im.
A first course in modular forms, F. Diamond and J. Shurman.
Elementary theory of L-functions and Eisenstein series, H. Hida.
Introduction to elliptic curves and modular forms, N. Koblitz.
Modular forms, T. Miyake.
Introduction to modular forms, S. Lang.
Modular functions and modular forms, J.S. Milne.
Galois representations and modular forms, K. Ribet.
Lectures on modular forms and Hecke operators, K. Ribet and W. Stein.
Introduction to the arithmetic theory of automorphic functions, G. Shimura.
Modular forms: a computational approach, W. Stein.
Quaternion algebras, J. Voight.
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. 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. I expect undergraduate 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 required to fill in any gaps in your background.
Disability Accomodations
Please contact 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.
