M619: Complex Variables II

A complex elliptic curve is the quotient of the complex plane by a lattice. Such an object is (topologically) a torus and (algebraically) the solution set to a cubic equation. Moreover, the points on this object have a natural group law. The interplay between these analytic, geometric and algebraic aspects makes the study of elliptic curves extraordinarily rich and rewarding. Finally, the moduli space of all elliptic curves is a fascinating object in its own right.

In this course we'll cover complex elliptic curves and their moduli, and their higher-dimensional analogues ("abelian varieties"). In doing so, we'll encounter modular forms and theta functions. Important abstract ideas in complex analytic geometry (e.g., moduli spaces and line bundles) become concrete in the case of abelian varieties.

The objects we'll encounter are ubiquitous in mathematics. For instance, one can use theta functions to give a proof of Picard's theorem on the image of a holomorphic function; to count the representations of an integer as a sum of squares; or to solve the one-dimensional heat equation. More generally, complex abelian varieties are a continuing source of interesting problems in analysis, geometry and number theory.

Logistics

Requirements

Each is worth half of your grade:

Help

Questions directed to j.achter@colostate.edu will be answered swiftly. Office hours are listed here.