Math 7359 (Elliptic Curves and Modular Forms), Fall 2023



Course Information
Instructor Class Times Office Hours
Evan Dummit
edummit at northeastern dot edu 		MR 6:00pm-7:40pm
Hayden 321 		MWR 3:00pm-4:00pm
Lake 571
This course will provide a graduate-level introduction to elliptic curves and modular forms. Here is the course syllabus.
Course Motivation: Elliptic curves are algebraic curves of genus 1 that arise in a wide variety of contexts in mathematics and their study involves techniques from nearly every discipline: algebra, analysis, geometry, topology, and (of course) number theory. Modular forms are analytic functions on the complex upper half-plane satisfying a certain functional equation, and although they are intrinsically analytic objects, they turn out to have surprisingly deep connections to the (seemingly far more) algebraically-flavored elliptic curves.

The connection between elliptic curves and modular forms, which falls under the broader heading of "modularity", is central to Wiles's proof of Fermat's Last Theorem, and one of the end goals of the course is to elucidate some of the major ideas of this connection, along with highlighting other modern developments in number theory related to elliptic curves and modular forms.
Course Topics: We begin with an overview of the addition law on elliptic curves, and then reformulate these ideas in the language of divisors using the Riemann-Roch theorem. We then study general structural properties of elliptic curves over arbitrary fields: isogenies, differentials, the Tate module, the Weil pairing, endomorphisms, and automorphisms. Next, we study elliptic curves over two more specialized types of fields: over finite fields Fq, which involves more number-theoretic techniques, and over the complex numbers C, which involves complex-analytic and geometric techniques.

We then review some of the classical theory of modular forms with a primary aim of discussing modularity of elliptic curves, focusing in particular on the relationship between the L-function of an elliptic curve and modular forms, which will tie together essentially all of the material from the course. Time permitting, we will discuss some of the modern developments in the study of elliptic curves and modular forms (e.g., Wiles's proof of Fermat's Last Theorem, the Birch and Swinnerton-Dyer conjecture and work of Gross/Zagier and Kolyvagin, Bhargava's work on average n-Selmer ranks, or other topics of student or instructor interest).
Textbook: I will write course notes (see below) in lieu of an official textbook, since I will be drawing material from a variety of sources. Silverman's "Arithmetic of Elliptic Curves" and Diamond/Shurman's "A First Course in Modular Forms" are good references for the material. Serre's "A Course in Arithmetic" and Koblitz's "Introduction to Elliptic Curves and Modular Forms" are also good references for some of the material, although they present it differently.
Background: There are no formal prerequisites, but students should have comfort with algebra at the beginning graduate level (Math 5111 or 5112) and complex analysis at the undergraduate level (Math 4555). I will freely refer to some results from commutative algebra, algebraic geometry, and complex analysis, but the goal is to make the course as self-contained as possible.
Grades: Grades will be based on attendance and participation (50%), and on occasional homework assignments (50%), which will be collaborative and may involve roundtable discussions of the solutions. Depending on student and instructor interest, there may also be student presentations on course-related topics during the semester.


Lecture Notes
Current version of the lecture notes (through Lecture 24 on 12/7):
Notes, Lectures 1-24


Lecture Slides
These are the slides for the Zoom lectures.
Date Material
Thu, Oct 5th Lecture 9: Divisors
Thu, Oct 12th Lecture 10: Riemann-Roch and Applications
Mon, Oct 16th Lecture 11: Differentials
Thu, Oct 19th Lecture 12: Riemann-Roch and Ramification
Mon, Oct 23rd Lecture 13: Riemann-Hurwitz and Isogenies
Thu, Oct 26th Lecture 14: More Isogenies
Mon, Oct 30th Lecture 15: Dual Isogenies
Thu, Nov 2nd Lecture 16: The Weil Conjectures and Tate Module
Mon, Nov 6th Lecture 17: The Weil Pairing and the Weil Conjectures
Mon, Nov 20th Lecture 20: Elliptic Functions (lecture in-class with slides)
Mon, Nov 27th Lecture 21: Elliptic Curves via ℘ (lecture in-class, partially with slides)


Homework Assignments
Homework assignments will be posted here. There will be one assignment every 2-3 weeks. A recommended amount of problems to aim for solving on each assignment appears at the top of the assignment. You are not expected to solve all of the problems, although it is certainly worth your time to try to solve all of them.

Homework #1, due Fri Sep 29th.

Homework #2, due Fri Oct 20th.

Homework #3, due Fri Nov 3rd.

Homework #4, due Fri Nov 17th.

Homework #5, due Wed Dec 13th. [Note: For students who did not attend lecture regularly, additional problem 2 is mandatory.]