Math 252: Modular Abelian Varieties

Instructor Information

Instructor: William Stein (Benjamin Peirce Assistant Professor of Mathematics)
Office: Science Center 515 (by the balcony)
Office Hours: Monday 2-3PM, Friday 3-4PM
Phone: 617-495-1790 (office), 617-308-0144 (mobile)
Email: [email protected]
Web page:

Course Objectives

Andrew Wiles proved Fermat's Last Theorem by showing that most elliptic curves y2=x3+ax+b are "modular". In this course we will study modular elliptic curves along with their higher-dimensional generalizations, which are called "modular abelian varieties". I will not assume that you already know about modular forms and abelian varieties.

My main goal for this course is to describe how to construct an abelian variety from a modular form. I will then state the Birch and Swinnerton-Dyer Conjecture for modular abelian varieties and explain some of the main theorems and computational evidence for it, thus taking you to the forefront of current research. I will also discuss methods for computing with modular abelian varieties.


There will be homework (60%) and a final project (40%), but no exams.


The primary textbooks for the course are Seminar on Fermat’s Last Theorem (ed. Murty) and Arithmetic Geometry (ed. Cornell and Silverman), but we will look at many other articles and books. See the references section of the web site for comments, reviews, and scans of some of the relevant material.

Tables and Software

Detailed Outline

I will fill in the references for the topics listed below as the semester progresses. Each bullet represents approximately one lecture.

1. Modular Curves

2. Abelian Varieties

3. Jacobian Varieties

4. Modular Forms

5. Hecke Algebras

6. Attaching an Abelian Variety to a Modular Form

7. The Birch and Swinnerton-Dyer Conjecture