Math 790, Moduli of Elliptic Curves

Fall 2021

Instructor: Richard Hain

Overview: This course will be an elementary introduction to moduli spaces of elliptic curves, also known as modular curves. An elliptic curve can be defined as any of the following:

They are important in number theory, algebraic geometry, dynamical systems and mathematical physics. Not all elliptic curves are isomorphic. The space whose points correspond to isomorphism classes of all elliptic curves is the "moduli space of elliptic curves". It is also called the "modular curve". In this short course I will give a concrete description of the modular curve as a complex analytic orbifold (and possibly also as an algebraic stack). It is the quotient of the hyperbolic plane by SL2(Z). This construction is elementary and is a prototype for the construction of moduli spaces in general. I will also introduce certain "generalized functions" on the modular curve, which are known as "modular forms". These are of fundamental importance in number theory and other areas, such as string theory.

Background: students should know:

Lecture notes: Lectures on Moduli Spaces of Elliptic Curves, in Transformation groups and moduli spaces of curves, 95--166, Adv. Lect. Math. (ALM) 16, 2011. (available here)


  1. J.-P. Serre: Chapter VII of A course in arithmetic. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, 1973.

Problem sets and notes:

Return to: Richard Hain * Duke Mathematics Department * Duke University