Math 582E
Elliptic Curves and Elliptic Fibrations

Charles Doran

Winter 2007, Monday/Wednesday/Friday, 12:30
The classical theory of elliptic curves over the complex numbers lies squarely at the interface of algebraic geometry, complex manifold theory, and utomorphic function theory. With this beautiful subject as our starting point, we will explore the extensions of this theory that result from considering elliptic curves in families. This will involve a blend of algebraic, alalytic, and geometric methods, and we will investigate applications in other fields as well (e.g. to arithmetic and to physics).

Topics:

  1. Elliptic curves and lattices
  2. Weierstrass normal form
  3. Elliptic integrals and periods of elliptic curves
  4. Picard-Fuchs ordinary differential equations
  5. Kodaira's classification of singular fibers
  6. Elliptic surfaces
  7. The canonical bundle formula
  8. PSL(2,Z) and congruence module subgroups
  9. Uniformization of modular curves
  10. Elliptic modular surfaces
  11. Elliptic fibered K3 surfaces
  12. Periods for K3 surfaces
  13. Atkin-Lehner involutions
  14. The moonshine modular groups
  15. Modular families of K3 surfaces
  16. Arithmetic applications: Irrationality proofs and record curves
  17. Physical duality with bundles over elliptic curves
  18. Elliptic fibered threefolds
  19. Schoen's fiber product Calabi-Yau threefolds
  20. Periods from iterated fibrations

Prerequisites: Some basic knowledge of algebraic geometry (at the level of the two-quarters introductory course) OR familiarity with modular forms is sufficient.