Here is a basic overview ¡­

Here is a basic overview the course. More detailed information will appear over the next few weeks, and during the semester. Some changes may occur, in particular towards the end of the semester, depending on the pace of the lectures and interests of the audience.

• Review of algebraic curves: Differentials, divisors. The Riemann-Roch Theorem and The Hurwitz Formula.

• Elliptic curves: Weierstrass equations, discriminant and j-invariant. Group law. Divisors. Isogenies and the endomorphism ring. The invariant differential. Points of finite order, Tate modules. Automorphisms.

• Elliptic curves over finite fields: Number of rational points, Hasse bound. Frobenius. Weil conjectures. Hasse invariant.

• Elliptic curves over the complex numbers: Elliptic functions. Complex tori. Uniformization.

• Abelian Varieties: Basic properties. Projectivity. Isogenies, endomorphism ring, Tate modules. Jacobian varieties.