Elliptic curves and modular forms

2018–19–A

Dr. Daniel Disegni

Course topics

  1. Lattices. Doubly periodic functions.
  2. Riemann surfaces: definition and examples (Riemann sphere, roots, logarithms).
  3. Riemann surfaces: Euler characteristic, genus, Hurwitz formula. Riemann–Roch theorem (statement)
  4. Complex tori and elliptic curves.
  5. Morphisms between elliptic curves. The group law. Arithmetic of elliptic curves (a glimpse: Mordell’s theorem, L-functions, the Birch and Swinnerton-Dyer conjecture).
  6. The group SL(2,Z). The space of all lattices. The trefoil knot.
  7. Modular forms for SL(2,Z): finite-dimensionality. Eisenstein series.
  8. Theta series. Four squares. Even unimodular lattices and sphere packing in dimension 8.
  9. Hecke operators. `Reminders’ on the Riemann zeta function. L-functions. Modularity of elliptic curves (statement).
  10. The Ramanujan conjecture. Expander graphs. The j-invariant. Moonshine.
  11. Modular curves and their compactification.
  12. Complex multiplication.

Requirements and grading

University course catalogue: 201.2.5291