Elliptic curves and modular forms
Time and Place
יום ג 14:00 - 12:00
יום ה 13:00 - 11:00
- Lattices. Doubly periodic functions.
- Riemann surfaces: definition and examples (Riemann sphere, roots, logarithms).
- Riemann surfaces: Euler characteristic, genus, Hurwitz formula. Riemann–Roch theorem (statement)
- Complex tori and elliptic curves.
- Morphisms between elliptic curves. The group law. Arithmetic of elliptic curves (a glimpse: Mordell’s theorem, L-functions, the Birch and Swinnerton-Dyer conjecture).
- The group SL(2,Z). The space of all lattices. The trefoil knot.
- Modular forms for SL(2,Z): finite-dimensionality. Eisenstein series.
- Theta series. Four squares. Even unimodular lattices and sphere packing in dimension 8.
- Hecke operators. `Reminders’ on the Riemann zeta function. L-functions. Modularity of elliptic curves (statement).
- The Ramanujan conjecture. Expander graphs. The j-invariant. Moonshine.
- Modular curves and their compactification.
- Complex multiplication.