CLASS HOURS AND LOCATION: TTh 9:30-11:00 RLM 12.166.

UNIQUE NUMBER: 57255.

OFFICE HOURS: Wed 9:30 -- 11:00 or by appointment.

TEXTBOOK: No required textbook. Some references are listed below.

PREREQUISITES: Graduate Algebra and some exposure to algebraic curves (e.g. my course last Spring). Contact me if you have any questions about prerequisites.

GRADE POLICY: Each student will be required to take notes for a week and TeX them.

COURSE DESCRIPTION: The course will cover the theory of elliptic curves, modular curves and modular forms from an algebraic-geometric viewpoint. We will discuss the geometry of elliptic curves, describe modular curves as parameter spaces of families of elliptic curves with additional structure and study modular forms as functions (or sections of line bundles) on modular curves. As a goal, we will discuss the Modularity Conjecture and its role in Wiles's proof of Fermat's Last Theorem.

Some references (not required)

Published by Springer.

G. Cornell, J. Silverman and G. Stevens,
Modular forms and Fermat's last theorem,

J. Silverman, The Arithmetic of Elliptic Curves, and
Advanced Topics in the Arithmetic of Elliptic Curves.

F. Diamond, J. Shurman, A first course in modular forms.

Books and lecture notes by
J. Milne: Elliptic Curves. Modular Functions and Modular Forms.

And, for the fearless: N. Katz, B. Mazur, Arithmetic Moduli of Elliptic Curves. PUP.

- Math Overflow.
- Isogenies are homomorphisms on Math Overflow.
- A review of non-archimedean elliptic functions, Tate's paper on the Tate curve.
- references to equations for modular curves of small genus on Math Overflow.
- Eichler-Shimura on Math Overflow.
- Construction of modular curves on Math Overflow, including a warning by Brian Conrad on the limitations of the approach taken in this course.