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
- Time MWF 2:00-2:50 ENGRG E106
- Text
The main text for this course will be [D], augmented by [M] and other
sources.
Requirements
Each is worth half of your grade:
- Homework Homework will be assigned weekly, and is due at the beginning of class
on Fridays. You're encouraged to work with other students in the
class, but the work you actually turn in must be your own.
- Final project There will be a final project due at the end of the semester, with written and oral components. More details will be available later.
Help
Questions directed to
j.achter@colostate.edu
will be answered swiftly. Office hours are listed here.