## 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:

• the quotient of the complex numbers by a lattice (complex analysis)
• a flat 2-dimensional torus (differential geometry)
• a smooth cubic curve in the projective plane (algebraic geometry).
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:

• basic complex analysis (at the undergraduate level)
• basic algebraic topology (fundamental groups, covering spaces, homology, Euler characteristic). Concurrent enrollment in MATH 611 will suffice.

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)

References:

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:

• If you work through this problem set, you will understand why the quotient of the upper half plane by SL2(Z) is a Riemann surface and hence a smooth complex curve.
• If you work through this problem set, you will understand the fundamental domain of the action of SL2(Z) on the upper half plane and you will be able to see that SL2(Z) is generated by two matrices S and T that correspond to the transformations S(z)=-1/z and T(z)=z+1.